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This is a published textbook by Michael Spivak (Publish or Perish, Houston, 1999), not Phil's own writing. Volume One develops the language of differentiable manifolds: tangent bundles, tensors, vector fields and differential equations, integral manifolds, differential forms, integration and Stokes' Theorem, Riemannian metrics, Lie groups, and an excursion into algebraic topology. It sits in the Wedge Stuff folder, likely as a reference on the wedge product and forms.

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"A Comprehensive Introduction to. DIFFERENTIAL GEOMETRY i VOLUME ONE Third Edition Q MICHAEL SPIVAK PUBLISH ORPERISH, INC. $ Houston. Texas 1999 ACKNOWLEDGEMENTS Iamgreatly indebted to Ric/tam’ S.Palais without hisencouragement these volumes would have remained ashort setofmimeographed notes and Donald E.Knuth without hisTEX program they would never have become typeset books PREFACE The preface totheﬁrstedition, reprinted onthesL1CCeeding pages, excused thisbook’s deﬁciencies ongrounds that canhardly bejustiﬁed now that these “notes” truly have become abook. Atonetime Ihadoptimistically planned tocompletely revise allthismaterial forthemomentous occasion, butIsoon realized thefutility ofsuch a.nunder- taking. AsIexamined these ﬁvevolumes, written somany years ago, Icould scarcely believe that Ihad once had theenergy tolearn somuch material, or even recall how Ihadunearthed some ofit. SoIhave contented myself with thecorrection oferrors brought tomyatten- tionbydiligent readers, together with afewexpository ameliorations; among these istheinclusion ofatranslation ofGauss’ paper inVolume 2. Aside from that, thisthird andﬁnal edition differs from theprevious ones only inbeing typeset, andwith ﬁgures redrawn. Ihave merely endeavored totypeset these books inamanner beﬁtting asubject ofsuch importance and beauty. Asaﬁnal note, itshould bepointed outthatsince theﬁrstvolumes ofthis series made their appearance in1970, references inthetextto“recent” results should beplaced incontext. Preface tot/zeFirst Edition Howmass NOTES §AMEToBE andhowtheyglid not cgme tobeagbook Formany years Ihave wanted towrite theGreat American Differential Geometry book. Today adilemma confronts anyoneintent onpenetrat- ingthemysteries ofdifferential geometry. Ontheonehand, onecan consult numerous classical treatments ofthesubject inanattempt to form some idea how theconcepts within itdeveloped. Unfortunately, amodern mathematical education tends tomake classical mathematical works inaccessible, particularly those indifferential geometry. Onthe other hand, onecannowfind texts asmodern inspirit, andasclean in exposition, asBourbaki 'sAlgebra. Butathorough study ofthese books usually leaves oneunprepared toconsult classical works, andentirely ignorant oftherelationship between elegant modern constructions and their classical counterparts. Most students eventually find that this ignorance oftheroots ofthesubject has itsprice ——noonedenies that modern definitions areclear, elegant, andprecise; it's just that it's impossible tocomprehend how anyoneever thought ofthem. And even after onedoes master amodern treatment ofdifferential geometry, other modern treatments often appear simply tobeabout totally different subjects. Ofcourse, these remarks merely mean that nomatter howwell some ofthe present daytexts achieve their objective, Inevertheless feel that an introduction todifferential geometry ought tohave quite different aims. There aretwomain premises onwhich these notes arebased. The first premise isthat itisabsurdly inefficient toeschew themodern language ofmanifolds, bundles, forms, etc. ,which wasdeveloped precisely in order torigorize theconcepts ofclassical differential geometry. Rephrasing everything inmore elementary terms involves incredible x Preface totheFirst Edition contortions which arenotonly unnecessary, butmisleading. Thework ofGauss, forexample, which uses infinitesimals throughout, ismost naturally rephrased interms ofdifferentials, even ifitispossible torewrite itinterms ofderivatives .Forthis reason, theentire first volume ofthese notes isdevoted tothetheory ofdifferentiable manifolds, thebasic language ofmodern differential geometry. This language iscompared whenever possible with theclassical language, so that classical works canthen beread. Thesecond premise forthese notes isthat inorder foranintroduction todifferential geometry toexpose thegeometric aspect ofthesubject, anhistorical approach isnecessary; there isnopoint inintroducing thecurvature tensor without explaining howitwasinvented andwhat it hastodowith curvature .Ipersonally felt that Icould never acquire asatisfactory understanding ofdifferentiable geometry until Iread theoriginal works. Thesecond volume ofthese notes gives adetailed exposition ofthefundamental papers ofGauss andRiemann. Gauss’ work isnowavailable inEnglish (General Investigations ofCurved Surfaces; Raven Press) .There arealso twoEnglish translations ofRiema.nn's work, butIhave provided a(very free) translation inthesecond volume . Ofcourse, Idonotthink that oneshould follow alltheintricacies of thehistorical process, with itsinevitable duplications andfalse leads What isintended, rather, isapresentation ofthesubject along the lines which itsdevelopment Egg have followed; asBernard Morin said tome,there isnoreason, inmathematics anymore than inbiology, why ontogeny must recapitulate phylogeny. When modern terminology finally isintroduced, itshould beasanoutgrowth ofthis (mythical) historical development .Andallthemajor approaches have tobepresented, forthey were allrelated toeach other, andallstill play animportant role . Prejizce totheFirst Edition xz Atthis point Iamreminded ofapaper described inLittlewood' s Mathemati,c§an' sMiscellany. Thepaper began "The aimofthis paper is toprove ..."andittranspired only much later that this aimwasnot achieved (the author hadn't claimed that itwas) .What Ihave outlined above isthecontent ofabook therealization ofwhose basic plan andthe incorporation ofwhose details would perhaps beimpossible; what Ihave written isasecond orthird draft ofapreliminary version ofthis book. Ihave hadtorestrict myself towhat Icould write andlearn about within thepresent academic year, andallrevisions andcorrections have hadto bemade within this same period oftime. Although Imaysome daybeable todevote toitscompletion thetime which such anundertaking deserves, atpresent Ihave noplans forthis. Consequently, Iwould like tomake these notes available now, despite their deficiencies, andwith allthe compromises Ilearned tomake intheearly hours ofthemorning. These notes were written while Iwasteaching ayear course indif— ferential geometry atBrandeis University, during theacademic year 1969-7'0. Thecourse wastaken bysixjuniors andseniors, andaudited by afewgraduate students .Most ofthem were familiar with thematerial in Calculus onllanifolds, which isessentially regarded asaprerequisite. More precisely, thecomplete prerequisites areadvanced calculus using linear algebra andabasic knowledge ofmetric spaces .Anacquaintance with topological spaces iseven better, since itallows onetoavoid the technical troubles which aresometimes relegated totheProblems ,butI tried hard tomake everything work without it. Thematerial inthepresent volume wascovered inthefirst term, except forChapter 10,which occupied thefirst couple ofweeks ofthesecond term, andChapter 11,which wasnotcovered inclass atall. Wefound it necessary totake rest cures ofnearly aweek after completing Chapters 2, 3,and7.Thesame material could easily beexpanded toafull year course xii Preface totheFirst Edition inmanifold theory with apace that fewwould describe asexcessively leisurely. Iamgrateful totheclass forkeeping upwith myaccelerated pace, forotherwise thesecond half ofthese notes would nothave been written. Iamalso extremely grateful toRichard Palais, whose expert knowledge saved meinnumerable hours oflabor. WM ‘ ﬂzwteéé oyzdmt, /era TABLE OF CONTENTS Although thechapters arenotdivided intosections, thelisting foreach chapter gives some indication which topics aretreated, andonwhat pages. CHAPTER l.MANIFOLDS Elementary properties ofmanifolds ........... Examples ofmanifolds ................ Problems ...................... CHAPTER 2.DIFFERENTIAL STRUCTURES C°°structures .................... C°°functions .................... Partial derivatives .................. Critical points .................... Immersion theorems ................. Partitions ofunity .................. Problems ....................._ CHAPTER 3.THE TANGENT BUNDLE The tangent space ofR” ............... The tangent space ofanimbedded manifold ....... Vector bundles .____DD'D........... Thetangent bundle ofamanifold ........... Equivalence classes ofcurves, andderivations .-..... Vector ﬁelds ..................... Orientation ..................... Addendum. Equivalence ofTangent Bundles ...... Problems ...................... xiii....2Ol ....27 ....3l ....35 ....4O ....42 ....5O ....53 ....63 ....67 ....7l ....75 .._.77 ....82 ....84 ....89 ....95 xiv Contents CHAPTER 4.TENSORS The dual bundle .......................107 The differential ofafunction .................109 Classical versus modern terminology ..............lll Multilinear functions .....................ll5 Covariant andcontravariant tensors ..............ll Mixed tensors, andcontraction ................l2l Problems ..........................I27 CHAPTER 5.VECTOR FIELDS AND DIFFERENTIAL EQUATIONS Integral curves ........................135 Existence anduniqueness theorems ...............L39 Thelocal ﬁow ........................L43 One-parameter groups ofdiffeomorphisms ...........L48 Liederivatives ........................I50 Brackets ...........................L53 Addendum l.Differential Equations ..............L64 Addendum 2.Parameter Curves inTwo Dimensions .......167 Problems ..........................L69 CHAPTER 6.INTEGRAL MANIFOLDS Prologue; classical integrability theorems ............179 Local Theory; Frobenius integrability theorem ..........l90 Global Theory ........................l94 Problems ..........................l98 CHAPTER 7.DIFFERENTIAL FORMS Alternating functions .....................201 Thewedge product ......................203 Forms ............................207 Differential ofaform .....................210 Frobenius integrability theorem (second version) .........215 Closed andexact forms ....................218 ThePoincare Lemma .....................225 Problems ..........................227 Contents CHAPTER 8.INTEGRATION Classical line and surface integrals ... Integrals over singular k—cubes ..... The boundary ofachain ._..... Stokes’ Theorem ........... Integrals over manifolds ........ Volume elements ........... Stokes’ Theorem ........... deRham cohomology ........ Problems .............. CHAPTER 9.RIEMANNIAN METRICS Inner products ............ Riemannian metrics ......... Length ofcurves ........... The calculus ofvariations ....... The First Variation Formula andgeodesics The exponential map ......... Geodesic completeness ........ Addendum. Tubular Neighborhoods .. Problems .............. CHAPTER 10.LIEGROUPS Liegroups .............. Left invariant vector ﬁelds ....... Liealgebras ............. Subgroups andsubalgebras ...... Homomorphisms ........... One-parameter subgroups ....... Theexponential map ......... Closed subgroups .......... Left invariant forms ......... Bi—invariant metrics .......... Theequations ofstructure ....... Problems .............. xvi Contents CHAPTER ll.EXCURSION INTHE REALM OFALGEBRAIC TOPOLOGY Complexes andexact sequences .............. The Mayer—Vietoris sequence ............... Triangulations ...................... The Euler characteristic ,.,...........,... Mayer~Vietoris sequence forcompact supports ........ The exact sequence ofapair _.............. Poincare Duality .,...,.........,..... TheThom class ..................... Index ofavector ﬁeld ................... Poincare-Hopf Theorem ................. Problems ...._........__......... APPENDIX A ToChapter l-..................... Problems ....,.........,......_.. ToChapter 2...................... Problems ......,................. ToChapter 5...................... T0Chapters 7,9,l0 ................... Problem ......................... NOTATION INDEX ..................... 477 INDEX ........................... A Comprehensive Introduction Z0 DIFFERENTIAL GEOMETRY i" VOLUME ONE CHAPTER 1 MANIFOLDS The nicest example ofametric space isEuclidean n~space R",consisting of alln-tuples x=(xl,...,x") with each xi<5R,where Risthesetofreal numbers. Whenever wespeak ofR"asametric space, weshall assume thatit hasthe“usual metric” H d(-xay) = 2(yi—-xi): J :'=l unless another metric isexplicitly suggested. Forn=0wewillinterpret R0as thesingle point 06R. Amanifold issupposed tobe“locally? likeoneofthese exemplary metric spaces R". Tobeprecise, amanifold isametric space Mwith thefollowing property: Ifx<5M,then there issome neighborhood Uofxandsome integer n30such that Uishomeomorphic toR". The simplest example ofamanifold is,ofcourse, justIR"itself; foreach x6R"wecan take UtobeallofR". Clearly, R"supplied with anequiva- lentmetric (one which makes ithomeomorphic toll?."with theusual metric), isalso amanifold. Indeed, ahasty recollection ofthedeﬁnition shows that anything homeomorphic toamanifold isalsoamanifold—the speciﬁc met- ricwith which Misendowed plays almost norole, andweshall almost never mention it. [Ifyou know anything about topological spaces, you canreplace “metric space” by“topological space” inourdeﬁnition; thisnewdeﬁnition allows some pathological creatures which arenotmetrizable andwhich failtohave other properties onemight carelessly assume must bepossessed byspaces which are locally sonice. Appendix Acontains remarks, supplementing various chapters, which should beconsulted ifoneallows amanifold tobenon-metrizable.] The second simplest example ofamanifold isanopen ballinIR";inthis case wecan take Utobetheentire open ball since anopen ball inR“is homeomorphic toR".This example immediately suggests thenext: anyopen 1 2 Chapter J subset VofR"isamanifold—for each x6Vwecanchoose Utobesome open ballwith x<5UCV.Exercising amathematician’s penchant forgeneralization, /t\ tu V/, “¢\U weimmediately announce aproposition whose proof islefttothereader: An open subset ofamanifold isalso amanifold (called, quite naturally, anopen submanifold oftheoriginal manifold). The open subsets ofR"already provide many different examples ofmanifolds (just how many isthesubject ofProblem 24),though bynomeans all.Before proceeding toexamine other examples, which constitute most ofthischapter, some preliminary remarks need tobemade. Ifxisapoint ofamanifold M,andUisaneighborhood ofx(Ucontains some open setVwith x<5V}which ishomeomorphic toR"byahomeomor- phism (,6:U—>R",then ¢(V) CR"isanopen setcontaining ¢(x). Conse- 41 A quently, there isanopen ballWwith qb(x) <5WC¢(V). Thus xE¢_1(W) C VCU.Since ob:V—>R"iscontinuous, theset¢_1(l/V) isopen inV,andthus open inM;itis,ofcourse, homeomorphic toW,andthus toR".This compli- cated little argument justshows thatwecanalways choose theneighborhood U inourdeﬁnition tobeanopen neighborhood. Manyhtds 3 With alittle thought, itbegins toappear that, infact, Umust beopen. But toprove this,weneed thefollowing theorem, stated here without proof.* 1.THEOREM. IfUCIR“isopen andf:U—>IR”isone-one andcontinu- ous,then f(U)CR"isopen. (Itfollows thatf(V)isopen foranyopen VCU, sof_' iscontinuous, and fisahomeomorphism.) Theorem liscalled “Invariance ofDomain”, foritimplies thattheproperty of being a“domain” (aconnected open set)isinvariant under one-one continuous maps intoR".The proof thattheneighborhood Uinourdeﬁnition must be open isasimple deduction from Invariance ofDomain, lefttothereader asan easy exercise (itisalsoeasy toseethatifTheorem lwere false, then there would beanexample where theUinourdeﬁnition wasnotopen]. Wenextturnourattention totheinteger nappearing inourdeﬁnition. Notice thatnmay depend onthepoint x.Forexample, ifMCR3is M={(x,y,2):z=0}U{(x,y,z):x=0andz=1} =M1UM2, then wecanchoose n=2forpoints inM;andn=Iforpoints inM2.This 1=-..______,,/ M2 .-.-gv '1*.> 1.-==*-'-'$i_‘1L-E_'§;<-1§.),{\_ ‘5.3+, 1._--.?:.=::g_1,-,=:.,;;d _ :-.\__ .., *=I~'3*,-mt, ....~.~'>;-.§=j,,s-i;¥>=’2t.-E-1¢’§?§5*?E -ta;-"-or .1,-> .,..-- ._-:_~3,--=.-,-...‘ _.__;_. .,3 ,§,,._,’,;g-'-J§;:?Z1 5...1..-=,-..-.I'.“~ :'-v~I-_'=:_-;:t1.-.,.»,.;=.¥;;_’{_;_ r~'.5-'.=-3? -'-IF., . ':"'*;'E"f'f[1‘9_‘¢ :1‘,-.;-'-_f_-,,p_,T_ r.-::\g.¢:gr:?<‘ .-~ sn+"‘~"{*t,"=,?_!..¥:.-‘?'}'.*_:. 1&9.‘-r-;::1:: §§:?.=:~."<.»~1§‘>;:-i§€JF*.Er%:=*=’<=2-5 -". -.1*‘:'-i:'.=,"-Z/A”.-.11:--.-i»-.1legal:eaa*It’l=alai3=,'l?.*“‘"~ 3?" -.,. ,,-;<,.1,§.>».;-2:-'g .~;-§:.-;=-'=§;='.'.1.;-t27,_»z<,;;¢=_.-.,,,= .»’f~f»$=,_._,"%;;;.-.,-W "''-*~=».:»"..:-.-:\#.:-*‘r.¢r' -.__,_. example, bytheway, isanunnecessarily complicated device forproducing one manifold from two. Ingeneral, given M1and M2,with metrics_d1 anddz,we canﬁrstreplace each d,-with anequivalent metric d,-such thatd,-(x,y)<Ifor allx,y<5M,-;forexample, wecandeﬁne d, .df= OI’ dj=ITllI'l(d;‘, l). *All proofs require some amount ofmachinery. The quickest routes arcprobably pro- vided byVick, Homology Theory andMassey, Singular Homology Theory. Anold-fashioned, butpleasantly geometric, treatment may befound inNewman, Yiipology ofPZane Sets. 4 Chapter .7 Then wecandeﬁne ametric donM=M]UM2by d(x y)_d,-(x, y) iftherc issome Esuch thatx,y6M,- ’ I otherwise (weassume thatM1andM2aredisjoint; ifnot,theycanbereplaced bynewsets which are). Inthenew spaCe M,both M1andM2areopen sets. IfM1andM2 aremanifolds, Misclearly amanifold also. This construction canbeapplied toanynumber ofspaces-——even uncountably many; theresulting metric space is called thedisjoint union ofthemetric spaces Mi.Adisjoint union ofmanifolds isamanifold. Inparticular, since aspace with onepoint isamanifold, soisany discrete space M,deﬁned bythemetric d( )_{0 ifx=y x'y_ Iifxgéy. Although different n’smay berequired atdiffereilt points ofamanifold M, itwould seem thatonly onencanwork atagiven point x<5M.Fortheproof ofthisintuitively obvious assertion wehave recourse once again toInvariance ofDomain. Asaﬁrststep, wenote thatIR"isnothomeomorphic toRmwhen n.7£m,forifn:>m,then there isaone-one continuous map from Rminto anon-open subset ofIR".The further deduction, that thenofourdeﬁnition isunique ateach x<5M,islefttothereader. This unique niscalled the dimension ofMatx.Amanifold hasdimension 11oris11-dimensional orisan n-manifold ifithasdimension nateach point. Itisconvenient torefer totl1e manifold MasM"when wewant toindicate tl1at Mhasdimension n. Consider once more adiscreie space, which isa0-dimensional manifold. Theonly compact subsets ofsuch aspace areﬁnite subsets. Consequently, an uncountable discrete space isnot0'-compact (itcannot bewritten asacountable union ofcompact suhsets). The same phenomenon occurs with higher-dimen- sional manifolds, asweseebytaking adisjoint union ofuncountably many manifolds homeomorphic toIR". Inthese examples, however, themanifold is notconnected. Wewilloften need toknow thatthisistheonly wayinwhich 0"-compactness canfailtohold. 2.THEOREM. IfXisaconnected, locally compact metric space, then Xis 0*-compact. PROOF. Foreach x<5Xconsider those numbers r>0such that theclosed ball {yEXId(x,y) 51'} fwarzgiilds 5 isacompact set(there isatleast onesuch r>0,since Xislocally compact). Thesetofall such r>0isaninterval. If,forsome x,thissetincludes allr>0, then Xiso~compact, since O0 X=LJUEX1Mmw5nlrr=l Ifnot,then foreach x<5Xdeﬁne r(x)tobeone—half theleast upper bound of allsuch r. The triangle inequality implies that {yEX1d(X1,J#)5 r}C‘U’EXId(>~'2,J#) S1‘+d(X1,-\'2)}, sothat {yEXId(x1,y)5r— d(X1,x2)} C{y6Xrd(X2,J’) 5rl, which implies that m rUO2rwQ—%dunnl Interchanging x1andx2gives ta lHm)—HnN5gdUnnL sothefunction r:X—>11?.iscontinuous. This hasthefollowing important consequence. Suppose ACXiscompact. LetA’betheunion ofallclosed balls ofradius r(y)andcenter y,forally<5A.Then A’isalsocompact. The proof isasfollows. Letz1,z2,z3,... beasequence inA’. Foreach ithere isay,-<5Asuch that2;isintheballofradius r(y,-) with center y,-.Since Aiscompact, some subsequencc ofthey,-,which wemight aswellassume isthesequence itself, converges tosome point y<5A.Now theclosed ball Bofradius %r(y) and %r(y) r(y) y K 6 Chapter J center yiscompact. Since y,-—>yand since thefunction riscontinuous, eventually theclosed balls {y6XId(y,y.-) 5r(yi)} arecontained inB.Sothesequence 2;iseventually inthecompact setB,and consequently some subsequence converges. Moreover, thelimit point isactually intheclosed ballofradius r(y) andcenter y(Problem l0).Thus A’iscompact. Now letx0<5Xandconsider thecompact sets A1={X0} An-I-I =An!- Their union Aisclearly open. Itisalso closed. Toseethis, suppose that xis apoint intheclosure ofA.Thcn there issome y<5Awith d(x,y)<§r(x). BY(1), PU’)2rt-Y)——d(x,y) l\J-—-l\.>-—-mmto>r(x) -—-—r(x) =§r(x) >d(x,y). This shows thatify6An,then xEAn’,soxEA. Since Xisconnected, and A75Elisopen andclosed, itmust bethat X=A, which iso—compact. ¢$¢ After thishassle with point-set topology, wepresent thelong-promised exam- ples ofmanifolds. The only connected l~manifolds arethelineRand thecircle, orI-dimensional sphere, S‘,deﬁned by S‘={xe1R§2:d(.\-,0)= 1}. /Warzgiilds 7 The function f:(0,2rr) —>S1deﬁned byf(6) =(cos6,sin6) isahome- omorphism; itiseven continuous, though notone-one, on[0,2:-r]. Wewill often denote thepoint (cos6,sin 6)6S'simply by66[0,23¢]. (Ofcourse, it isalways necessary tocheck that useofthisnotation isvalid.) The function g:(—rr,rr) —>S',deﬁned bythesame formula, isalso ahomeomorphism; together with fitshows that S1isindeed amanifold. There isanother waytoprove this,better suited togeneralization. Thepro- jection Pfrom thepoint (0,1) onto thelineIR><{-1} CIR><IR,illustrated in (0,I) X _F<_><z./___.___ ______________ __ theabove diagram, isahomeomorphism ofS1—{(0,1)}onto IR><{-1}: thisis proved most simply bycalculating PISI—{(0,1)}—>IR><{-1} explicitly. The point (0,l)maybetaken careofsimilarly, byprojecting onto IR><{I},oritsuf- ﬁces tonote that S'is“homogeneous”—there isahomeomorphism taking any point into any other (namely, anappropriate rotation ofIR2]. Considerations similar tothese now show that then-sphere S"={x6IR"+' :d(x,0) =I} isann-manifold. The 2-sphere S2,commonly known as“the sphere”, isour ﬁrstexample ofacompact 2-manifold orsurface. From these fewmanifolds wecanalready construct many others bynoting thatifM;aremanifolds ofdimension IT;(i=1,2), then M]><M2isan(n1+n2)— manifold. Inparticular S]><»-- ><S'\..i.~..i.-/ ntirncs iscalled then-torus, while S1><S'iscommonly called “the torus”. Itisob- viously homeomorphic toasubset ofIR“,anditisalsohomeomorphic toa certain subset ofIR3which iswhat most people have inmind when theyspeak of 8 Chapter J “the torus”: This subset may beobtained byrevolving thecircle {(0,y,z) EIR3;(y—n2+Z2=1/4} around thez~axis. The same construction may beapplied toany I~manifoId I0 a I I I w_v I‘. ix,l contained in{(0,y,:)6IR3:y>0}.The resulting surface, called asurface of revolution, llascomponents homeomorphic either tothetorus ortothecylinder SIxIR,thelatter ofwhich isalsohomeomorphic totheannulus, theregion of theplane contained between twoconcentric circles. ._...~__~j I \II *-.I»-- II I I -____.-._-..~__-- _'____--_ I- ,~ p ‘ I ' _W:;_W_w_ _________ "' —U -_.- F-~_‘ '-~- _---' I ‘II""' \ H w.___r___-, Thc next simplest compact 2~manifold isthe2~holed torus. Toprovide amore ‘<I:>"<I:>' explicit description ofthe2~holed torus, itiseasiest tobegin with a“handle”, a space homeomorphic toatorus with ahole cutout;more precisely, wethrow }l4a2zy?JZ(is' 9 away allthepoints ononesideofacertain circle, which remains inourhandle, and which willbereferred toastheboundary ofthehandle. The 2-holed torus may beobtained bypiecing twoofthese together; itisalsodescribed as thedisjoint union oftwohandles with corresponding points ontheboundaries “identiﬁed”. The n~holed torus may beobtained byrepeated applications ofthisproce- dure. Itishomeomorphic tothespace obtained bystarting with thedisjoint T T —-I-‘T-7" ‘I }'| union ofnhandlcs and asphere with nholes, andthen identifying points on theboundary ofthe1"“handle with corresponding points ontheill‘boundary piece ofthesphere. There isone 2-manifold ofwhich most budding mathematicians make the acquaintance when they stillknow more about paper and paste than about 10 Chapter J metric spaces—the famous Mb'biz1J smjo, which you “make” bygiving astrip ofpaper ahalf twist before pasting itsends together. This canbedescribed ;i.'-\"'.-‘-.-‘-;.'- .::;';:':--- 1-‘-3-'-'-‘¢~=-1%€“~' :':§:';'3.-~ ... __.,..._._._.,...;_.:. -_ 5..,.._ analytically astheimage inIR3ofthefunction f:[0,21:]><(-1, 1)—>IR3deﬁned by f(6,r) =(2cos6 +Icos%cos6I, 2sin6 +Icosgsinél, rsing). unitvector Sir,% cef ircl o Q- \ / radius 2 cos1% 6,*--¢-_..._- cosg 1 ‘\ 6, J1 cos5cos9 egég el /v 2cos9 Ifwedeﬁne fon[0,2:1]><[-1, I]instead, weobtain theMobius strip with aboundary; asinvestigation ofthepaper model willshow, thisboundary is homeomorphic toacircle, nottotwodisjoint circles. With ourrecently intro- duced terminology, theMobius strip canalsobedescribed as[0,I]><(-1,I) with(0,1) and(1,-r)“identiﬁed”. !t!i!i y! -4-1-4-<-ﬁt-4-<44-<10 75 _1 ,*M”H.... I/Vehave notyethadtomake precise thisnotion of“identiﬁcation”, butour next example willforce theissue. Wewish toidentify each point x6S2with Zliaizgﬁilds" 11 itsantipodal point —-x6S2.The space which results, theprojective plane, P2, isalotharder tovisualize than previous examples; indeed, there isnosubset ofIR3which represents itadequately. The precise deﬁnition ofP2uses thesame trick thatmathematicians always usewhen they want twothings which arenotequal tobeequal. The points ofP2aredeﬁned tobethesets{p,—-p} forp6S2.Wewilldenote thisset by[p]6P2,sothat{—p] =[p]. Wethus have amap f:S2—>P2given byf(p)=[p],forwhich f(p)=f(q)implies p=:l:q. Wewillpostpone forawhile theproblem ofdeﬁning themetric giving thedistance between two points [p]and [q],butwecaneasily saywhat theopen setswillturn outtobe (andthisisallyouneed toknow inorder tocheck thatP2isasurface}. Asubset UCP2willbeopen ifandonly iff“'(U) CS2isopen. This justmeans that theopen setsofP2areofthe form f(V) where VCS2isanopen setwith theadditional important property thatifitcontains pitalsocontains —p, | an 2 »V-.7.-.. V(,___.’..‘ 12 Chapter J Inexactly thesame way, wecould have deﬁned thepoints oftheMobius strip Mtobe allpoints (s,r) 6(0,1) ><(-1, 1) together with allsets{(0,r), (1,—r)}, denoted by[(0,1)] or[(1,-r)]. There isamap f:[0,1] ><(—1,1)—> Mgiven by _on) Ks¢Ql f“S’i‘) _I[(S,I)] ifs=0or1, and UCMisopen ifand only iff“(U) C[0,1] ><(-1, 1)isopen, sothat theopen setsofMareoftheform f(V)where Visopen andcontains (s,-1) whenevcr itcontains (s,I)fors=0orI. ..V V Togetanidea ofwhat P2looks like, wecanmake things easier forourselves byﬁrst throwing away allpoints ofS2below the(x,y)-plane, since they are identiﬁed with points above the(x,y)-plane anyway. This leaves theupper hemisphere (including thebounding ciiclc), which ishomeomorphic tothedisc 02={X6IR2:d(x,0)51}, and wcmust identify each p6S‘with —-p 6S‘. Squaring things offa bit,thisisthesame asidentifying points onthesides ofasquare according totheschenie shown below (points onsides with thesame label areidentiﬁed insuch 21way thattheheads ofthearrows areidentiﬁed with each other), The dotted lines inthis picture arethekeytounderstanding P2. Ifwedistort the A \ \ \ \ \ NB \ XB \\ \ \ A A T‘\ -ts.'_\-i~::_<;.;;; ->_-.55:",5 :,:,_‘}_ _. 1-_I.'\. K‘A AA ‘ \ "1';-12% 1!B 175%!‘-;.~"?_ (2 U-\ I I 1 :"“:."- .4. g,-§,. »l.1..;-t is... -- I\‘?'-stirs"? M ist;t:5‘.=.>.t':..I.. 1M\ -<*'-:4 4%.‘?-4-":;~‘-'1' 1*-'-;&‘§7=--. I ;:'..-!-1--"'lB “"'"9;§ ‘=-=_‘>z§*;:~s.:*- n','?$.=s- ‘E=-:1-=‘_’»=_l?'ﬁi;R .=--"1--.=;;;,, .. |>§...~.‘ I-~.1*.BB I ~-.;:- \ B B \3':*.?t‘.?%s=-§ I '\'ls" <{~§;3f~ﬁ / .- -'."A "<1" -- ‘Q’-=-;‘;.=:%~=,ii-VMarzyiilds 13 region between them abitweseethat thefront part ofBfollowed bythe hack partofA,attheupper left,istobeidentiﬁed with thesame thing atthe lower right, inreverse direction; inother words, weobtain aMobius stripwith A \ A \ B[ ~, if \ \ \\ \ \\ \ \ \\ \\ \\ \ \\ \ [B \/A9\ \A A aboundary (namely, thedotted line, which isasingle circle). IfthisMobius strip isremoved, weareleftwith twopieces which canberearranged toform something homeomorphic toadisc. The projective plane isthus obtained from §$i“5' _._-,_-|_--W.,(91>» .... thedisjoint union ofadiscandaMobius strip with aboundary, byidentifying points ontheboundary andpoints ontheboundary ofthedisc, both ofwhich arecircles. Thus tomake amodel ofP2wejusthave tosewacircular piece of cloth andacloth Molnius strip together along their edges. Unfortunately, alittle experimentation willconvince you that thiscannot bedone (without having the twopieces ofcloth pass through each other). The subset ofIR3obtained astheunion oftheMobius strip and adisc, al- though nothoineoniorpliic toP2,canstillhedescribed mathematically interms U:ofP.There isclearly acontinuous function f1P—>IRwhose image isthis subset; moreover; although fisnotone-one, itislocally one-one; that is;every point p6P2hasaneighborhood Uonwhich fisone-one. Such afunction f 14 Chapter J iscalled atopological immersion (thesingle word “immersion” hasamore spe- cialized meaning, explained inChapter 2).We can thus saythat P2can be topologically immersed inR3,although nottopologically imbedded (there isno homeomorphism fliom P2toasubset ofR3]. InR4,however, witl1 anextra dimension toplay around with, thedisccanbeadded soasnottointersect the Mobius strip. Another topological immersion ofP2inR3canbeobtained byﬁrst immers- ingtheMobius strip sothat itsboundary circle liesinaplane; thiscanbedone inthefollowing way, The ﬁgures below show thattheMobius strip may beob- tained from anannulus byidentifying opposite points oftheinner circle. (This isalso obvious from theFact that thell/Iobius strip istheprojective plane with a disc removed.) Tl1is inner circle canhereplaced byaquadrilateral. When the A‘ A2—>A'E::i:::'i::::::|A2—> T‘/'1 Ail,LA: A11‘A2 AI A21? B |/4] "HIE IIIII“ ''''“Bi""'- /12 A2 —> 2;’ }B—> —> A1112 resulting ﬁgure isdrawn upinto 3-space and theappropriate identiﬁcations are made weobtain the“cr0ss—cap”. The cross-cap together with thedisc atthe bottom isatopologically immersed P2. CD 1‘2> }l4a22%1Zds 15 The one gap inthepreceding discussion isthedefinition ofa metric forP2. The missing metric can besupplied byanappeal toProblem 3-l,which will later beused quite often, andwhich thereader should peruse sometime before rt-acling Chapter 3.Roughly speaking, itshows thatthings likeP2,which ought Inbemanifolds, are. (Those who know about topological spaces willrecognize inasadisguised case oftheUrysolm Metrization Theorem.) Forthepresent, however, wewillobtain ourmetric byatrick that simultaneously provides an imbedding ofP2inR4. Consider thefunction f:S2—>R4deﬁned by f<x,y,z) =<yz,xz,xy,x’ +2112+3:2). (Ilcarly f(]J) =f(-11). V\lemaintain that f(p) =f(q) implies that p=:|:q. 'l'nprove this, suppose that f(x, y,z) =f(a, b,c).Wehave, ﬁrst ofall yz=be (1) xz=ac xy=ab. Ifa,b,c ¢0,thisleads to bx)1 I ~—--'- 2 £1 <> __Cx G Now (x+y+z)2=x2+y2+z2+2(xy+xz+yz) =1+2(xy +xz +yz), sowealsohave (x+y+z)2=(a+b+c)2, hence 6) a+b+c=iU+y+Q. Using (2),thisgives b b (1 Q (1 16 Chapter J soA’=:|:a. Similarly, weobtain y=:|:b, 2=:|:b, with thesame sign (which comes from (3))holding forallthree equations. Inthiscase wehave proved our contention without even using thefourth coordinate off.Now suppose a=0. IfX7’:0,then (l)would immediately give y=z=0,sothat (x,y,z) =(:|:i,0,0). Buty=2:=0implies (by(l)again) that bc=0,sob=0orc=0and (G,b,C)=(0,:|:i,0)Or(0,0,:|:l). These equations clearly contradict x2+2y2+322=a2+2b2+3:32. Thus x=0also, and wehave G-t>"~¢-/l\-I() yz=bc' y2+z2=l yz+3:2=2b2+362 (6) b2+('2=1. But(6)implies that 2;-2+3:2=2y2+3(1-yz) =3—y{ andsimilarly forbandc,so(5)gives 3-yz=3-b2 (7) y=:|:b. Now(4)gives (8) z=:l:(' (this holds even ify=b=0,since then 2,C=zbl). Clearly, (4)alsoshows that thesame sign holds in(7)and which completes theproof. Since f(]J) =f(q) precisely when p=:|:q,wecandeﬁne fl-:P2—>R4by fitpii=ftp). This map isone-one andwecanuseittodeﬁne themetric inP2: ¢?tt1»1,tq1) =d(f([11]),f([q])) =d(f(11),f(q))- Man§‘bZa’.s' 1'7 Then onecancheck that theopen setsareindeed theones described above. Bytheway, themap g:P2—>R3deﬁned bytheﬁrst 3components off, g([X.y,Z]) =(yz.Xz,Xy) isatopological immersion ofP2inR3. The image inR3isSteiner’s “Roman surface”. “)5?” T W 9 \’Vith thenew surface P2atour disposal, wecan create other surfaces in thesame way asthen-holed torus. Forexample, toahandle wecanattach aprojective space with ahole cutout, or,what amounts tothesame thing, a Mobius strip. The closest wecan come topicturing thisisbydrawing across- cap sticking onatorus. Wecan also join together apair ofprojective planes with holes cutout,which amounts tosewing twoMobius strips together along their boundary. Although thiscanbepictured astwocross—caps joined together, ithasanicer, and famous, representation. Consider thesurface obtained from thesquare with identiﬁcations indicated below; itmay also beobtained from thecylinder [0,1]><SIbyidentifying (0,x) 6[0,1]><S1with (I,x'), where X’ isthereflection ofxthrough aﬁxed diameter ofthecircle. Notice that the\ W 18 Chapter J identiﬁcations onthesquare force P],Pg,P3,and P4tobeidentiﬁed, sothat theset{P1, P2,P3,P4} isasingle point ofour new space. The dotted lines P1 P3 ,_ _,4 .\ ~ \ B B ' (0.10 - (1.x’) ‘"0|\-IinJP{P12-P2} {P32-P4} below, dividing thesides into thirds, form asingle circle, which separates the surface into twoparts, one ofwhich isshaded. Ag Ag /"13 A2 /41 B3K B2 -'32'—> B3T3‘. ~‘\“~ B] A B] A3A2A1 ‘B3 Rearrangement ofthe two parts shows that this surface isprecisely two Mobius strips with corresponding points ontheir boundary identiﬁed. The description interms of[0,1]><Slimmediately suggests animmersion ofthe surface. Turning one end ofthecylinder around and pushing itthrough itself orients theleft—hand boundary sothat (0,x)isdirectly opposite (1,x’),towhich itcan then bejoined, forming the“Klein bottle”. , ab i“ ii‘- }l4anﬂbZd.s 19 Examples ofhigher-dimensional manifolds willnotbetreated innearly such tlvtail, but, inaddition tothefamily ofn-manifolds S",wewillmention the related family of“projective spaces”. Projective n-space P“isdeﬁned asthe collection ofallsets{p,—p} forp<5S".The description oftheopen setsinP" isprecisely analogous tothedescription forP2.Although these spaces seem to Ibrm afamilv asregular asthefamily S”,wewill seelater that thespaces P" lbreven ndiﬁ"er inavery important way from thesame spaces foroddn. One further deﬁnition isneeded tocomplete thisintroduction tomanifolds. Wehave already discussed some spaces which arenotmanifolds only because they have a“boundary”, forexample, theMobius strip and thedisc. Points ntithese “boundaries” donothave neighborhoods homeomorphic toR",but they dohave neighborhoods homeomorphic toanimportant subset ofR".The (closed) half-space IHI"isdeﬁned by IHI"={(xl,...,x") ER”zx"Z0}. Amanifold-with-boundary isametric space Mwith thefollowing property: IfX<5M,then there issome neighborhood Uofxandsome integer n30such that Uishomeomorphic toeither R”orllil". Apoint inamanifold-with-boundary cannot have aneighborhood homeo- morphic toboth R"and lHl"(Invariance ofDomain again); wecantherefore distinguish those points x6Mhaving aneighborhood homeomorphic tollil". The setofallsuch xiscalled theboundary ofMand isdenoted by3M. IfMis actually amanifold, then 3M =lb.Notice that ifMisasubset ofR”,then 8M isnotnecessarily thesame astheboundary ofMintheoldsense (deﬁned for anysubset ofR“); indeed, ifMisamanifold-with-boundary ofdimension <:n, then allpoints ofMwillbeboundary points ofM. Ifmanifolds-with-boundary arestudied asfrequently asmanifolds, itbecomes bothersome tousethislong designation. Often, theword “manifold” isused for“manifold-with-boundary”. Amanifold inour sense isthen called “non- bounded”; anon-bounded compact manifold iscalled a“closed manifold”. We willstick totheother terminology, butwillsometimes use“bounded manifold” instead of“manifold-with-boundary”. 20 Chapter J PROBLEMS 1.Show that ifdisametric onX,then both ail=d/(1 +d)and d-"== min(i,d) arealso metrics and that they areequivalent tod(i.e., theidentity map 1:(X,d) —>(X,d) isahomeomorphism). 2.If(X;,d,-)aremetric spaces, fori 6I,with metrics d;<1,andX;{'1X;=lb for1'75j,then (X,d) isametric space, where X=U,~X;, and d(x,y) = d;-(x,_i') ifx,y <5X;forsome 1',while d(x,y) =1otherwise. Each X;isan open subset ofX,and Yishomeomorphic toXifand only ifY=U,Y; where theY;aredisjoint open sets and Y;ishomeomorphic toX;foreach 1'. The space (X,d)(oranyspace homeomorphic toit)iscalled thedisjoint union ofthespaces X;. 3.(a)Every manifold islocally compact. (b)Every manifold islocally pathwise connected, andaconnected manifold is pathwise connected. (c)Aconnected manifold isarcwise connected. (Apath isacontinuous image of[0,I],butanarcisaone-one continuous image. Adiﬁicult theorem states thatevery path contains anarebetween itsendpoints, butadirect proof of arcwise—connectedness canbegiven formanifolds.) 4.Aspace Xiscalled locally connected ifforeach x<5Xitisthecase that every neighborhood ofxcontains aconnected neighborhood. (a)Connectedness does notimply local connectedness. (b)Anopen subset ofalocally connected space islocally connected. (c)Xislocally connected ifand only ifcomponents ofopen sets areopen, so evcry neighborhood ofapoint inalocally connected space contains anopen connected neighborhood. (d)Alocally connected space ishomeomorphic tothedisjoint union ofitscom- ponents. (e)Every manifold islocally connected, and consequently homeorrlorphic to thedisjoint union ofitscomponents, which areopen submanifolds. 5.(a)The neighborhood Uinourdeﬁnition ofamanifold isalways open. (b)The integer ninourdeﬁnition isunique foreach .1‘. 6.(a)Asubset ofann-manifold isann-manifold ifand only ifitisopen. (b)ifMisconnected, then thedimension ofMat2:isthesame forallxEM. 7.(a)ifUCRisaninterval and f:U—>Riscontinuous and one-one, then fiseither increasing ordecreasing. }l4arzﬂb:'d.s 21 (h)The image f(U) isopen. (The map fisahomeomorphism. -\-..._.4 8.l"orthisproblem, assume (l)(The Generalizedjordan Curve Theorem) IfACR”ishomeomorphic toSP] ,then R"-—Ahas2components, and Aistheboundary ofeach. (2)IfBCR“ishomeomorphic toD"={x(-3R"Id(x,0) 51},then R"-—Bisconnected. ,._._,_._,__-2.:In-II!_r"--IP:One component ofR"——A(the“outside ofA”)isunbounded, andtheother “inside ofA”)isbounded. h)lfUCR"isopen, ACUishomeomorphic toS"'"1 and f:U—> lllt”isone-one and continuous (sothat fisahomeomorphism onA),then flinside ofA)=inside off(A). (First prove C.) (r)Prove Invariance ofDomain, 9.(a)Give anelementary proof that R‘isnothomeomorphic toR"forn>1. (h)Prove directly from theGeneralizedjordan Curve Theorem that Rmisnot homeomorphic toR"formyen. 10.Intheproof ofTheorem 2,show that thelimit ofaconvergent subsequencc olithe 2;isactually intheclosed ballofradius r(y) andcenter y. 11.Every connected manifold (which isametric space) hasacountable base foritstopology, and acountable dense subset. P12.(a)Compute thecomposition f=S1-—{(0,1)} —> R‘><{-1} —>R‘ explicitly forthemap Ponpage 7,and show that itisahomeomorphism. (l3)Dothesameforf:S"-'~{(0,...,0, 1)}->1R"*'. 13.(a)The text describes theopen subsets ofP2assetsoftheform f(V), where VCS2isopen and contains -pwhenever itcontains p.Show that this lastcondition isactually unnecessary. (b)The analogous condition isnecessary fortheMobius strip, which isdiscussed immediately afterwards. Explain how thetwocases difier. 14.(a)Check that themetric deﬁned forP2gives theopen sets described in thetext. (b)Check that P2isasurface. 22 Chapter J 15.(a)Show thatP‘ishomeomorphic toSl. (b)Since wecan consider S“_l CS",and since antipodal points inS"*l are stillantipodal when considered aspoints inS",wecanconsider P""l CP"in anobvious way. Show that P”—P"*1 ishomeomorphic tointerior D"={xE R”:d(x,0) <I}. 16.Aclassical theorem oftopology states that every compact surface other than S2isobtained bygluing together acertain number oftoriand projective spaces, and that allcompact surfaces-with-boundary areobtained from these bycutting outaﬁnite number ofdiscs. Towhich ofthese “standard” surfaces arethefollowing homeomorphic? Q( l‘N Arts"v‘ A:i;.;;. 1'7.LetCCRCR2betheCantor set.Show thatR2—Cishomeomorphic tothesurface shown atthetopofthenext page. Maizyblds 23 ,-----. _,_____,..# thiscircle isnotinthesurface these circles are __<7 , inthesurface l8.Alocally compact (but non-compact) space X“has oneend” ifforevery compact CCXthere isacompact Ksuch that CCKCXand X-—Kis connected. ()R" hasoneendifn>1,butnotifn:1. (I1)R"~{0}does not“have oneend” soR"-—{0}isnothomeomorphic toRm.p\\,_| l9.This problem isasequel totheprevious one; itwillbeused inProblem 24. AnendofXisafunction swhich assigns toeach compact subset CCXa non-empty component e(C) ofX-—C,insuch away that C;CC3implies *?(C2) C8(5))- (a)IfCCRiscompact, then R-—Chasexactly 2unbounded components, the “left” component containing allnumbers <some N,the“right” onecontaining allnumbers >some N.IfsisanendofR,show thats(C) iseither always the “left” component ofR—C,oralways the“right” one. Thus Rhas2ends. (b)Show thatR"hasonly oneendeforn>1.More generally, Xhasexactly oneendsifandonly ifX“has oneend” inthesense ofProblem l8. (c)This part requires some knowledge oftopological spaces. Let8(X)bethe setofallends ofaconnected, locally connected, locally compact Hausdorff space X.Deﬁne atopology onXU€(X) bychoosing asneighborhoods N¢(s0) ofanendsqthesets Ng(e@) =s0(C) U{ends s:e(C) =s@(C)}, forallcompact C.Show that XUE?(X)isacompact I-Iausdorﬁi space. What isRU€(R), and R"U€(R") forn>1? 20. Consider thefollowing three surfaces. (A)The inﬁnite-holed torus: ( ''' 24 Chapter J ()The doubly inﬁnite-holed torus: '''<<:¥f:;i/H313,’/T\_:/::; ''' Obit()Theinﬁnitejailcellwindow: E H_1it?nhg?it;1h¥_H_._-- __...___-___ —_ —_ —_ sll;Ql llll ._i..-— i-1-i i-L-L " Q-L-Ii _i..-.-/(‘ma '"_i'“?\ C ”T-7 ' "T-I ( ell;;lf,llelte.Citritit "K;"' 7'1‘./-r (a)Surfaces (A)and(C)have oneend, while surface (B)does not. (b)Surfaces (A)and (C)arehomeomorphic! Hint: The region cutoutbythe lines inthepicture below isacylinder, which occurs attheleftof(A).Now draw intwomore lines enclosing more holes, andconsider theregion between the twopairs. .]’t'ltJlll,Jil|ti.t._._Jltti, C“"]€1c1t‘AlWEgg? F-*—'='=.\-ayy ,?1\1v1vra,m,, A~|v»‘itsnslieg “*—-?—*1l I ‘ "-—"__' ,"'?'__' W %l) t ll lr llPF“t Lé }l4anQ’bZds' 25 21.(a)The three open subsets ofR2shown below arehomeomorphic. ‘I 4‘ II. -, I III III1' -_-.------,.--- ...-.--,II -H-.__ ‘-5 -‘H -‘"- ' I ' \ -' I I I. - | -II0'_-‘_.'. 3 .', : 2:9II'u.l' 1-7"‘, i.'!-" "v" \ir|iini:te swiss cheese"..---.= i’,,_.r"».. _.---._. _.-'-.: ""..- =.I1..-=..-"',.--.. I0I i‘ 0nI ‘ 1‘-' .- ~~... III ., III---------------------------------------------- ,- _- _- - I I O I I I I I I I I O (b)The points inside thethree surfaces ofProblem 20arehomeomorphic. 22.(a)Every open subset ofRishomeomorphic tothedisjoint union ofinter- vals. (b)There areonly countably many non-homeomorphic open subsets ofR. 23.Forthepurposes ofthisproblem wewilluseaconsequence oftheUiysohn Metrization Theorem, that foranyconnected manifold M,there isahomeo- morphism ffrom Mtoasubset ofthecountable product R><R>< . (a)IfMisaconnected non-compact manifold, then there isacontinuous function f:M—>Rsuch that f“goes toooatoo”, i.e.,if{xn} isasequence which iseventually inthecomplement ofevery compact set,then f(x;;)—>00. (Compare with Problem 2-30.) (b)Given ahomeomorphism f:M—>R><R>< andag:M—>Rwhich goestoooatoo,deﬁne f;M_>R><(R><R><---)byf(x)=(g(x),f(x)). Show that j:(M)isclosed. (c)There areatmost cnon-homeomorphic connected manifolds (where c=2“° isthecardinality ofR). 24.(a)Itispossible forR2-AandR2—Btobehomeomorphic even though A and Barenon-homeomorphic closed subsets. (b)IfACR2isclosed andtotally disconnected (theonly components ofAare points), then 8(R2—A)ishomeomorphic toA,Hence R2—AandR2—Bare non-homeomorphic ifAandBarenon-homeomorphic closed totally discon- nected sets. (c)The derived setA’ofAisthesetofallnon-isolated points. Wedeﬁne Al") inductively byAl‘) =A’andA(""") =(A("))’. Foreach nthere isasubset A, ofRsuch that A,,(")consists ofonepoint. 26 Chapter 1 *(d) There areenon-homeomorphic closed totally disconnected subsets ofR2. Hint: LetCbetheCantor set,andc;<c;<c3< asequence ofpoints inC.Foreach sequence n;<H3< ,onecanaddasetAn,such thatits n,-‘hderived setis{c;}. (e)There arecnon-homeomorphic connected open subsets ofR2. 25. (a)Amanifold-with-boundary could bedeﬁned asametric space Mwith thepI‘0pCTty that foreach x<5Mthere isaneighborhood Uofxand an integer n30such that Uishomeomorphic toanopen subset ofllll”. (b)IfMisamanifold-with-boundary, then 8Misaclosed subset ofMand 3M and M—3M aremanifolds. (c)IfC;-,1‘ <5Iarethecomponents of3M, and I’CI,then M—U,-EPC; is amanifold—with-boundary. 26.IfMCR"isaclosed setandann-dimensional manifold-with—boundary, then thetopological boundary ofM,asasubset ofR”,is3M. This isnot necessarily trueifMisnotaclosed subset. 27-(a)Every point (a,b,c)onSteiner’s surface satisﬁes b2c2+a2c2 +a2b2 = abc. (b)If(a,b,c) satisﬁes thisequation and095D=\/b2c2 +a2c2 +a2b2, then (a,b,c) isonSteiner’s surface. Hint: Letx=bc/D, etc. (c)The set{(a,b,c) <5R3:b2c2 +a2c2 +a2b2 =abc} istheunion ofthe Steiner surface andoftheportions (-00, -1/2)and(1/2,oo)ofeach axis. CHAPTER 2 DIFFERENTIABLE STRUCTURES We arenow ready toapply analysis tothestudy ofmanifolds. The neces- sary tools of“advanced calculus”, which thereader should bring along freshly sharpened, arecontained inChapters 2and3ofCalculus onManﬂblds. Wewill usefreely thenotation and results ofthese chapters, includirzg some problems, notably 2-9, 2-I5, 2-25, 2-26, 2-29, 3-32, and3-35; however, wewill denote theidentity map from IR”toIR"byI,rather than byIT(which willbe used often enough inother contexts), sothatIi(x)mxi. Onageneral manifold Mthenotion ofacontinuous function f:M—>IR makes sense, butthenotion ofadifferentiable function f:M—>IRdoes not. This isthecase despite thefact that Mislocally likeIR",where differentia- bility offunctions canbedeﬁned. IfUCMisanopen setandwechoose ahomeomorphism ¢:U—>IR",itwould seem reasonable todeﬁne ftobe differentiable onUiff0¢“': IR"—>IRisdiﬁereiitiable. Unfortunately, if 31/:V—>IR"isanother homeomorphism, and UOVqéI5,then itisnot necessarily truethat fo1/1"‘:IR”—>IRisalsodifferentiable. Indeed, since f<>v»“' =f<>¢"'<><¢<>i/f‘), wecanexpect fo1/1"‘ tobedifferentiable forallfwhich make forb“! differ- entiable only if<1’:o11/“I:IR"—>IR"isdifierentiable. This iscertainly notalways 11/ ¢ IR" IR" ¢-w“) 27 28 Chapter 2 thecase; forexample, oneneed merely choose <1’:tobehow, where h:IR"—>IR" isahomeomorphism thatisnotdifierentiable. Ifweinsist ondeﬁning differentiable functions onanymanifold, there isno wayoutofthisimpasse. Itisnecessary toadorn ourmanifolds with alittle additional structure, theprecise nature ofwhich issuggested bytheprevious discussion. Among allpossible homeomorphisms from UCMonto IR”,wewish toselect acertain collection withtheproperty that¢oib“' isdiflerentiable whenever qb,11/ areinthecollection. This isprecisely what weshall do,butafewreﬁnements willbeintroduced along theway. First ofall,wewillbeinterested almost exclusively infunctions f:IR"—>IR" which areC°°(that is,each component function flpossesses continuous partial derivatives ofallorders); sometimes wewillusethewords “diiierentiable” or “smooth” tomean (I'°°. Moreover, instead ofconsidering homeomorphisms from open subsets U ofMonto IR",itwillsuﬁice toconsider homeomorphisms x:U—>x(U) CIR" onto open subsets ofIR". The useoftheletters x,y,etc.,forthese homeomorphisms, henceforth ad- hered toalmost religiously, ismeant toencourage thecasual confusion ofapoint pEMwith x(p) <5IR",which has“coordinates” x'(p),...,x" (p). The only time thisnotation willbeconfusing (and itwillbe)iswhen wearereferring to themanifold IR",where itishard nottolapse back intothepractice ofdenoting points byxandy.Wewilloften mention thepair (x,U),instead ofxalone, justtoprovide aconvenient name forthedomain ofx. IfUand Vareopen subsets ofM,twohomeomorphisms x:U—>x(U) C IR"andy:V—>y(V) CIR"arecalled C°°-related ifthemaps Jim-"';x(Unv)_>y(UnV) xoy"":y(UflV)—> it-(Um/) areC°°. This make sense, since x(UﬁV) andy(UOV) areopen subsets ofIR". Also, itmakes sense, andisautomatically true, ifU('1V=Q. Afamily ofmutually C°°-related homeomorphisms whose domains cover M iscalled anatlas forM.Aparticular member (x,U)ofanatlas Aiscalled achart (fortheatlas A),oracoordinate system onU,fortheobvious reason that itprovides away ofassigning “coordinates” topoints onU,namely, the coordinates x'(p), ...,x"(p) tothepoint p6U. Wecaneven imagine amesh ofcoordinate lines onU,byconsidering the Dﬂérerzfiable iS'lruclm'es 29 inverse images under xoflines inIR"parallel tooneoftheaxes. -r ’-".¢l_',"'~-,Z'“:¢;'_ifi ,,, _ T ."'5-~,:"' '"i "' iII T - .t..:..! .E.‘ ‘ .- .',. A .~ ....':. r.'...'..|..'..:..:..>..:..\. ..:..'...'..\..'..:..:..'..; The simplest example ofamanifold together with anatlas consists ofIR"with anatlas Aofonly onemap, theidentity I:IR"—>IR”.Wecaneasily make the atlas bigger; ifUandVarehomeomorphic open subsets ofIR",wecanadjoin anyhomeomorphism x:U—>Vwith theproperty that xand x""‘ areC°°. Indeed, wecanadjoin asmany such x’saswelike—it iseasy tocheck that they areallC°°-related toeach other. The advantage ofthisbigger atlas ‘ll isthatthesingle word “chart”, when applied tothisatlas, denotes something which must bedescribed incumbersome language ifonecanrefer only toA. Aside from this, ‘LtdiiTers only superﬁcially from A;onecaneasily construct ‘Lt from A(and onewould befoolish nottodosoonce andforall).What hasjust been said fortheatlas {I}applies toanyatlas: l.LEMMA. IfAisanatlas ofC°°-related charts onM,then Aiscontained inaunique maximal atlas A’forM. PROOF. LetA’bethesetofallcharts ywhich areC°°—i-elated toallcharts x<5A.Itiseasy tocheck thatallcharts inA’areC°°-related, soA’isanatlas, and itisclearly theunique maximal atlas containing A.'1' Wenowdeﬁne aC°°manifold (ordifferentiable manifold, orsmooth manifold) tobeapair (M,A),where Aisamaximal atlas forM.Thus, about thesimplest example ofaC°° manifold is(lR",‘ll), where ‘ll(the “usual C°°-structure for IR””) isthemaximal atlas containing {I}.Another example is(IR,V)where V contains thehomeomorphism xi—>x3,whose inverse isnotC°°,together with allcharts C°°-related toit.Although (IR,11)and(IR,V)arenotthesame, there isaone-one onto function f:IR—>IRsuch that A-EU ifandonlyif xof<-EV, namely, theobvious map f(x)=x3.Thus (IR,‘l1) and(IR,V)arethesort ofstructures onewould want tocall“isomorphic”. The term actually used is so Chapter 2 “diffeomorphic”: twoC°°manifolds (M,A) and (N,.fB) arediffeomorpliic if there isaone-one onto function f:M—>Nsuch that L,x<-3.23 ifandonlyif xofi-EA. The map fiscalled adiffeomorphism, andf“lisclearly adiffeomorphism also. Ifwehadnotrequired ouratlases tobemaximal, thedeﬁnition ofdiffeo- morphism would have hadtobemore complicated. Normally, ofcourse, wewillsuppress mention oftheatlas foradifferentiable manifold, andspeak elliptically of“the differentiable manifold M”; theatlas forMissometimes referred toasthedg'ﬂ'ereuiiabZe structure forM.Itwill always beunderstood thatIR"refers tothepair (IR", U). Itiseasy toseethat adiffeomorphism must becontinuous. Consequently, itsinverse must also becontinuous, sothat adiffeomorphism isautomatically ahomeomorphism. This raises thenatural question whether, conversely, two homeomorphic manifolds arenecessarily diffeomorphic. Later (Problem 9-24) wewillbeable toprove easily that IRwith anyatlas isdiffeomorphic to(IR,U). Aproof ofthecorresponding assertion forIRZismuch harder, theproof forIR3 would certainly betoodifficult forinclusion here, andtheproof oftheessential uniqueness ofC°°structures onIR"forn35requires very diflieult teehniques from topology. Inthecaseofspheres, theprojections P]andP2from thepoints (0,...,0,1) and(0,...,0, -1)ofS"""'areeasily seen tobeC°°-related. They therefore cletermine anatlas—the “usual C°°structure forS"“"'”. This atlas may alsobe described interms ofthe2nhomeomorphisms ﬂiS”“’ {'){x 6IR“Ix’>0}—> lR”“’ g,-:S”""'!'1{x <5IR"Ix’<0}—> IR""" deﬁned byf-(x) =g,-(x) =(x’,. ..,x’"",x’+', ...,x"), which areC°°-related toP1andP2.There are,uptodiffeomorphism, unique differentiable structures onS"forn56.Butthere are28diffeomorphism classes ofdifferentiable structures onS-’,and over 16million onS3'.However, weshall notcome close toproving these assertions, which arepartoftheﬁeld called “differential topology”, rather then differential geometry. (Perhaps most astonishing ofall isthequite recent discovery that IR4hasadifferentiable structure that isnot cliffeomorphie totheusual differentiable structure!) Other examples ofdifferentiable manifolds willbegiven soon, butwecan already describe adifferentiable structure A’onanyopen submanifold Nof Dﬂéreutiable Struclu res 31 adifferentiable manifold (M,A); theatlas A’consists ofall(X,U)inAwith UCN. _]ust asdiffeomorphisms areanalogues forC°° manifolds ofhomeomor- phisms, there areanalogues ofcontinuous maps. Afunction fIM—>Nis called differentiable ifforevery coordinate system (x,U)forMand(y,V) forN,themap yofox“12 IR"—>IR”’isdifferentiable. More particularly, f ?~—----ir iscalled differentiable atp6MifyOfox“1isdifferentiable atx(p) for coordinate systems (x,U)and (y,V) with pEUand f(p) 6V.Ifthisis true foronepairofcoordinate systems, itiseasily seen tobetrue foranyother pair. Wecanthus deﬁne differentiability offonanyopen subset M’CM; asonewould suspect, thiscoincides with differentiability oftherestricted map f|M’:M’—>N.Clearly, adifferentiable map iscontinuous. Adifferentiable function f:M—>IRrefers, ofcourse, totheusual differen- tiable structure onIR,and hence fisdifferentiable ifand only iff0x“! is differentiable foreach chart x.Itiseasytoseethat (l)afunction f;IR"—>IR"‘isdifferentiable asamap between C°°manifolds ifandonly ifitisdifferentiable inthcusual sense; (2)afunction f:M—>IR"’isdifferentiable ifand only ifeach ff:M—>IR’" isdiﬁerentiable; /—~\/—\,-PUD""'\-1-I)acoordinate system (x,U)isadiffeomorphism from Utox(U); afunction f:M—>Nisdifferentiable ifandonly ifeach y’ofis differentiable foreach coordinate system yofN; (5)adifferentiable function f:M—>Nisadiffeomorphism ifandonly if fisone-one onto andf“I:N—>Misdifferentiable. The differentiable structures onmany manifolds aredesigned tomake certain functions differentiable. Consider ﬁrsttheproduct M;><M2oftwodifferen- 32 Chapter 2 tiable manifolds Mi, and thetwo“projections” Jr,-:M]><M2—>M;deﬁned byJT,:(p], P2)=p,-.Itiseasy todeﬁne adifferentiable structure onM1><M2 which makes each indifferentiable. Foreach pair(x,-,U,-)ofcoordinate systems onM,-,weconstruct thehomeomorphism X1 XX2: U] XU2 %' lRnl+n2 deﬁned by 1'1><~\'2(P1= P2)=(-\'1(P1),X2(P2)), i-6-, X1><X2'-=(-1'10 Tfiixz °K2)- Then weextend thisatlas toamaximal one. Similarly, there isadifferentiable structure onP"which makes themap f:S"—>IP"(deﬁned byf(p) ==[p]={p,-p}) differentiable. Consider anycoordinate system (x,U)forS”,where Udoes notcontain -pifitcon- tains p,sothat f|U isone-one. The map xo(f|U)“1 isahomeomorphism onf(U) CIP",and anytwo such areC°°-related. The collection ofthese homeoinorpliisms canthen beextended toamaximal atlas. Toobtain differentiable structures onother surfaces, weﬁrstnote that aC°° manifold-with-boundary canbedeﬁned inanobvious way. Itisonly necessary toknow when amap f:Ilil"—>IR”istobeconsidered differentiable; wecallf differentiable when itcanbeextended toadifferentiable function onanopen neighborhood ofIlil". A“handle” isthen aC°°manifold-with-boundary. Adifferentiable structure onthe2-holed torus canbeobtained by“matching” thedifferentiable structure ontwohandles. The details involved inthisprocess arereserved forProblem 14-. Todealwith C°°functions effectively, oneneeds toknow thatthere arelotsof them. The existence ofC°°functions onamanifold depends ontheexistence of C°°functions onIR”which are0outside ofacompact set.Webriefly recall here thenecessary facts about such C°°functions (c.f.Calculus onMamjiilds, pg.29). Dﬂérenliable Slruclures 33 (l)The function htIR—>IRdeﬁned by ...x2 l h(x)=le 1/xfo \ l/0 X=0 -_-.;-.-- . | —l l isC°°, and h(”)(0) =Oforalln. (2)The function j:IR—>IRdeﬁned by ---2 ---2-- -t . j(x)={e if1)-e"""') x<E(-l,1) ‘A A 2 0 x¢(-1.1) _, , isC°°. Similarly, there isaC°°function k:IR—>IRwhich ispositive on(0,5) and0 elsewhere. ) :k 5 (3)The function l:IR—>IRdeﬁned by 1 l ~»-</.5»)/(/.*IRdeﬁned by 4: =0 go)=jt.»-’/8)---jtx"/8) "E>°t--4: isC°°;itispositive on(-2, e)><---x(-—-e,2)and0elsewhere. OnaC°°manifold Mwecannow produce many non-constant C°°func- tions, The closure {x:f(x) 750}iscalled thesupport off,anddenoted simply bysupport f(orsometimes supp f). 2.LEMMA. LetCCUCMwith Ccompact and Uopen. Then there isaC°°function f:M—>[0,1] such thatf=1onCandsupportf CU. (Compare Case2oftheproof ofTheorem l5.) 34 Chapter 2 PROOF. Foreach p6C,choose acoordinate system (x,V)with VCUand x(p) =0.Then x(V) D(--8,2) >< ><(—e,e) forsome e>0.The function gox(where gisdeﬁned in(4))isC°°onV.Clearly itremains C°°ifweextend ittobe0outside ofV.Letfpbetheextended function. Thefunction fpcanbe constructed foreach p,andispositive onaneighborhood ofpwhose closure is contained inU.Since Ciscompact, ﬁnitely many such neighborhoods cover C, andthesum, j},+ +-f,,,,, ofthecorresponding functions hassupport CU. OnCitispositive, soonCitis36forsome6 >0.Letf=lo([,,, +---+j:,,,,), where lisdeﬁned in(3).'1' Bytheway, wecould have deﬁned C’manifolds foreach r31,notjustfor “r=00”. (Afunction f:IR"—>IRisC’ifithascontinuous partial derivatives uptoorder r).A“CUfunction” isjustacontinuous function, soaC'3manifold is justamanifold inthesense ofChapter I.Wecanalsodeﬁne analytic manifolds (afunction f:IR"—>IRisanalytic ata<5IR"iffcan beexpressed asa power series inthe(xi—uf)which converges insome neighborhood ofa).The symbol C°’stands foranalytic, anditisconvenient toagree thatr<00<cu foreach integer r30.Ifat<)8,then thecharts ofamaximal C*6atlas areall C°'-related, hutthisatlas canalways beextended toabigger atlas ofC°‘-related charts, asinLemma l.Thus, aC*6structure onMcan always beextended toaC°’structure inaunique way; thesmaller structure isthe“stronger” one, theC0structure (consisting ofallhomeomorphisms x:U—>IR")being the largest. The converse ofthistrivial remark isahard theorem: Foror31, every C°’structure contains aC*6structure foreach )8>or;itisnotunique, of course, butitisunique uptodiffeomorphism. This willnotbeproved l1ere.* Infact, C°‘manifolds foror5:500willhardly ever bementioned again. One rcmark isinorder now; theproof ofLemma 2produces anappropriate C“ function fonaC“manifold, for05or5oo.Ofcourse, forat=cutheproof flior aproof see unkres, Elenzenlagi Diﬂermlial Y5/1ul0_g)'. Dﬂrreulfable -$'lruclures 35 fails completely (and theresult isfalse-an analytic function which is0onan open setis0everywhere). With differentiable functions now atourdisposal, itisﬁtting that webegin differentiating them. What weshall deﬁne arethepartial derivatives ofadif- ferentiable function f:M—>IR,with respect toacoordinate system (x,UAt thispoint classical notation forpartial derivatives issystematically introduced, soitisworth recalling alogical notation forthepartial derivatives ofafunction f:IR"—>IR.Wedenote byD,-f(a) thenumber 1- f(a]:--~:ai+h:,j"1an)“f(a) rm 2~ . lz—)-O l? The Chain Rule states thatifg:IR“—>IR"andf:IR"—>IR,then D1‘(f0gm-Zv.~f<gt~>> ~D1'8"(¢1)-fzl Now, forafunction f:M—>IRandacoordinate system (x,U)wedeﬁne %tp>=P==Dilf0x*’><x>. (orsimply =_-D;(f0x""l) 0x,asanequation between functions). Ifwe deﬁne thecurve c,-:(—-e, e)—>Mby c.-t1=>-x"‘txtp> +<0.---./1.--so». 2*---- then thispartial derivative isjust Hm1'tc.-<1i>>- ftp),l1—)-U li soitmeasures theratechange offalong thecurve c,~;infactitisjust(foc;)’(0). Notice that _ _ .1ni=j fix‘ fix-l(p) 1loits721. Ifxhappens tobetheidentity map ofIR",then D;f(p) =3f/8x’(p), which istheclassical symbol forthispartial derivative. 36 Chapter 2 Another classical instance ofthisnotation, often notcompletely clariﬁed, is theuseofthesymbols 3/6r and 6/66inconnection with “polar coordinates”. Onthesubset AofIREdeﬁned by A=lR”-{(x,y)6lR2:y=0andx_20} ‘ s; =IR2-L wecanintroduce a“coordinate system” P:A—>IR2by P(X,J’) -"-=(f(X=y),9(X,J*)), where r(x, y)=--Vx”+y”and 6(x, y)istheunique number in(0,21?)with x=rtx.2)cosetx.Y) ‘ml y="tr,y)sin60¢,y)- "-"(mil This really isacoordinate system onAinoursense, with itsimage being the set{rIr>0}><(0,211). (Ofcourse, thepolar coordinate system isoften 2::~~~~~~~~~~~ (r,9) "6-axis”ii. * - W -—.:__ “r-axis” notrestricted tothesetA.One candelete anyrayother than Lif6(x, y) isrestricted tolieintheappropriate interval (69.60 +27:); many results are essentially independent ofwhich lineisdeleted, andthissometimes justiﬁes the sloppiness involved inthedeﬁnition ofthepolar coordinate system.) Wehave really deﬁned Pasaninverse function, whose inverse P“' isdeﬁned simply by P"'(r, 6)=(rcos6, rsin6). Dlﬁéfﬂfllldblﬁ Slruclures 37 From thisformula wecancompute 6f/firexplicitly: (f0P“’)(r, 6)=f(rcos6,rsin6), so gtxiy) =Ditf0P“"')(P(X,y)) =Dif(P"‘(P(X=y)) -Di[P""‘]‘(P(X,y)) +1>iftP*‘tPtx.n> -1->i[P*‘1ttPt>-.i»)> bytheChain Rule =D1f(X,y) -cos60¢.y)+-D2f(x!y)-SiI16(x!y)‘ This formula justgives thevalue ofthedirectional derivative offat(x,y), along aunit vector v=(cos6(x, y),sin 6(x, y))pointing outwards from the origin to(x,y).This istobeexpected, because cl,theinverse image under P / }sin6{x,y) (‘xi J’); ‘—-v—’ Cl"»" COS9{Ii}') 1'a{xI-l’) ofacurve along the“r-axis”, isjustalineinthisdirection. Asimilar computation gives %(-Y,J’)=D1f(X,J’)l*'*'(x>J’) Sin60¢,J’)l+D2f(r, J')[r(X= y)¢0s9(X. y)l- The vector w=(—sin6(x, y),cos6(x, y))isperpendicular tov,andthus the direction, atthepoint (x,y),ofthecurve C3which istheinverse image under P ofacurve along the“6-axis”. The factor 1-(x, y)appears because this curve V ¥- U 1: _'7---.- --7 ‘X. as ﬂ@m2 goes around acircle ofthat radius as6goes from 0to2:-r,soitisgoing r(x,y) times asfastasitshould goinorder tobeused tocompute thedirectional derivative offinthedirection w.Note that8f/86 isindependent ofwhich lineisdeleted from theplane inorder todeﬁne thefunction 6unambiguously. Using thenotation 8f/fixforD1f,etc., andsuppressing theargument (x,y) everywhere (thus writing anequation about functions), wecanwrite theabove equations as if__if ax. 5;--5cos6 +$s1n6 6 6%=%(—r sin6)+$1‘ cos6. Inparticular; these formulas also telluswhat fix/8:‘ etc., are, where (x,y) denotes theidentity coordinate system ofIR2.Wehave 6x/6r =cos6,etc.,so ourformulas canbeputintheform 2__3f6x +fiffiy 81'TfixBr 3yfir w,wm+y@ asC6x50ca;as Inclassical notation, theChain Rule would always bewritten inthisway. Itis apleasure toreport thathenceforth thismay always bedone: 3.PROPOSITION. If(x,U)and(y,V)arecoordinate systems onM,and f:M—>IRisdifferentiable, then onUOVwehave af"af6x-7 “I W"Zwa_i»="1:1 PROOF. It’stheChain Rule, ofcourse, ifyoujustkeep your cool: $02) =Di-(f0y“")(y(P)) =D;([f 0Xfll°[X0J#”’l)(y(P)) =ZD.i(f0-»-""‘><txOi»""‘1<i»>>> -D.-[X0i»""‘1-"<i»>jnl Dﬂérenlz'able Sl?’Z£ClLt?'6.s‘ 39 I1 =Z12,-ifOx""'><xtp>>»1>.-[x»‘ Oi»*'1ti»>>j=1 "af a1,=j;]w(P)-,,i,,,,(P)- '»' Atthispoint wecould introduce the“Einstein summation convention”. No- ticethatthesummation inthisformula occurs fortheindex j,which appears both “above” (in6X1’/6y’) and “below” (in6f/6x*’).There arescads offor- mulas inwhich thishappens, often with hoards ofindices being summed over, andtheconvention istoomit theZsign completely—double indices (which byluck, thenature ofthings, andfelicitous choice ofnotation, almost always occur above andbelow) being summed over. Iwon’t usethisnotation because whenever Ido,Isoon forget I’msupposed tobesumming, and because bydo- ingthings “right”, onecanavoid what Elie Cartan hascalled the“debauch of indices”. Wewilloften write formula (l)intheform a“6x=la 6y" 6y‘6x7’ here 6/6yfisconsidered asanoperator taking thefunction fto6f/6y’.The operator taking fto6f/6yf(p)isdenoted by ” -.i 6 6 61 6 aiip’ tus ax Zyi(p) 3 y_,, F216 62:1,, Forlater usewerecord aproperty ofZ=6/6x5|P;itisa“point-derivation”. 4.PROPOSITION. Foranydifferentiable f,giM—>IR,andanycoordinate system (x,U)with p<-5U,theoperator E=6/6x"|,, satisﬁes Etfg)=f(P)@(g) +f(f)g(P)- PROOF. Left tothereader. '1' If(x,U)and(x’,U’)aretwocoordinate systems onM,then><nmatrix 6x” (',",";,"_,r(P)) 40 Chapter 2 isjust theJacobian matrix ofx’0A'““1 atx(p). Itisnon-singular; infact, its inverse isclearly 6x’ (3-;;}'(P)) - Now ifftM"—>NmisC°°and(y,V)isacoordinate system around f(p), therank ofthem><J‘?matrix 3i(£a_.?_Q(,,,)xi clearly does notdepend onthecoordinate System (x,U)or(y,V).Itiscalled therank offatp.The point piscalled acritical point offiftherank off atpis<m(thedimension oftheimage N);ifpisnotacritical point off, itiscalled aregular point off.Ifpisacritical point off,thevalue f(p) is called acritical value off.Other points inNareregular values; thus q6N isaregular value ifandonlyifpisaregular point offforevery p<5f"'(q). This istrue, inparticular, ifq9!f(M)—a non-value offisstilla“regular value”. Iff:IR—>IR,then xisacritical point offifandonly iff"(x)=0.It ispossible forallpoints oftheinterval [a,b]tobecritical points, although this canhappen only iffisconstant on[a,b]. Iff:IR2—>IRhasallpoints as critical values, then D1f=D;f=0everywhere, sofisagain constant. On theother hand, afunction f:IR3—>IR2may have allpoints ascritical points without being constant, forexample, f(x,y)=A-.Inthiscase, however, the image f(lR2) =IR><{0}CIR2isstilla“small” subset ofIR”.The most important theorem about critical points generalizes thisfact. Tostate it,wewillneed some terminology. Recall thatasetACIR"has“measure zero” ifforevery e>0there isa sequence B],B3,B3,...of(closed oropen) rectangles with O0 ACUBH rim] andco Zv(B,,) <e, ﬂ=l wl1e1'e v(B,,) isthevolume ofB".Wewant todeﬁne thesame concept fora subset ofamanifold. Todothisweneed alemma, which inturn depends on alemma from Calculus onMamjiilds, which wemerely state. D;ﬂi=:renlz'able Slruclures 41 5.LEMMA. LetACIR"bearectangle and letf:A—>IR"beafunction such that |D,-f'| 5KonAfori,j=l,,..,n. Then |f(X) '"f(J/)| .€"’K|X-*J#| forallx,y 6A. 6.LEMMA. Iff:IR"—>IR"isC1andACIR"hasmeasure 0,then f(A) has measure 0. PROOF. Wecanassume thatAiscontained inacompact setC(since IR"isa countable union ofcompact sets). Lemma 5implies that there issome Ksuch that lf(X) :-f(J’)| 5H’-KIX -~yl forallx,y<5C.Thus ftakes rectangles ofdiameter clinto setsofdiameter 5?I2Kd. This clearly implies that f(A)hasmeasure 0ifAdoes. '1' Asubset AofaC°°:2-manifold Mhasmeasure zero ifthere isasequence ofcharts (x;,U,-), with ACU,Ur,such that each setx,-(A {'1U,-) CIR"has measure 0.Using Lemma 6,itiseasy toseethat ifACMhasmeasure 0,then x(A ('1U)CIR”hasmeasure 0foranycoordinate system (x,U). Conversely, ifthiscondition issatisﬁed andMisconnected, orhasonly countably many components, then itfollows easily from Theorem l-2that Ahasmeasure 0.(But ifMisthedisjoint union ofuncountably many copies ofIR,and Aconsists of onepoint from each component, then Adoes nothave measure 0).Lemma 6 thusimplies another result: 7.COROLLARY. Iff:M—>NisaC'function between twon-manifolds and ACMhasmeasure 0,then f(A) CNhasmeasure 0. PROOF. There isasequence ofcharts (xi,U,-)with ACU,U,’and each set x,-(A {'1U,-)ofmeasure 0.If(y,V)isachart onN,then f(A) {'1V=U,f(A {'1 U,-)F)V.Each set ytft/1r)U.-)nV)-yQfQx""'tx<A n11.)) hasmeasure O,byLemma 6.Thus y(f(A)f)V) hasmeasure 0.Since U,-) iscontained intheunion ofatmost countably many components ofN,itfollows that f(A)hasmeasure 0.'1' 42 crtepter 2 8.THEOREM (SARD’S THEOREM). Iff:M—>NisaCImap between n-manifolds, and Mhasatmost countably many components, then thecritical values offform asetofmeasure 0inN. PROOF. Itclearly suffices toconsider thecasewhere MandNareIR".But thiscaseisjustTheorem 3.14ofCalculus onManﬁbltls. '1' The stronger version ofSard ’sTheorem, which wewillnever use(except once, inProblem 8-24), states* that thecritical values ofaCI‘map f:M"—>N"' areasetofmeasure 0ifkZl+max(n —m,0). Theorem 8istheeasy case, andthecase m>nisthetrivial case (Problem 20). Although Theorem 8will bevery important later on,forthepresent w_earemore interested inknowing what theimage off:M—>Nlooks likelocally, interms oftherank koff atp<5M.More exact information canbegiven when factually hasrank lc inaneighborhood ofp.Itshould benoted that fmust have rank 3lcin some neighborhood ofp,because some lc><lcsubmatrix of(6(yf of)/6x‘.) hasnon-zero determinant atp,andhence inaneighborhood ofp. 9.THEOREM. (l)Iff:M”—>N’"hasrankkatp,thenthere issome coor- dinate system (x,U)around pandsome coordinate system (y,V)around f(p) with yof ox““’ intheform J/<>f°>¢“"(¢1’=---#1")=((11,---r¢1”,1l/”+’(ﬂ).---rt1/’"(a))- Moreover, given anycoordinate system y,theappropriate coordinate system onNcanbeobtained merely bypermuting thecomponent functions ofy. (2)Iffhasrank kinaneighborhood ofp,then there arecoordinate systems (x,U)and (y,V)such that yof<r>x""(a’,...,a") =(a1,...,a",0,...,0). Remark: The special case M=IR",N=IR’"isequivalent tothegeneral theo- rem, which gives only local results. Ifyistheidentity ofIR’",part (l)says that byﬁrstperforming adiffeomorphism onIR",andthen permuting thecoordi- nates inIR’",wecaninsure that fkeeps theﬁrstkcomponents ofapoint ﬁxed. These diffeomorphisms onIR"and IR“areclearly necessary, since fmay not even beone-onc onIRk><{0}CIR",anditsimage could, forexample, contain onlypoints with ﬁrstcoordinate 0. *Foraproof, seeMilnor, Yiyioltzgy From flu’D;'f’t'eitlial1le l'icut)l)0iit! orSternberg, Lectures on Dﬂrrenlial Geometry. Dwérerztiable Structures 43 Inpart (2)wemust clearly allow more leeway inthechoice ofy,since f(]R") may notbecontained inanyk-dimensional subspace ofIR". PROOF. (1)Choose some coordinate system uaround p.Byapermutation of thecoordinate functions u‘andy’wecanarrange that (1) d¢t( (p))%0 ¢,5=1,...,k. Deﬁne x°‘=y°’0f o.'=1,...,k x"'=u' r=k+1,...,n. Condition (1)implies that _ 80>Of)8 I (2) det )=det “ This shows that x=(x011*‘) 0atisacoordinate system insome neighbor- hood ofp,since (2)andtheInverse Function Theorem show that x014*‘isa difieomorphism inaneighborhood ofu(p).Now€ ‘IL5:’ q=x_1(a1,...,a") means x(q) =(a1,_..,a"), hence x‘.(q) =ai, hence yaOf(q) =00‘ of=limiku’(q)=a’ r=k+1,...,n, so y<>f<>>F‘(a‘,---,0") =y<>f(q) forq=x“(a‘,---,0") 2 (a]I“'3ak7i,i)' (2)Choose coordinate systems xand vsothat vof0x'1hastheform in Since rank f=kinaneighborhood ofp,thelower square inthematrix is0 _ '1 aw’on__>< Dk+1l1l/kl-1 Dmlnlfm 44 Chapter 2 must vanish inaneighborhood ofp.Thus wecanwrite 1,!/'(a)=1,l}"(a1,...,a]‘) r=k+1,...,m. Deﬁne yﬂ‘ Z UH yr=Ur____1"(}r O(v1’___,vk). Since (3) yov_'(b1,...,b”')=y(q) forv(q):(b1,...,b'") Z(b1J"-sbka bk-{-1 “'11-[}k+I(b]:---:bk); ---2 bm “Th-Z}m(bI:"'1bk)): (31?)1( ) hasnon—zero determinant, soyisacoordinate system inaneighborhood off(p).Moreover,theJacobian matrix yofox_1(a1,...,a“) =y-»v"‘ ov-»fox"‘<a’,....a") =y@~v"‘<-:2.-.,a".1//*+‘<a),...,t1/’"<a>) =<a‘,...,a’2 t1»"+’<a)-t1?"+'(a’,-_.,a"), 1//*"<a)-r?*"<a’,...,a")) bi/(3) =<a‘,...,a",0,...,0)- -:- Theorem 9acquires aspecial form when therank offisnorm: 10.THEOREM. (l)Ifm5nand f:M"—>Nmhasrank matp,then for any coordinate system (y,V)around f(p),there issome coordinate system (x,U)around pwith y0f<>x_](a1,...,a”) =(a1,...,a”'). Dgjizrenriable Structures 45 (2)If225mandf:M"—>N’"hasrank natp,then foranycoordinate system (x,U)around p,there isacoordinate system (y,V)around f(p)with yof ox_1(a',...,a") =(a1,...,a",0,...,0). PROOF (I)This ispractically aspecial case of(l)inTheorem 9;itisonly necessary toobserve that when k=m,itisciearly unnecessary, intheproof of thiscase, topermute they‘inorder toarrange that Q det( (p))¢0 a:,ﬁ=1,...,m; onlytheu"need bepermuted. (2)Since therank offatanypoint must be5n,therank offequals rt insome neighborhood ofp.Itisconvenient tothink ofthecase M:R" andN=ll?"andproduce thecoordinate system yforRmwhen wearegiven theidentity coordinate system forIR".Part (2)ofTheorem 9yields coordinate systems ¢forIR"and 1,0forRmsuch that 11/ofo¢_1(a],...,ﬂ") =(a1,...,a",O,...,0). Even ifweclonotperform 471ﬁrst, themap fstilltakes IR”intothesubset , , ms")=fc¢<1s">> 1111"‘R ¢ R f 11/ _._~e?.=‘v=; =e.==~ -4 )' >Ale’-I-'.'-.1I_-I1". 35.1"‘ /‘1//<f<1s")> f(lR”) which 1,11takes toR"x{O}C1R’"—the points ofIR”justgetmoved to thewrong place inIR"><{O}. This canbecorrected byanother map on1R’". Deﬁne Itby l(b1,...,bm)= (¢-1(b‘,...,b"),b"+‘,...,1;-"'). Then 1011/@f(a‘,..-,0") =10¢Q/<>¢*‘<b‘,...,b") for(b1,...,b“)=¢(a) =i(b‘,...,b",o,...,o) =(¢-‘(b‘,...,b"),0,...,0) =(a1,...,a“,O,...,O), soIt011/isthedesired y.Ifwearegiven acoordinate system xonIR"other than tl1eidentify, wejustdeﬁne Mb‘,-..,b"')=<1-<¢"‘(b’, ...,b")>,b"+‘, ...,b'"); itiseasily checked that y==ito11/isnow thedesired y.*1‘ 46 Chapter 2 Although pisaregular point offincase (l)ofTheorem l0andacritical point incase (2)(if11<m),itiscase (2)which most interests us.Adifferentiable function f:M”—>N’"iscalled anirmnersion iftherank offisrt,the dimension ofthedomain M,atallpoints ofM.Ofcourse, itisnecessary that mZ:1,and itisclear from Theorem l0(2] that animmersion islocally one-one (soitisatopological immersion, asdeﬁned inChapter l).Ontheother hand, adifferentiable map fneed notbeanimmersion even ifitisglobally one-one. The simplest example isthefunction fIR—>IRdeﬁned byf(x)=x3,with f’(O) =O.Another example is e_"'—2 x>O X g(x)= 0 JL‘=0 t_t -e"x—2 x<O. / Amore illuminating example isthefunction hiIR—>R2deﬁned by ho-)=(go),tgo-)1); ” although itsimage isthegraph ofanon-differentiable function, thecurve itself manages tobedifferentiable byslowing down tovelocity 0atthepoint (0,0). One caneasily deﬁne asimilar curve whose image looks likethepicture below. 75 Three immersions ofIRinR2areshown below. Although thesecond and third immersions ,5;and ,5;areone-one, their images arenothomeomorphic W l 132(3) 1)wZ.’re2z:!:fabZe Siructzncs 47 toIR.Ofcourse, even iftheone-one immersion f:P—>Misnotahomeo- morphism onto itsimage, there iscertainly some metric andsome differentiable structure onf(P)which makes theinclusion map 2':f(P)—>Manimmer- sion. Ingeneral, asubset M1CM,with adifferentiable structure (not nec- ' 'bl 'hh 'M'h'ts asubset ofM)iscalled an cssartly compati ewit temetric 1ineri as , immersed submanifold ofMiftheinclusion map 1':M1—>Misanimmersion. Thefollowing picture, indicating theimage ofanimmersion B3:IR—>S‘><S‘, 1.!»second time around shows that M;may even beadense subset ofM. Despite these complications, ifM1isak—dimensional immersed submanifold ofM”andU;isaneighborhood inM;ofapoint peM1,then there isa coordinate system (y,V)ofMaround p,such that U101/={qEM=y"+‘(q)=---=y"(q)=0}; this isanimmediate consequence ofTheorem lO(2), with f=1'.Thus, if g:M1—>NisC°°(considered asafunction onthemanifold M1)inaneigh- borhood ofapoint p6M1,then there isaC°°function §onaneighborhood VCMofpsuch that g=§ofonVOM;—we candeﬁne _ , y°’(q')=y°'(q) <r=-1,---,/<s(q)=g(q), where {y"'(q')=0 r=k+l,...,n. 48 Chapter 2 Ontheother hand, even ifgisC°° onallofM,wemay notbeable to deﬁne §onM.Forexample, thiscannot bedone ifgisoneofthefunctions 19?‘:5.-(M)—>R.One other complication arises with immersed submanifolds. IfM1CMis animmersed submanifold, and ftP—>MisaC°° function with f(P) CM1, itisnotnecessarily true that fisC°°when considered asamap into M1, with its C°°structure. The following ﬁgure shows thatfmight noteven becontinuous f(P) f P M|CM=]R2 M1 asamap into M1.Actually, thisistheonly thing that cangowrong: ll.PROPOSITION. IfM; CMisanimmersed manifold, ftP—>Mis aC°° function with f(P) CM1, and fiscontinuous considered asamap into M1,then fisalsoC°°considered asamap into M1. PROOF. Let1':M1—>Mbetheinclusion map. Wewant toshow that2'*10f isC°°ifitiscontinuous. Given p6P,choose acoordinate system (y,V)forM around f(p)such that U1={qE V1J’k+'(q)='-'=J’“(6l)=9} isaneighborhood off(p) inM]and (y'|U;,...,y"|U1)isacoordinate system OfM1 011U1. " -.| "---.,_ | I‘ U, i Dwé:-'mz!iabZe Structures 49 Byassumption, 1"]0fiscontinuous, so f10r'(open set) isanopen set. Since U;isopen inM], thismeans that f"1(U|) CPisopen. Thus ftakes some neighborhood ofpePintoU1.Since allylofareC°°, andy1,...,yk areacoordinate system onU1,thefunction fisC°° considered asamap into M1. '1' Most ofthese diﬁiculties disappear when weconsider one-one immersions f1P—>Mwhich arehomeomorphisms onto their image. Such animmersion iscalled animhedding (“embedding” fortheEnglish). Animmersed subman- ifold M;CMiscalled simply a(C°°)submanifold ofMiftheinclusion map 1':M1—>Misanimbedding; itiscalled aelosed submanifold ofMifM1is alsoaclosed subset ofM. @ .. aclosed .-- submanifold There isone way ofgetting submanifolds which isvery important, and gives thesphere .S‘"_1 CIR"~—{0}CIR",deﬁned as{x:lxlz=l},asaspecial case. 12.PROPOSITION. IffIM"—>Nhasconstant rank konaneighborhood off‘!(y),then f"'(y)isaclosed submanifold ofMofdimension n—-k(or isempty). Inparticular, ifyisaregular value offIM"—>N“, then f-1(y) isan(H~—m)—dimeiisional submanifold ofM(orisempty). PROOF. Left tothereader. '2' Itistobehoped that however abstract thenotion ofC°° manifolds may appear, submanifolds ofRNwillseem likefairly concrete objects. Now itturns outthat ezieiji (connected) C°°manifold can beimbedded insome RN,sothat manifolds canbepictured assubsets ofEuclidean space (though thispicture isnotalways theniost useful one). Wewill prove this fact only forcompact manifolds, butwefirstdevelop some ofthemachineiy which would beused in 50 Chapter 2 thegeneral case, since wewill need itlater onanyway. Unfortunately, there are many deﬁnitions andtheorems involved. If(9isacover ofaspace M,acover (9’ofMisareﬁnement of(9(or “reﬁnes (9”)ifforevery Uin(9’there issome Vin(9with UCV(thesetsof(9’ are“smaller” than those of(9)—a subcover isavery special case ofareﬁning cover. Acover (9iscalled locally ﬁnite ifevery pEMhasaneighborhood W which intersects only ﬁnitely many sets in(9. 13.THEOREM. If(9isanopen cover ofamanifold M,thenthere isanopen cover (9"ofMwhich islocally ﬁnite andwhich refines (9.Moreover, wecan choose allmembers of(9’tobeopen setsdiffeomorphic toIR". PROOF V\=’ecanobviously assume thatMisconnected. ByTheorem 1.2,there arecompact sets C1,Cg, C3,. ..with M=C1UCgUC3U---.Clearly C;has anopen neighborhood U;with compact closure. Then F;UCghas anopen neighborhood U;with compact closure. Continuing inthisway weobtain open setsU,-,with compact and CU,-+1, whose union contains allC,-, and hence isM.LetU_; =U0=El. .. ﬂ/ // Now Mistheunion fori>lofthe“annular” regions A;=U,-~— U,-_1. Since each A,iscompact, wecanobviously cover A;byaﬁnite number ofopen sets, each contained insome member of(9,and each contained inV;=U,-+1 -U,-_g. \'Vecan also choose these open setstobediffeomorphic toR“. Inthisway we obtain acover (9"which reﬁnes (9and which islocally ﬁnite, since apoint inU,- isnotin forj;">_2+i'. *9 Dwéreiztiable Structures 51 Notice that if(9isanopen locally ﬁnite cover ofaspace Mand CCMis compact, then Cintersects only ﬁnitely many members of(9.This shows that anopen locally finite cover ofaconnected manifold must becountable (like the cover constructed intheproof ofTheorem 13). l4.THEOREM (THE SHRINKING LEMMA). Let(9beanopen locally finite cover ofamanifold M.Then itispossible tochoose, foreach Uin(9,an open setU’with FCUinsuch away that thecollection ofallU’isalso an open cover ofM. PROOF V\lecanclearly assume that Misconnected. Let(9={U1, Ug,U3,...}. Then c,=U;--(UgUU3U---) isaclosed setcontained inU1,and M=C1UU2UU3U---.Let U1’bean open setwith C;CU,’CU,’CU1.Now C'2=U2-"(UiUU3U--') isaclosed setcontained inU3,and M=U,’UC2UU3U---.LetU;bean open setwith CgCU5CU5CUg.Continue inthisway. ForanypEMthere isalargest nwith pEU",because (9islocally ﬁnite. Now pEU,"UU§'U--tUU,’;U(U,,+1UU,,+;;U---); itfollows that peU,"UU§U---, I since replacing U,,+,- byU,,+,- cannot possibly eliminate p.'3‘ 15.THEOREM. Let(9beanopen locally ﬁnite cover ofamanifold M.Then there isacollection ofC°°functions (pg: M—>[0,l],oneforeach Uin(9, such tliat (l)support ¢UCUforeach U, (2)Z¢U( p)=lforallpEM(this sum isreally aﬁnite sum insome U neighborhood ofp,by(l)). 52 Chapter 2 PROOF Case .7.Each Uin(9hascompact closure. Choose theU’asinTheorem 14.Apply Lemma 2toU’CUCMtoobtain aC°°function (try; M—>[0,1] which islonFandhassupport CU.Since theU’cover M,clearly Z1,lrU>Oeverywhere. Ue(9 Deﬁne ‘PU=Zti/UUGO Case 2.General case. This case can beproved inthesame way, provided that Lemma 2istrue forCCUCMwith Cclosed (but notnecessarily compact) and Uopen. Butthisisaconsequence ofCase It Foreach p6Cchoose anopen setUpCUwith compact closure. Cover M--Cwith open sets V0,having compact closure and contained inM--C. The open cover {U,,, V0,}hasanopen locally ﬁnite reﬁnement (9towhich Case.7 applies. Let f=Z(pg, where (9’={UE(9:UCUpforsome p}. U60’ This sum isC°°,since itisaﬁnite sum inaneigliborliood ofeach point. Since ZU¢U(p) =lforallp,and¢U(p) =0when UCV0,,clearly f(p) =l forallp6C.Using thefact that (9islocally ﬁnite, itiseasy toseethat supportf CU.6+ l6.COROLLARY. If(9isany open cover ofamanifold M,then there isa collection ofC°°functions ¢,-:M—>[0,1] such that (1)thecollection ofsets{p:¢,-(p)qéO}islocally ﬁnite, (2)Z,-¢,-(p) =Iforallp6M, (3)foreach 1'there isaU6(9such thatsupport ¢,-CU. (Acollection {¢,-I M—>[0,1]}satisfying (l)and (2)iscalled apartition ofunity; ifitsatisﬁcs (3),itiscalled subordinate to(9.) Itisnow fairly easy toprove thelasttheorem ofthischapter. l7.THEOREM. IfM”isacompact C°°manifold, then there isanimbed- cling ftM—>RNforsome N. 1)i'ﬂi:rei2t2'a/ile Structures 53 PROOF There areaﬁnite number ofcoordinate systems (x1,U1), ...,(xk, Uk) with M=U;U---UU1,. Choose U,’asinTheorem l4,andfunctions 1,[r;:M—> [0,1] which arelon andhave support CU,-.Deﬁne fiM—>RN, where N=nk+k,by f=(tl/1 -X1,---=1!/it-Xi<,1l/1,---,tl/i<)- This isa1iimmersion, because anypoint pisinU,-’forsome 2',and onU,-’,where 1/1;-.=1,theNXn_]acobian matrix 8f°‘ _ _ 8x‘?W CO1'lltE1II1S ll'lC P?XPII'nI'1ltI'].X £3 Z 8x, 8x, Itisalso one-one. Forsuppose that f(p)=f(q). There issome 2'such that pEU,-’,Then (Zr,-(p) =1,soalso 11/,-(q) =1.This shows that wemust have qeU,-.Moreover, ti/s-Xr'(P) ==11/:-r<:(¢1), sop=q,since xiisone-one onU,-.'1' Problem 3-33 shows that, infact, wecanalways choose N-_-=2n+1. PROBLEMS 1.(a)Show that being C°°—related isnotanequivalence relation. (b)Intheproof ofLemma l,show that allcharts inA’areC°°-related, as claimed. 2.(a)IfMisametric space together with acollection ofhomeomorphisms x:U—>IR"whose domains cover Mand which areC°°-related, show that thetiateach point isunique without using Invariance ofDomain. (b)Show siinilai'ly that 3Miswell-defined foraC°°manifold-with-boundary M. 3.(a)AllC°°functions arecontinuous, andthecomposition ofC°°functions isC°°. (b)Afunction flM—>NisC°°ifand only ifgofisC°° forevery C°° function g:N—>1R. 4.How many clistinct C°°structures arethere onIR?(There isonly oneupto diffeoinorphism; that isnotthequestion being asked.) 54 C/iapter 2 5.(a)IfNCMisopen and A’consists ofall(x,U)inAwith UCN,show that A’ismaximal forNifAismaximal forM. (b)Show thatA1’canalsobedescribed asthesetofall(x|V ON,VON)for (x,V)inA. (c)Show that theinclusion 1':N—>MisC°°, and that A’istheunique atlas with thisproperty. 6.Check thatthetwoprojections P1andP2on.S‘"“] areC°°related tothe2n homeomorphisms ﬂ-andgt. 7.(a)IfMisaconnected C°°manifold and p,q EM,then there isaC°° curve c:[0,1] —>Mwith c(0)=pandc(l)==q. (b)Itiseven possible tochoose ctobeone-one. 8.(a)Show that (M1 xMg) ><M3isdiffeomorphic toM1><(Mg xM3) and that M1><Mgisdiffeomorphic toM2><M1. (b)The diffcrentiable structure onM1><M2makes the“slice” maps Pl FT (PI: P2*—+(151,192) ofM1,M;—>M1><Mgdifferentiable forallp16M1,192EMg. (c)More generally, amap f:N—>M1><MgisC°°ifandonly ifthecompo- sitions 7:1Of:N—>M1and rt;0f:N—>MgareC°°. Moreover, theC°° structure wehave defined forM1><Mgistheonlyonewith thisproperty (d)Iff,-: N—>M;areC°°(i=1,2),canonedetermine therankof(f],fg)I N —>M1><Mgatpinterms ofthe ranks off;atp?Forf,-:N;—>M,-,show that fl><f2IN1>< N2—>M1><M2,d@ﬁI1eC1 byf1><f2(p1,P2) =(f1(p1)»f2(p2)). isC°°anddetermine itsrank interms oftheranks ofj}. 9.Letg:S"—>IP”bethemap pi——>[p].Show that ftP“—>MisC°°ifand onlyiffog: S”—>MisC°°. Compare therank offandtherank offcg. 10.(a)IfUCIR"isopen and ftU—>IRislocally C°° (every point hasa neighborhood onwhich fisC°°),then fisC°°. (Obvious.) (b)Iff:11-11"—>IRislocally C°°, then fisC°°, i.e., fcan beextended toa C°°function onaneighborhood ofll-ll". (Not soobvious.) 11.Iff:ll-11"—>IRhastwoextensions g,htoC°°functions inaneighborhood of1H1“,then Dy-g andD,-Ii arethesame atpoints ofIR”-1 ><{O}(sowecanspeak ofD,-f atthese points). 12.IfMisaC°°inanifold-witli—boundary, then there isaunique C°°structure on8Msuch thattheinclusion map i:8M—>Misanimbedding. 1)wereutiabZe Structures 55 13.(a)LetUCM"beanopen setsuch thatboundary Uisan(ii—l)—dimen- sional (differentiable) submanifold. Show that Uisanit—dimensional manifold- with-boundary. (Itiswelltobear inmind thefollowing example: ifU={xE IR":d(x,O) <1or1<d(x,O) <2},then Uisamanifold-with-boundary, but 3Uqéboundary U.) (b)Consider theﬁgure shown below. This ﬁgure may beextended byputting at smaller copies ofthetwoparts ofS’intotheregions indicated byarrows, and thenrepeating thisconstruction indeﬁnitely. Theclosure Softheﬁnalresulting ﬁgure isknown asAlexandeﬁs Horned Sphere. Show that Sishomeomorphic toS2. (Hint: The additional points intheclosure arehomeomorphic totheCantor set.) IfUistheunbounded component ofIR3—S,then S=boundary U,but Uisnota2-dimensional manifold-with-boundary, sopart (a)istrueonly for differentiable submanifolds. 14.(a)There isamap f:IR2—>IR2such that (l)f(x,0) =(x,0) forallx, (2)f(x,y) CH2fory3O, (3)f(x,y) c1R2-H2fory<0, 56 C/zapter .2 O0 (4)frestricted totheupper half-plane orthelower half-plane isC,butf itself isnotC°°. ' (b)Suppose Mand NareC°°manifolds-with-boundary and fI3M —>3N isadiffeomorphism. LetP=MU;Nbeobtained from thedisjoint union ofMandNbyidentifying xE3Mwith f(x) E3N.If(x,U)isacoordinate system around pE3Mand (y,V)acoordinate system around f(p),with f(U f'13M) =V('13N, and (yof)|U I")3M =x|U ('18M, wecan deﬁne a homeomorphism from UUVCPtoIR"bysending Uto1H1"byxand Vto thelower half-plane bythereflection ofy.Show that thisprocedure does not / T I \ ﬁes “’;¥&;,,1%gd/i ofy deﬁne aC°°structure onP\ (c)Now suppose that there isaneighborhood Uof3M inMand adiffeo- morphism ct:U—>3Mx[0,I),such that o:(p) =(p,0) forallp68M, anda similar diffeomorphism ,5:V—>3Nx[0,l).(Wewillbeabletoprove later that such diffeomorpliisms always exist). Show that there isaunique C°°structure \its-Q onPsuch thattheinclusions ofMandNareC°°andsuch thatthemap from UUVto8Mx(--l, l)induced byorand13isadiffeomorphism. (d)Byusing twodifferent pairs (or,,5),define twodifferent C°°structures onIR2, consiclcred astheunion oftwo copies of11-112with corresponding points on311-112 iclentiﬁed. Show thattheresulting C°°manifolds arediffeomorphic, butthat thediffeomorphism cannot bechosen arbitrarily close totheidentity map. Dgjﬂerentiabte Structures 57 15.(a)Find aC°°structure on11-111x11-11’which makes theinclusion into IR2 aC°°map. Can theinclusion beanimbedding? Aretheprojections oneach factor C°°maps? (b)IfMand Naremanifolds-with-boundary, construct aC°° structure on MxNsuch thatallthe“slice maps” (deﬁned inProblem 8)areC°°. 16.Show that thefunction fIIR—>IRdeﬁned by e_1"x x>0 f<x>={Ox50 isC°°(theformula e_1"‘2 isusedjusttogetafunction which is>0forx<0, ande_"""| could beusedjustaswell). 17.Lemma 2(asaddended bytheproof ofTheorem 15)shows thatifC1andCg aredisjoint closed subsets ofM,then there isaC°°function ftM->[0,1] such that C1Cf"1(0) and CgCf_’(1). Actually, wecan even ﬁnd fwith C1=f"l(0) andCg=f"1(l). The proof turns outtobequite easy, once you know thetrick. (a)Itsuffices toﬁnd, foranyclosed CCM,aC°°function fwith C=f"1(0). (b)Let{U1} beacountable cover ofM-C,where each U1isoftheform U1=x"’({a e1R”:|a|< 1}) forsome coordinate system xtaking anopen subset ofM——Conto IR". Let f,-:M—>[0,1] beaC°°function with f,->0onU1andf,---=0onM-—U,-. Functions like Elf; 32)’,-ir, i."""?{-, 3x1 3x13x willbecalled mixed partials of)’,-,oforder 1,2,. ...Let oz;=supofall mixed partials off1, ...,f,-ofall orders 51'. Show that O0 frf'-"=ZE i=1 isC°°, and C=f"'1 18.Consider thecoordinate system (yl,yz)for1R2deﬁned by y‘(cub)=a y2(a,b) =a+1’). 58 @@m2 (a)Compute Bf/3y1(a,b) from thedeﬁnition. (b)Also compute itfrom Proposition 3(toﬁnd 3]’/Ely’ ,write each 1’interms ofylandyz). Notice thatElf/Eiyl qé8f/_3I’ even though y’=11;theoperator 8/Ely’ depends onyand1',notjustony’. 19.Compute the“Laplacian” a2 a2 xﬁai interms ofpolar coordinates. (First compute 3/3x interms of3/Br and 3/39; then compute 32/8x2 from this). Answer: %[%(r9fi,_) +5’-%(%%)]. 20.1ff:M" —>N’" isC’and m>it,then f(M) hasmeasure 0(provided that Mhasonly countably many components). 21.The following pictures show, forit=1,2,and3,asubdivision of[0,1] x [0,1]into 22"squares, A,,,1,. ..,An,22JI; square An’), islabeled simply k.The numbering isdetermined bythefollowing conditions: (a)The lower leftsquare isA,,,1. (b)The upper leftsquare isA,,,2z~. (c)Squares /1,,’1,and A,,,;,,_1_1 have acommon side. (d)Squares A,,_41_1_1, A,,,41_1_2, A,,,41_1_3, A,,,41_1_4 arecontained inA,,_1,;_1_1. IE ' 3 II III lill "'2 IEM‘III " Deﬁne f:[0,1]—>[0,I]x[0,1]bythecondition-HIEIIEHHHEIIIEIIIEEEIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII k l kf(l) EAn‘); TOT all "F if 5 Show that fiscontinuous, onto [0,1]x[0,1],andnotone-one. Dgjjfrreritiabte Structures 59 22.Forp/2" E[0,1],deﬁne f(p/2") EIRZasshown below. 1 ,f(2) 15.2-1 sh W.-:~:=_-"' .='.v-*4....=..11.-~"— l";_-._-1':-,=-,¢,;-_?;=,1j,_'.'u *' .:,- "‘ : #2,: .*--7'- ;-1q..-.-.'-.-;»-.\.-.i:- . £111.».u;,1';§-37'," 1_'.=;;:€'i.".!-'.*'.Z".?-I‘ .*."-l‘.~_~“—;s'* 7 .r:r=t~*.-.1sv f—)3l3 t'_f£=_1:-iii-?:?e1:4!‘£‘E=‘E.i1_ ,~:’s:‘1?as?~.L ‘Q rc..|..-.. » .1-.. .'~'--1'-:3;-1 \. '.’—:=.';-1'-Yl.1.-:11: ms?--1 '.-<3-.:-rs.,: {-"t-‘-i.=-..r-:-1.-1,-=4-‘gt .Li-T ‘wanes?A-no .~~r~ B=fmat)at-> (a)Show thatfisuniformly continuous, sothatithasacontinuous extension g:[0,1] —>IR2. Show thatgisone-one, andthatitsimage willnothave measure 0iftheshaded triangles arechosen correctly. (b)Consider thehomeomorphic image ofS1obtained byadding, below the image ofg,asemi-circle with diameter thelinesegment AB. \'Vhat does the inside ofthiscurve look like? 23.Letc:[0,1] —>IR"becontinuous. Foreach partition P:{r0,...,:,,} of [0,1],deﬁne k ac.P)=Zdtcv.->.c<:.-_1)>.i=1 The curve cisrectiﬁable if{£(c, P)}isbounded above (with length equal to sup{€(c, P)}). Show that theimage ofarectiﬁable curve hasmeasure 0. 24.(a)IfMisaC°°manifold, asetM1CMcanbemade into ak-dimen- sional submanifold ofMifand only ifaround each point inM1there isa coordinate system (x,U)onMsuch that M1PIU={p:x"+’(p) = = X”(P) =0}- (b)The subset M1canbemade into aclosed submanifold ifandonly ifsuch coordinate systems exist around every point ofM. 25.The set{(x,lxl); xEIR}isnottheimage ofanyimmersion ofIRintoIR2. 26.(a)IfUCIR"isopen and f:U~—>]R"_" isC°°, then thegraph of f={(p,f(p)) EIR”: pEU}isasubmanifold ofIR". (b)Every submanifold ofIR"islocally ofthisform, after renumbering coor- dinates. (Neither Theorem 9nor10isquite strong enough. Youwillneed 60 Chapter 2 theimplicit function theorem (Calculus onManifolds, pg.41). Theorem 10ises- sentially Theorem 2-13 ofCalculus onManifolds; comparison with theimplicit function theorem willshow howsome information hasbeen allowed toescape.) 27.(a)Animmersion from onerz-manifold toanother isanopen map (the image ofanopen setisopen). (b)IfMand Nareti-ITII-1I1lfOlClS with Mcompact and Nconnected, and f:M—>Nisanimmersion, then fisonto. 28.Prove Proposition l2:Iff:M”—>Nhasconstant rank konaneigh- borhood off_1(y), then f_'(y)isa(closed) submanifold ofMofdimension rz~k(orisempty). 29.LetfllF°2—>IR3bethemap e([x.y.z]) =(yz.Xr='.xy) deﬁned inChapter l,whose image istheSteiner surface. Show that gfails to beanimmersion at6points (theimage points arethepoints atdistance :l:l/2 oneach axis). There isaway ofimmersing P2inIR3,known asBoy’s Surface. SeeHilbert and Cohn-Vossen, Geometry andtheImagirtatiort, pp.317-321. 30.Acontinuous function f:X—>Yisproper iff“’(C) iscompact for every compact CCY.The limit set1_.(f) offisthesetofallyEYsuch thaty=limf(x,,) forsome sequence x1,x2,x3,... EXwith noconvergent subsequencc. L(f) =I5ifandonly iffisproper. f(X) CYisclosed ifand only if1_.(f) Cf(X). There isacontinuous f:IR—>IR2with f(IR)closed, but1_.(f) 750. Aone-one continuous function f:X—>Yisahomeomorphism (onto its image) ifand only ifL(f) Of(Y) -=Q1. (e)Asubmanifold M1CMisaclosed submanifold ifand only iftheinclusion map i:M1—>Misproper. (f)IfMisamanifold, there isaproper map f:M—>IR;thefunction fcan bemade C°°ifMisaC°°manifold.oaie 31.(a)Find acover of[0,1] which isnotlocally ﬁnite butwhich is“point- ﬁnite”: every point of[0,I]isinonlyﬁnitely many members ofthecover. (b)Prove theShrinking Lemma when thecover (9ispoint-ﬁnite andcountable (notice thatlocal-ﬁniteness isnotreally used). (c)Prove theShrinking Lemma when (9isa(not necessarily countable) point- ﬁnite cover ofanyspace. (You willneed Zorn’s Lemma; consider collections C’ D;'ﬂZ22'entiabZe Structures 61 ofpairs (U,U’)where UE(9,U’CU,andtheunion ofall U’for(U,U’)EC’, together with allother U6(9covers thespace.) 32.(a)IfM1CMisaclosed submanifold, UI)M1isanyneighborhood, andf:M1—> IRisC°°, then there isaC°°function f:M—>IRwith )7: f onM1, and with support JFCU. (b)This isfalse ifM=IRand M1=(0,1). (c)This isfalse ifIRisreplaced byadisconnected manifold N. Remark: Itisalsofalse ifM=IR2,M1=N=S1,and f=identity; infact, in thiscase, fhasnocontinuous extension toamap from IR2toS1,buttheproof requires some topology. However, fcanalways beextended toaC°°function inaneighborhood ofM1(extend locally, and usepartitions ofunity). 33.(a)The setofallnon-singular 22><nmatrices with realentries iscalled GL(n, IR),thegeneral linear group. ItisaC°°manifold, since itisanopen subset ofIR"2. The special linear group SL(r1,IR), orunimodular group, isthe subgroup ofallmatrices with det=l.Using theformula forD(det) inCalcuius onAdarzgfialds, pg.24,show that SL(n, IR)isaclosed submanifold ofGL(n, IR)of dimension I22—l. (b)The symmetric I2><nmatrices may bethought ofasIR"("+l)/2. Deﬁne 1,!/:GL(n,IR) —>(symmetric matrices) by1,1/(A) =A-A‘,where A‘isthe transpose ofA.The subgroup v,(r“‘(I) ofGL(n,IR) iscalled theorthogonal group O(n). Show that AEO(n) ifandonly iftherows [orcolumns] ofAare orthonormal. (c)Show thatO(n) iscompact. (d)ForanyA6GL(n,IR), define RA: GL(n,IR) —>GL(n,IR) byR,4(B) =BA. Show that RAisadiffeomorphism, andthat 11/oRA=11/forallA6O(n). By applying thechain rule, show thatforAEO(n) thematrix 3:1 3if (%k7(A)) hasthesame rank as(%ﬁ(I)) . (Here xklarethecoordinate functions inIR"2, and1,lr'7then(n+ l)/2component functions of11/.)Conclude from Proposition I2thatO(n) isasubmanifold of GL(n,IR). (e)Using theformula 11/"1'(A)=Zn.-1.0,”. (A—-=(at,-I).k 62 Chapter 2 show that _ Hi; k=1 75j 3U - /Cz:' ' ~”%.v<A>= ¥*’. ax 20;] /C=1 21 0 otherwise. Show that therank ofthismatrix isn(n+l)/2 atI(and hence atAforall AEO(n).) Conclude that O(n) hasdimension n(n—~1)/2. (F)Show thatdetA=:l:lforallAEO(n). The group O(n)r‘1SL(n,IR) iscalled thespecial orthogonal group SO(n), ortherotation group R(n). 34.LetM(m,n) denote thesetofallm><:2matrices, andM(m,n;k) theset ofallmxnmatrices ofrank k. (a)Forevery X0eM(m,n;k) there arepermutation matrices PandQsuch that PXOQ =(ﬁg 13;) , where A0isk><kandnon~singular. (b)There issome 2>0such thatAisnon-singular whenever allentries of A--A0are<:g, ABPXQ=(C D)(c)If where theentries ofA—A0are<e,then Xhasrank kifandonly ifD-.-.= CA_lB. Hint: If1;,denotes thekxkidentity matrix, then 1;,0 AB_ A B XI,,_.;, CDTXA+C XB+D ' (d)M(m,n;k) CM(m,n) isasubmanifold ofdimension k(m +n-~k)forall k35m,n. CHAPTER 3 THE TANGENT BUNDLE Apoint v6IR“isfrequently pictured asanarrow from Otov.Butthere are many situations where wewould liketopicture thissame arrow asstarting U U atadifferent point pEIR": ll/p+UP *1’ U JJ4 .0 _; J- J» a J Forexample, suppose c:IR—>IR”isadifferentiable curve. Then c’(:) = (cl’(r), ...,c"’(t)) isjust apoint ofIR",butthelinebetween c(r)andc(r)+c'(r) istangent tothecurve, andthe“velocity vector” or“tangent vector” c’(l) of thecurve ciscustomarily pictured asthearrow from c(t) toc(t) +c’(£)_ c(r)+c’(t) (‘(1') 0Ci(f) 63 64 Chapter 3 Togive thispicture mathematical substance, wesimply describe the“arrow” from ptop+vbythepair (p,v).The setofallsuch pairs isjustIR”><IR", which wewillalsodenote byTIR”, the“tangent space ofIR"”; elements ofTIR" arecalled “tangent vectors” ofIR".Wewilloften denote (p,v)ETIR" byvp (“the vector vatp”); inconformity with thisnotation, wewilldenote theset ofall1[p,v) forvEIR"byIR”,,. Attimes, itismore convenient todenote a member ofTIR" byasingle letter, likev.Torecover theﬁrstmember ofapair veTIR", wedeﬁne the“projection” map rt:IR“><IR“—>IR"byrr(a,b) =a. Foranytangent vector v,thepoint rr(v) is“where it’sat”. The setrt-'( p)may bepictured asallarrows starting atp.Alternately, P itcanbepictured more geometrically asaparticular subset ofIR"><IR",the onevisualizable case occurring when n=1.This picture gives risetosome TR“‘M7;)“ “IEYXW i Ii ‘I I. . II ,. .. l___l_ug___JlwI:‘ Ill w llllii \I_.:__l_‘; ll II"I "I.r1~"r .,;I , "'\"*v— I.I ll}ill g U31,‘ I g p.‘:‘3‘I‘H1‘I“ p lilllwl i,‘;¥l*! Nil‘, 71' 13"I I,__, P te1'minology—we calln"'( p)theﬁbre over p.This fibre canbemade intoa TheYimgent Bundle 65 vector space inanobvious way: wedeﬁne (p,v)®(p,w)=(p,v+w) a.(pvv) 2(pza 'v)- (Theoperations EBand ~should really bethought ofasdeﬁned on Urr'l(p) Xrr_l(p), and IRXTIR", respectively. Penn Usually wewilljustuseordinary +and»instead of®and~.) Iff:IR"—>IR”'isadifferentiable map, andpEIR",then thelinear trans- formation Df(p): IR"—>IR”'may beused toproduce alinear map from IR”,—>IR"';(,,) deﬁned by up*—>[Df(P)(v)]Itp>- This map, whose apparently anomalous features willsoon bejustiﬁed, isde- noted byf,.,,; thesymbol ﬁtdenotes themap ﬁt:TIR” —>TIR”' which isthe union ofallftp. Since _)’...,,(v) isdeﬁned tobeavector EIR’"jr(,,), thefollow- ingdiagram “commutes” (thetwopossible compositions from TIR" toIR”'are equal), f ' TIR” Z1 TIR”' rrj in rr0f,.1f0rt. R».._..,.L_. Rm Thus, ﬁ,,hasthemap f,aswell asallmaps Df(p), built into it. This isnottheonly reason fordeﬁning 1’...inthisparticular way, however. Suppose thatg:IR""—>IRI‘isanother differentiable function, sothat, bythe chain rule, (1) D(gOf)(p)=Dg(f(p)) <>Df(p)- Byourdeﬁnition, glh 7: This looks horribly complicated, but,using (l),itcanbewritten g*(f..(v,,)) =(gOf)*(v,,); as czwpms thuswehave g*°fi==(g°f)*- This relation would clearly fallapart completely ifft(vp) were notinIR’"f(,,); with ourpresent deﬁnition ofﬁk,itismerely anelegant restatement ofthe chain rule. Henceforth, wewillstate almost allconcepts about_]acobian matrices, like rank orsingularity, interms ofj’,,,rather than Df.The “tangent vector” ofa curve c:IR—>IR"canbedeﬁned interms ofthisconcept, also. The tangent vector ofcatImaybedeﬁned as Ci(i)c(r) ElR”¢(:)- [Ifchappens tobeoftheform 5Pm ctr)-=<1.r<o>1‘<>r f:RaR cm=“’m then C’(t)c(-I‘) =(1:}”(£))c(r)§ thisvector liesalong thetangent linetothegraph offat(I,f(£))_] Notice thatthetangent vector ofcattisthesame as C*(ll') Z ‘T(CH0): ---ICnl(t))C(.l'): where 1,=(I,I)isthe“unit” tangent vector ofIRatI. 0 l I I;~c '1'0 0- ) Ifg:IR"—>IR”'isdifferentiable, then gocisacurve inIR”'. The tangent The Yizngent Bundle 67 vector ofg0catXis (g°C)=k(lt) 2g=|=(c=r(lt)) =g,,,(tangent vector ofcatr). ‘U c(t) L9 8(¢l[f)) llg*(U) Consider nowann—dimensional manifold Mandanimbedding 1':M—>IR”. Suppose wetake acoordinate system (x,U)around p.Then 1'ox“l isamap from IR"toIRNwith rank n.Consequently, (iox“‘),,.(IR",,(,,)) isann—dimen— sional subspace ofIRN;(,,,. This subspace doesn’t depend onthecoordinate RN IR” .‘K; x(P) system x,forifyisanother coordinate system, then (foy_'),, -.=(iox_l ox0y_l),,, ‘T(I.°x_])* 0(XOy_1)=l= and (“T°J’_')*.vu>)¢ Rnytpi —>Rnxtm isanisomorphism (with inverse (yOX_1)*.»<( P)}. 68 Chapter 3 There isanother way toseethis, which justiﬁes thepicture wehave drawn. Ifc:(-—e,e) —>IR"isacurve with c(O) =x(p), then or=iox“! ocisa curve inIRNwhich liesinz'(M), andevery differentiable curve int'(M) isofthis Ci(0)x(p) x(p) k,’f/»- I. form (Proof P).Now a*(l0) 2(1.°x_l)* °c*(l0)= sothetangent vector ofevery ozisin(iox“'),,.(IR”,,(,,)). Moreover, every vector inthissubspace isthetangent vector ofsome oz,since every vector inIR",,(,,, is thetangent vector ofsome curve c.Thus, ourn—dimensional subspace isjust thesetofalltangent vectors ati'(p)todifferentiable curves ini(M). Wewill denote thisn-dimensional subspace by(M,1‘),,. Wenowwant tolookatthe(disjoint) union T(M,i')= U(M,r),ctow)XRNcTIRN. p€M Wecandeﬁne a“projection” map rt:T(M,i) —>M by rt'(v) =pifvE(M,t')P. Asinthecase ofTIR", each “ﬁbre” rr_'( p)hasavector space structure also. Beyond thiswehave tolook alittle more carefully atsome speciﬁc examples. Consider ﬁrst themanifold M=S1and theinclusion i:S‘—>IR2. The curve c(9)=(cos9,sin 9)passes through every point ofS‘,and c’(9) --=(-—sin9,cos9)750. Foreach p=(cos9,sin 9)eS‘,letup.-.-=(-sin9,cos 9),,(itclearly doesn’t matter which oftheinﬁnitely many possible 9’swechoose). Then (Sl,1),,con- TheYitrzgerzl Bundle 69 sistsofallmultiples ofthevector up.Wecantherefore deﬁne ahomeomorphism up up up ST 7 P “P fl;T(Sl,t') —>SI><IR1byf,(}tu,,) -=(p,§l), which makes thefollowing dia- gram commute. r(.s',t) .-iii‘-.»->5‘ ><n‘ X % [rt’(a,b)=a] 51 Ifwedeﬁne the“ﬁbres” ofrt’tobethesetsrt’_] (p),then each ﬁbre hasavector space structure inanatural way. Commutativity ofthediagram means that fl takes ﬁbres into ﬁbres; clearly flrestricted toaﬁbre isalinear isomorphism onto theimage. Now consider themanifold M=S2and theinclusion it5'2CIR3. Inthis case there isnomap f2:T(S2,t') —>S2><IR2with theproperties ofthemap f1. Ifthere were, then, foraﬁxed vector vgé0inIR2,thesetofvectors {f2_](vp) IPE52} would beacollection ofnon-zero tangent vectors, oneateach point ofS2, which varied continuously. Itisawell-known (hard) theorem oftopology that thisisimpossible (you can’t comb thehair onasphere). /-5‘\§§\.--»\_\°&, 70 Chapter 3 There isanother example where wecanprove that noappropriate homeo- morphism T(M, 1')—>MXIR2exists, without appealing toahard theorem of topology. Themap 1'willjustbetheinclusion M—>IR3where MisaMobius strip, tobeprecise, theparticular subset ofIR3deﬁned inChapter l—M isthe image ofthemap f:[O,2rt] X(-1, 1)—>IR3deﬁned by f(9,t) =(2cos9 +Icos%cos9, 2sin9 +tcosgsin9, rsing). Ateach point p--=(2cos9,2sin9,0) ofM,thevector l"oi --—o+ A 0 0, (9,)4 I S"T 2rt 2, Y. * v,=(-2sin9,2cos9,0),, =f.((1,0)(,_0,) isatangent vector. The same istrueforallmultiples off,.((O, l)(9,0)), shown asdashed arrows inthepicture. Notice that f#((OI 1)(0,0)) =lDf(0»0)(O» 1)l(2,o,o) 3 =l:l(0=0)] =(1=0=O)t2,0,0), 3! (2,o.0) while ft=((0,1)(2=t,0)) =|:E;—{(2F»0)jI =(-1»0»0)(2,0,0)- (2,o,0) This means thatwecannever picknon-zero dashed vectors conlinuousbw onthe setofallpoints (2cos9,2sin9,0); Ifwecould, then each vector would be /.((0.M@>><t.t>) forsome continuous function It:[0,2rt] —>IR.This function would have tobe non-zero everywhere andalsosatisfy l(2rt) ==—~Jt(O), which itcan’t (byaneasy theorem oftopology). The impossibility ofchoosing non-zero dashed vectors continuously clearly shows thatthere isnowaytomap T(M, t‘),ﬁbre byﬁbre, TheYitrzgent Bundle 71 homeomorphically onto MXIR2. Wethus have another case where T(M ,1‘) does not“look like” aproduct MXIR". Foranyimbedding 1':M—>IRN, however, thestmcture ofT(M, 1')isalways simple locally: if(x,U)isacoordinate system onM,then rt_l(U), thepart ofT(M ,1’)over U,canalways bemapped, ﬁbre byﬁbre, homeomorphically onto UXIR".Infact, foreach p6U,theﬁbre (M,f)p Cql.1E1.lS OX_l)*x(p) (Rnx(p)) Zmp (lRnx(p)), where theabbreviation mphasbeen introduced temporarily; wecantherefore deﬁne f;tt-‘((1)-> U><rt" by f(mp(vX(p))) =(P-11)- Instandard jargon, T(M, i)is“locally trivial”. This additional feature qualiﬁes T(M, t‘)tobeincluded among anextremely important class ofstructures: AnIt-dimensional vector bundle (orn-plane bundle) isaﬁve-tuple Em(E,rt, B,®,G)), where (l)Eand Barespaces (the“total space” and “base space” ofE, respectively), (2)rt:E—>Bisacontinuous map ontoB, (3)®and (Daremaps EB:(Jrt"'(p)Xrt_'(p)—>E, G):IRXE—> E, p€B with ®(rt_'(p) Xrt'l(p)) Crt"(p) and ®(IR Xrt"(p)) C rt_'(p), which make each ﬁbre rt_'(p)intoann—dimensional vector space over IR, such thatthefollowing “local triviality” condition issatisﬁed: Foreach pEB,there isaneighborhood Uofpandahomeomor- phism I:rt"(U) ~—>UXIR“which isavector space isomorphism from each rt_'(q) onto qXIR",forallqEU. 72 Chapter 3 Because thislocal triviality condition really isalocal condition, each bundle 5=(E,rt, B,®,®) automatically gives risetoabundle EIA over anysubset ACB;tobeprecise, 5|/1=(J1'_l(/1), ttltt-'(/1), A,@|Um,71'_l(p) ><7l'_l(p), ®|11t><7l'_](/l)). Notation ascumbersome asallthisinvites abuse, andweshall usually refer simply toabundle rt:E—>B,oreven denote thebundle byEalone. For vectors v,w Ert_'(p)andaEIR,wewilldenote @(v, w)and(E)(a, v)byv+w, anda-vorav,respectively. I Thesimplest example ofann-plane bundle isjustXXIR"with rt:XXIR"—> Xtheprojection ontheﬁrstfactor, andtheobvious vector space structure on each ﬁbre. This iscalled thetrivial n—plane bundle over Xandwillbedenoted bye"(X). The “tangent bundle” TIR” isjust e"(IR"). The bundle T(S',:') considered before isequivalent toe'(S1). Equivalence ishere atechnical term: Two vector bundles E1=rt1:E1—>BandE2= rtg:E2—>Bareequivalent (E12E2)ifthere isahomeomorphism h:E1—>E2 which takes each ﬁbre rt1“'( p)isomorphically onto rt;f'( p).The map his called anequivalence. Abundle equivalent toe"(B) iscalled trivial. (The local triviality condition forabundle Ejustsaysthat§lUistrivial forsome neighborhood Uofp.) The bundles T(S2,t‘)andT(M ,1')arenottrivial, butthere isaneven simpler example ofanon-trivial bundle. The Mobius strip itself (not T(M,i)) can be 1-nnsidered asa1-dimensional vector bundle over S1,forMcanbeobtained from [0,l]XIRbyidentifying (0,a) with (l,—a), while S1canbeobtained from Oi“ ..,,inN 1 Y % __/__ /B /,5{0,l TheYlmgenl Bundle 73 [0,1] byidentifying 0with I;themap rtisdeﬁned byrt(t‘,a) =Ifor0<1I<:1 and rt({(O,a),(l,—a)}) ={O,l}.The diagram above illustrates local triviality near thepoint {O,I}ofS1.Suppose thats:S1—>Misacontinuous function with rtos=identity ofM(such afunction iscalled asection). Such amap M I) ecorresponds toacontinuous function Ii:[0,l]—>IRwith 5(0)=—.'§(l). Since E must be0somewhere, thesection smust be0somewhere (thatis,8(9)Err“'(9) must betheOvector forsome 9ES1).This surely shows that Misnotatrivial bundle. Anequivalence isobviously theanalogue ofanisomorphism. The analogue ofahomomorphism isthefollowing.* Abundle map from E1toE2isapairof continuous maps (_)F,f), with f2:E1—>E2and f:B1—>B2,such that (l)thefollowing diagram commutes mlItB,in B2, (2))5:rt1_1(p) —>rtfl (f(p)) isalinear map. The pair (ft,f)isabundle map from TIRI‘ toTIR' foranydifferentiable flIR,‘—>IR’.IfMi’ CIRI‘and Ni” CIR’aresubmanifolds, 1':M—>IR,‘and j:N—>IR’aretheinclusions, andthemap fsatisﬁes f(M) CN,then f, *There areactually several possible choices, depending onwhether oneisconsider- ingallbundles atonce, ﬁxed bundles over various spaces, oraﬁxed base space with varying bundles. Thus fmay berestricted tobeanisomorphism onﬁbres andfto betheidentity orahomeomorphism. The relations between some ofthese cases are considered intheproblems. 74 Chdpler 3 takes T(M,i) toT(N,j); toseethis,just remember that vET(M,z'),, isthe tangent vector ofacurve cinM,soﬂ.(v) isthetangent vector ofthecurve f0cinN,andconsequently f,.(v) ET(N,j). Inthiswayweobtain abundle map from T(M,z') toT(N,j). Actually, itwould have sufﬁced tobegin with aC°°function f:M—>N,since fcanbeextended toIRI‘locally Infact, thisconstruction could begeneralized much further, tothecase where 1'andj aremerely imbeddings oftwoabstract manifolds MandN,andf:M—>N isC°°; wejustconsider thefunction jofo1'“!:i(M) —>i(N) andextend it locally toIRA‘.The case which wewant toexamine most carefully isthesimplest: where M=Nand fistheidentity, while 1'andjaretwoimbeddings ofM inIRA‘andIR’,respectively. Elements ofT(M, 1'),areoftheform (iox_')...(w) \/1, .. forwElR"x(,,), while elements ofT(M, j),,areoftheform (jox“1).,.(w) for wElR",,(,,). Ifwe map tiox-‘).(w) t-><1<>x“)..<w) weobtain abundle map from T(M,t')|U toT(M,j)|U, which isobviously an equivalence. The map (M,z'),,—>(M,j),,induced onﬁbres isindependent of thecoordinate system x,forif(y,V)isanother coordinate system, then tiOy"')...<w) =(iOx*')*(<x Oy*‘)..<w)) <1Oy—])*(w) =<1ox-‘).(<x Qy"‘).<w))- Wecantherefore putallthese maps together, and obtain anequivalence from T(M,z') toT(M,j). Inother words, thedependence ofT(M,i) on1'isal- most illusory; wecould abbreviate T(M, i)toTM,ifweagreed thatTMreally denotes anequivalence class ofbundles, rather than onebundle. That isthe TheTitrzgenl Bundle 75 sortofthing analgebraist might do,anditisundoubtedly ugly. What wewould liketodoistogetasingle bundle foreach M,insome natural way, which has alltheproperties anyoneofthese particular bundles T(M, 1')has.Can wedo this? Yes, wecan. When wedo,TIR" willbedifferent from ourolddeﬁni- tion (namely, e”(lR”)), and sowill ftforfIIR”—>lR'", soinstating ourresult precisely wewillwrite “old ft.”when necessary. l.THEOREM. Itispossible toassign toeach n-manifold Mann-plane bun- dleTMover M,andtoeach C°°map f:M—>Nabundle map (f,,f), such that: (1)If1:M—>Mistheidentity, then 1...:TM—>TMistheidentity. If g:N—> P,then (gof),..==g*<>f*. (2)There areequivalences I":TIR" —>e"(lR") such thatforevery C°°func- tion f:IR“—>IR“thefollowing commutes. TIR"_w-_->f* TlR"" I'll jtm ' EDGE?!) EMURM) (3)IfUCMisanopen submanifold, then TUisequivalent to(TM)|U, andforf:M—>Nthemap (f|U),,: TU—>TNisjust therestriction ofﬁnMore precisely, there isanequivalence TU2(TM )|Usuch that thefollowing diagrams commute, where 1':U—>Mistheinclusion.* TU 5* >TM TUM-»--->(f'U)* TN (TM)|U TM PROOF. The construction ofTM isaningenious, though quite natural, sub- terfuge. Wewillobtain asingle bundle forTM,buttheelements ofTMwilleach belarge equivalence classes. *When using thenotation ft,itmust beunderstood thatthesymbol “f”really refers to atriple (f,M,N)where f:M—>N.The identity map IofUtoitself andtheinclusion map 1':U—>Mhave tobeconsidered asdifferent, since themaps 1,:TU—>TUand 1',..:TU—>TMarecertainly different (they map TUintotwodifferent sets). 76 Chapter 3 The construction ismuch easier tounderstand ifweﬁrstimagine that weul- ready hadourbundles TM. Then if(x,U)isacoordinate system, wewould have amap xi:TU —>T(x(U)), and thiswould beanequivalence (with inverse (x“'),,.). Since TUshould beessentially (TM )|U, and T(x(U)) should essentially bex(U) XIR",apoint e6rt“‘( p)would betaken byxi.tosome (x(p),v).Here visjust anelement ofIR"(and every vwould occur, since x... maps rt‘!(p)isomorphically onto {p}XIR"). Ifyisanother coordinate system, then y,,.(e) would be(y(p),w)forsome wEIR".Wecaneasily ﬁgure outwhat therelationship between vandwwould be;since (x(p),v)istaken to(y(p), w) byy,,0x,,."l =(y0x_l),,, and (yox"l)* issupposed tobetheold(y0x_l),.., wewould have (E1) w=D(yox_l)(x(p))(v). This condition makes perfect sense without anymention ofbundles. Itisthe clue which enables ustonow deﬁne TM. Ifxandyarecoordinate systems whose domains contain p,andv,wEIR", wedeﬁne (x,v)'5'(y,w) if(a)issatisﬁed. Itiseasytocheck (using thechain rule) thatt;isanequivalence relation; the equivalence class of(x,v)willbedenoted by[x,v],,.These equivalence classes willbecalled tangent vectors atp,andTMisdeﬁned tobethesetofalltangent vectors atallpoints pEM;themap rttakes *5»equivalence classes top.We deﬁne avector space structure onrt"!(p)bytheformulas [x,11],,+[x,w]p=[x,v+w],, a-[x,v]P '-=[x,a -v],,; thisdeﬁnition isindependent oftheparticular coordinate system xory,because D(y ox_1)(x(p)) isanisomorphism from IR“toIR”. Our deﬁnition ofTM provides aone-one onto map (b) Ix:rt_1(U) —>UXIR”, namely [x,11],,|—->(q,v). Wewant thistobeahomeomorphism, sowewant If‘(A)tobeopen forevery open ACUXIR",andthuswewant anyunion ofsuch setstobeopen. There isametric with exactly these setsasopen sets, butitisalittle ticklish toproduce, soweleave thisonepart oftheproof toProblem l. Wenow have abundle rt:TM —>M.Wewilldenote theﬁbre rt'1( p) byM,,,inconformity with thenotation lR"p, though TM, might bebetter. If TheItngerzl Bundle 77 f:M—>N,and(x,U)and(y,V)arecoordinate systems around pandf(p), respectively, wedeﬁne (C) ft([X, vlp)=ly.D(y°fOX-”)(X(P))(v)lf<,,)- Ofcourse, itmust bechecked that thisdeﬁnition isindependent ofxand y (thechain rule again). Condition (1)ofourtheorem isobvious. Toprove (2),wedeﬁne I"tobet1,where Iistheidentity map ofIR”andtx isdeﬁned in(b);itistrivial, though perhaps confusing tothenovice, toprove commutativity ofthediagram. Condition (3)ispractically obvious also. Infact, theﬁbre ofTUover pEU isalmost exactly thesame astheﬁbre ofTMoverp;theonlydifference isthat each equivalence class forMcontains some extra members, since inMthere aremore coordinate systems around pthan there areinUCM.*1‘ Henceforth, thebundle rt:TM—>Mwillbecalled thetangent bundle ofM. lf2':M—>IRI‘isanimbedding, then TMisequivalent toT(M,z'). Infact, if (x,U)isacoordinate system around p,and Iistheidentity coordinate system ofIRA’,then r1..([r,v],,) =[I.D(1'<= x“)o=<p))<v)1.-<,,> by(C) t"=t1 I (ftp).Dtf<>x"‘)(x(p))(v)) E(M.1'),,; thecomposition t“z',.iseasily seen tobeanequivalence. ButT(M, t’)willplay nofurther roleinthisstory—the abstract substitute TMwillalways beused instead. Having succeeded inproducing abundle over each M,which isequivalent to T(M, 1'),wenext askhow fortuitous thiswas. Can oneﬁnd other bundles with thesame properties? The answer isyes,andweproceed todeﬁne twodifferent such bundles. Fortheﬁrst example, weconsider curves c:(-—e,t:) —>M,each deﬁned onsome interval around O,with c(O) =p.If(x,U)isacoordinate system around p,wedeﬁne xocland xoCg,mapping IRtoIR”,c’-‘~10 ifandonl if: __ I1’2 Y have thesame derivative atO. Theequivalence classes, forallpEM,willbetheelements ofournewbun- dle,T’M. Forf:M—>Nthere isamap fntaking the§equivalence class 78 Chapéer3 ofctotheI-'3)equivalence class offoc.Without bothering tocheck details, wecanalready seethat thisexample is“really thesame” asTM— th'»-*~¢ 'al l f"1[x,v]p corresponds to: 6Pcqinv cnce (?aSSnO C1]/’where yisacurve 1nRwith 3/(O)=v; under thiscorrespondence, fgcorresponds tof,.. Inthesecond example, things arenotsosimple. Wedeﬁne atangent vector atptobealinear operator ifwhich operates onallC°°functions fandwhich isa“derivation atp”: ﬂfg)=f(p)f(s) +g(p)f(f)- Wehave already seen that theoperators E=3/Bx’. IFhave thisproperty. For these operators, clearly €(f)=€(g) iff=ginaneighborhood ofp.This condition isactually tmeforanyderivation E.For,suppose thatf=0ina neighborhood ofp.There isaC°°function h:M—>IRwith h(p)==1and support hCf"'(0).Then 0:3(0) =€(fh) =f(0)€(h)+h(0)€(f) =0+€(f). Thus, iff=ginaneighborhood of0,then 0=€(f* g)e--e€(f)* €(g). Iff isdeﬁned only inaneighborhood ofp,wemay usethistrick todeﬁne €(f); choose htobe1onaneighborhood ofp,with support hCf"1(0),anddeﬁne €(f) as€(fh). The setofallsuch operators isavector space, butitisnotapriori clear what itsdimension is.This comes outofthefollowing. 2.LEMMA. LetfbeaC°°function inaconvex open neighborhood UofO inIR",with f(0) =0.Then there areC°°functions g,-:U—>Rwith () f(xla"-ax”)=Z:?=]x|'gi(xla-"axn) f0rx€Ua (9)3;-(0)=D=-f(0)- (The second condition actually follows from theﬁrst.) PROOF. ForxEU,leth,¢(t) *-=f(tx); thisisdeﬁned for05t51,since Uis convex. Then|—-| 1 1H _ f<x)=r<x)~r<0)= Lh;<:)d:-= LZ12.-/(ix)-xvii.:'=-I Therefore wecanletg(x) =fgD,-f(tx) dz‘.*1‘ The Istrzgenl Bundle 79 3.THEOREM. The setofalllinear derivations atpEM"isann—dimen— sional vector space. Infact, if(x,U)isacoordinate system around p,then 3 3 P span thisvector space, andanyderivation Ecanbewritten " -8£2 Li.g;ax)af , (soEisdetermined bythenumbers €(x')). PROOF. Notice that 8(1): £(1-1);-.1-£(1)+1-8(1), so8(1) =O.Hence €(c) =c-6(1) =Oforanyconstant function conU. Consider thecase where M=R"and p=0.Assume Uisconvex. Given f onU,choose g;asinLemma 2,forthefunction f-—-f(O).Then ch M)=W-—/<0»=~@(ZPsi)£§0§'§£.Zii“§u“S.E0..>i=1 lP1==wuUaw»+FwM@m n _ =£(I’)—,.(0) +0.8, This shows that8/8IiI0span thevector space; they areclearly linearly inde- pendent. Itisasimple exercise tousethecoordinate system xtotransfer this result from IR”toM.*1‘ From Theorem 3wecanseethat, once again, abundle constructed from all derivations atallpoints ofMis“really thesame” asTM. Wecanlet H -8E=2a’F correspond to[x,a]p; i=1 P theformula a "af a ai-='T2ai¢=‘(p)ai’P j=l y P 80 Chapter 3 derived inChapter 2,shows that H,-8 H,-8 , .1- R,-8yjE0 ll-‘I-1I1ClOnlYlf br-‘Z0 1:] P i=1 P i=1 andthisisprecisely theequation which says that (x,a) '5'(y,b). Itiseasily checked that under thiscorrespondence, themap which corresponds toftcan bedeﬁned asfollows: [f.=(5)l(g) =~‘3(3Of)- H Notice thatifxdenotes theidentity coordinate system onIR",then Zal3%; corresponds toapwhen weidentify TR" with s"(]R"). 5:1 Wewillusually make nodistinction whatsoever between atangent vector v6MPand thelinear derivation itcorresponds to,that is,between [.>:,a]p and 3,1 .i=1 8x‘ P’ consequently, wewillnothesitate towrite v(f) foradifferentiable function f deﬁned inaneighborhood ofp.Infact, atangent vector isoften most easily described bytelling what derivation itcorresponds to,andthemap f,isoften most easily analyzed from therelation (f1=v)(g) =vlg°f)- ItisCustomary todenote theidentity coordinate system onIR‘byr,andto write d 1. 8-u— I‘ -—- dr,0°8: thisisabasis forRm. Ifc:IR—>Misadifferentiable curve, then d-— MC*(dl :0) E C00) iscalled thetangent vector tocat:0.Wewilldenote itbythesuggestive symbol dc dz‘,0 T/ze Yiuzgenl Bundle 31 This symbol willbesubjected tothestandard abuses oneﬁnds (unexplained) in calculus textbooks: thesymbol d-6-{E will often stand for -E , dt dt, thesubscript “I”now denoting aparticular number tEIR,aswellastheidentity coordinate system. Asyoumight wellexpect, itisnoaccident thatoursecond andthird examples turned outtobe“really thesame” asTM. There isageneral theorem that all “reasonable” examples willhave thisproperty, butitisalittle delicate tostate, andquite amess toprove, soithasbeen quarantined inanAddendum tothis chapter. The tangent bundle TM ofaC°°manifold hasalittle more structure than anarbitrary n—plane bundle. Since TM locally looks likeU><R“,clearly TM isitself amanifold; there is,moreover, anatural waytoputaC°°structure onTM. Ifx:U—>IR"isachart onM,then every element uE(TM)|U is uniquely oftheform H -8v"-12:0’? , p=rr(v). [=1 xP Letusdenote albyjCi(U). Then themap v+~><x‘<n<v)>.....x"<r<v)).>1=‘<v).....x"<v)) eR2" isahomeomorphism from (TM)|U tox(U) ><R“. This map, (xorr,J'c), is simply themap x,,.when weidentify TUwith U><IR"inthestandard way. If (y,V)isanother coordinate system, and H . 8 ‘U= W J=1 then, aswehave already seen, ll Fl. .3};-l . . __ 6"=2¢1'ﬁ(p)= Z60.-ty’ Ox‘)(x(P))-:'=l i=1 Thisshowsthatif(t,a).-=(H,...,:",a',.. .,a")eR2",then yinO(x*)-I ((#3) =(y<>x"‘(r). XL,a"Di(y‘ @X"‘)(r). Z121a"Di-ty“ <>x"‘)(r))- Thisexpression shows thaty,,.<>(x,,.)"' isC°°. 82 Chapter 3 ‘Wethus have acollection ofC°°-related charts onTM, which canbeex- tended toamaximal atlas. With thisC°° structure, thelocal trivializations x...areC°°. Ingeneral, a vector bundle rt:E—>Biscalled aC°°vector bundle ifEand BareC°° manifolds and there areC°° local trivializations inaneighborhood ofeach point. Itfollows that rt:E—>BisC°°. Recall thatasection ofabundle 21':E—>Bisacontinuous function s:B—>E such thatrtas=identity ofB;forC°°vector bundles wecanalsospeak ofC°° sections. Asection ofTM iscalled avector ﬁeld onM;forsubmanifolds M ofIR",avector ﬁeld may bepictured asacontinuous selection ofarrows tangent toM.The theorem that youcan’t comb thehair onasphere just states that ‘pu- I.é-".-N there isnovector ﬁeld onS2which iseverywhere non-zero. Wehave shown that there donotexist twovector ﬁelds ontheMobius strip which areeverywhere linearly independent. Vector ﬁelds arecustomarily denoted bysymbols likeX,Y,orZ,and the vector X(p)isoften denoted byX_,,(sometimes Xmay beused todenote a single vector, insome Mp). Ifwethink ofTMasthesetofderivations, then for anycoordinate system (x,U),wehave H - 8X(p)=Z:a’(p)-—,— forallpeU.i=1 8xP Thefunctions a‘.arecontinuous orC°°ifandonlyifX:U—>TMiscontin- uous orC°°. IfXandYaretwovector ﬁelds, wedeﬁne anew vector ﬁeld X+Yby (X+Y)(p)=—-Xtp)+Y(p)- Similarly, iff:M—>IR,wedefine thevector ﬁeld fXby (fX)(p) ==f(PlX(Pl- TheYiuzgent Bundle 83 Clearly X+Yand fX areC°° ifX,Y,and fareC°°. OnUwecanWrite n ‘a _ XZE 611%, i=1 8x thesymbol 8/8x’_ now denoting thevector ﬁeld H 8 P 8x’- Iff:M—>E1isaC°°function, and Xisavector ﬁeld, then wecandeﬁne anewfunction X(f): M—>IRbyletting Xoperate onfateach point: it/nip) =X,,</1. Itisnothard toCheck that ifXisaC°° vector ﬁeld, then X(f) isC'°° for every C°°function f;indeed, iflocally n _ 8 X(P)=Za‘(P)§i=1 then n _ -8f X = '—.ax: 1 which isasum ofproducts ofC°°functions. Conversely, ifX(f) isC°° for evegi C°°function f,then XisaC°°vector ﬁeld (since X(xl) =al). Let37denote thesetofallC°°functions onM.Wehave justseen thataC°° vector ﬁeld Xgives risetoafunction X:F—>.77.Clearly, Yo"+2)=it/)+1?(g) X(fe) fX(e) +eX(f); thus Xisa“derivation” ofthering 3'7.Often, aC°°vector ﬁeld Xisidentiﬁed with thederivation X.The reason forthisisthat ifA:3'7—>J‘?isanyderiva- tion, then A==Xforaunique C°° vector ﬁeld X.Infact, weclearly must deﬁne Xp(f) ==A(fl(P)= andtheoperator XPthusdeﬁned isaderivation atp. 84 Chapter 3 The tangent bundle isthetrue beginning ofthestudy ofdifferentiable mani- folds, andyoushould notread further until yougrok it.*The next fewchapters constitute adetailed study ofthisbundle. One basic theme inallthese chap- tersisthat anystructure onecanputonavector space leads toastructure on anyvector bundle, inparticular onthetangent bundle ofamanifold. Forthe present, wewilldiscuss justonenew concept about manifolds, which arises in thisvery wayfrom thenotion of“orientation” inavector space. The non—singular linear maps f:V—>Vfrom aﬁnite dimensional vector space toitself fallintotwogroups, those with detf>0,andthose with detf<O; linear transformations intheﬁrstgroup arecalled orientation preserving and theothers arecalled orientation reversing. Asimple example ofthelatter is themap f:IR"—>IR"deﬁned byf(x) =(x1,. ..,x"_1, -—x") (reﬂection inthe hyperplane x"--=0).There isnoway topass continuously between these two groups: ifweidentify linear maps R"—>R"with n><nmatrices, and thus with lR”2, then theorientation preserving and orientation reversing maps are disjoint open subsets ofthesetofallnon-singular maps (those with detqéO). The terminology “orientation preserving” isabitstrange, since wehave notyet deﬁned anything called “orientation”, which isbeing preserved. The problem becomes more acute ifwewant todeﬁne orientation preserving isomorphisms between tw0different (butiS0m0Tphic) vector spaces VandW;thisclearly makes nosense unless wesupply Vand Wwith more structure. Toprovide thisextra structure, wenote that twoordered bases (v1,. ..,vn) and(v"1,...,v',,)forVdetermine anisomorphism f:V—>Vwith f(v,-) =v’,-; thematrix A=(a,-j) offisgiven bytheequations I’! r 2:1);: ajfllj. i=1 Wecall(111,... ,v,,)and(v"1,...,v",,) equally oriented ifdetA>0(i.e.,iffis orientation preserving) andoppositely oriented ifdetA<O. The relation ofbeing equally oriented isclearly anequivalence relation, divid- ingthecollection ofallordered bases intojusttwoequivalence classes. Either of these twoequivalence classes iscalled anorientation forV.The class towhich (v1,...,vn)belongs willbedenoted by[vi,...,vn],sothatif,uisanorientation ofV,then (v;,...,v,,) Ep.ifand only if[v;,...,v,,] --=,u..Ifp.denotes one *Acultword ofthesixties, “grok” wascoined, purportedly asaword from theMartian language, byRobert A.Heinlein inhispop science ﬁction novel Stranger inaStrange Land. ltssense isnicely conveyed bythedeﬁnition inTheAmerican Heritage Dictionary: “Tounderstand profoundly through intuition orempathy”. T/ze Yhngent Bundle 85 I—~—1) >8 ) 0 v; wl w3 l IU1 U3,‘ 1 ‘' U2A \ iwiU21 w, ,,,""* planeof w;andwg 111 Examples ofequally oriented ordered bases inR,R2,andR3. orientation ofV,theother willbedenoted by--u,andtheorientation [e1,...,en] forR”willbecalled the“standard orientation”. Now if(V,,u,)and(W,v)aretwon-dimensional vector spaces, together with orientations, anisomorphism f:V—>Wiscalled orientation preserving (with respect topaand U)if[f(v;), ...,f(v,,)] =vwhenever [v1,...,v,,]=,u,;ifthis holds foranyone(v;,...,vn),itclearly holds forall. Forthetrivial bundle e”(X) ==X><R"wecanputthe“standard orientation" [(x,e;), ...,(x,e,,)] oneach ﬁbre {x}xR".Iff:e"(X) —>e"(X) isanequiva- lence, andXisconnected, then fiseither orientation preserving ororientation reversing oneach ﬁbre, forifwedeﬁne thefunctions a,-,;:X—>Rby II f(x,@r-J =Zap-(X) -(>~'.@;), i=1 then det(a,-j): X—>Riscontinuous and never O.Ifrt:E—>Bisanon- trivial rz-plane bundle, anorientation trofEisdeﬁned tobeacollection of orientations upforIF‘(p)which satisfy thefollowing “compatibility condition” foranyopen connected setUCB: lft:rt"!(U)—>U><R" isanequivalence, andtheﬁbres ofU><R" are given thestandard orientation, then tiseither orientation preserving ororientation reversing onallﬁbres. Notice thatifthiscondition issatisﬁed foracertain t,andt’:rr"l(U)—>UxR" isanother equivalence, then t’automatically satisﬁes thesame condition, since 86 Chapter 3 t’01-] IUxR”—>U><R"isanequivalence. This shows thattheorientations up deﬁne anorientation ofEifthecompatibility condition holds foracollection ofsetsUwhich cover B. Ifabundle Ehasorientation ,u:.:{pp}, ithasanother orientation -/.t= {-p,,,}, butnotevery bundle hasanorientation. Forexample, theMobius strip, considered asa1-dimensional bundle over SI,hasnoorientation. For, although theMobius strip hasnonon-zero section, wecanpick twovectors from each ﬁbre sothatthetotality Alooks liketwosections. Forexample, wecanletAbe[0,1]><{-1,1} with (0,a) identiﬁed with (1,-a); then Ajust looks liketheboundary oftheMobius strip obtained from [0,1]><[-1, 1].If "\Owehad compatible orientations up,wecould deﬁne asection stS‘->Mby choosing s(p) tobetheunique vector s(p) EAr'1n"‘1 (p)with [s(p)] =up. Abundle iscalled orientable ifithasanorientation, andnon-orientable oth- erwise; anoriented bundle isjust apair (E,pt)where ptisanorientation forE. This deﬁnition canbeapplied, inparticular, tothetangent bundle TM ofa C°°manifold M.Inthiscase, wecallMitself orientable ornon-orientable de- pending onwhether TMisorientable ornon-orientable; anorientation ofTM isalsocalled anorientation ofM,and anoriented manifold isapair (M,/st) where ptisanorientation forTM. The manifold R"isorientable, since TR” 2s"(R"), onwhich wehave the standard orientation. The sphere S"*1CR"isalsoorientable. Toseethiswe P.v=w TheItngent Bundle 87 note that foreach pES”"1 thevector w-=p,,Ee"(R") 2TR” isnotin t',,,(S""1,,) ETR“,, (Problem 2l), soforv1,...,v,,..; ES""l,, wecan deﬁne (v1,...,v,,..;) E,u,,ifand only if(w,t',,,(v,),...,i,,(v,,_,)) isinthestandard orientation ofR",,. Theorientation p,m{;.t,,:pES""'} thusdeﬁned iscalled the“standard orientation” ofS“"1. The torus S1XSIisanother example ofanorientable manifold. This can beseen bynoting thatforanytwomanifolds M;andMgtheﬁbre (M1 xM2),, ofT(M, xMg) canbewritten asV1,,®V1,,where (JT,'),,,,I Mp—>(M,-),, isan isomorphism andthesubspaces V,-,,vary continuously (Problem 26). Since TS‘ istrivial, thisshows that T(SI><S1)isalsotrivial, andconsequently orientable. Any n—holed torus isalso orientable—the proof ispresented inProblem l6, which alsodiscusses thetangent bundle ofamanifold-with-boundary. The Mobius strip Misthesimplest example ofanon-orientable 2—manifold. Fortheimbedding ofMconsidered previously wehave already seen thatonthe 1.__ .. ----—- ~ i +o(9,0)3, r2:1" ,5, v...1 _ subset S={(2cos9,2sin 9,0)} CMthere arecontinuously varying vectors v,,, butthatitisimpossible tochoose continuously from among thedashed vectors w,,=f.,.((0,1)(9,0)) andtheir negatives. Ifwehadorientations ,u,,forpES, then wecould simply choose w,,if[v,,,w,,]-=;.t,,and-w,, otherwise. The projective plane 1P2must benon-orientable also, since itcontains the Mobius strip (foranyorientable bundle E=rt:E—>B,therestriction EIB’ toanysubset B’CBisalsoorientable). Non—orientability of]P’2canbeseen inanother way, byconsidering the“antipodal map” A:S2—>S2deﬁned by A(p) =--p. This map isjust therestriction ofalinear map A-:R3—>R3 deﬁned bythesame formula. The map A,.: S2,, —>S2A(,,) isjust (p,v) |-> (,:f(p), A-(v)), when S2,,isidentiﬁed with asubspace of{p}><R3.The map A isorientation reversing, soif11,-=(p,u,-)ES2,,,thebases ($211222: and areoppositely oriented. This shows thatifptisthestandard orientation ofS2 and [‘Ug,Ug] 6;.t,,,then [A,,.v1,A,.vg] 6-;.tA(,,). Thus themap A:S2—>S2is 88 Chapter 3 “orientation reversing” (thenotion ofanorientation preserving ororientation reversing map f1M—>Nmakes sense foranyimbedding fofone oriented manifold intoanother oriented manifold ofthesame dimension). From thisfact itfollows easily that 1P2isnotorientable: If1P2had anorientation v={um} and g:S2—>1P2isthemap p|—->[p],then wecould deﬁne anorientation {ﬂap} onS"byrequiring gtobeorientation preserving; themap Awould then beorientation preserving with respect to/1,which isimpossible, since ll=pa or-—;.t. Forprojective 3—space 1P3thesituation isjust theopposite. Inthiscase, the antipodal map A:S3—>S3isorientation preserving. Ifg:S3—>1P3isthe map p|—->[p],weobviously candeﬁne orientations 11,,for1P3byrequiring g tobeorientation preserving. Ingeneral, these same arguments show that1P“is orientable fornoddandnon—orientable forneven. There isamore “elementary” deﬁnition oforientability, which does notuse thetangent bundle ofMatall.According tothisdeﬁnition, Misorientable if there isasubset A’oftheatlas AforMsuch that thedomains ofall(x,U)EA’cover M, forall(x,U) and (y,V)EA’,/—\,_i\J|—-I‘-/‘--._.-I I. 8det(l.)>0 on UOV.8x1 Anorientation itofTMallows ustodistinguish thesubset A’asthecollection ofall(x,U)forwhich x,,,:TMIU —>T(x(U)) 2x(U) xIR"isorientation preserving (when x(U) ><R"isgiven thestandard orientation). Condition (2) holds, because itisjust thecondition that (yox"‘),,.: T(x(U)) —>T(x(U)) isorientation preserving. Conversely, given A’wecan orient theﬁbres of TM|Uinsuchawaythatx,isorientation preserving, andobtain anorientation ofTM. Although ouroriginal deﬁnition iseasier topicture geometrically, the determinant condition willbevery important later on. TheYisrzgent Bundle 89 ADDENDUM EQUIVALENCE OFTANGENT BUNDLES The factthatallreasonable candidates forthetangent bundle ofMturn out tobeessentially thesame isstated precisely asfollows. 4.THEOREM*. Ifwehave abundle T'M over Mforeach M,andabundle map (f;;,f) foreach C°°map f:M—>Nsatisfying ’C5ﬁ."::)ofTheorem l, )ofTheorem 1,forcertain equivalences t"", )ofTheorem l,forcertain equivalences T’U2(T"M)|U, then there areequivalences eM: TM—>T"M such that thefollowing diagram commutes forevery C°°map f:M—>N. TMi-» TN ..,) jaT"M Q-L T'N PROOF. The details ofthisproof aresohorrible thatyoushould probably skip it(and youshould deﬁnitely quit when you getbogged down); thewelcome symbol *2»occurs quite aways on.Nevertheless, theidea behind theproof is simple enough. If(x,U) isachart onM,then both (TM)|U and (T"M)|U “look like” x(U) ><IR",sothere ought tobeamap taking theﬁbres ofoneto theﬁbres oftheother. What wehave tohope isthatourconditions onTM and T'M make them “look alike” inasuﬁiciently strong way forthisidea toreally work out. Those who have been through thissortofrigamarole before know (i.e., have faith) that it’sgoing towork out; those forwhom thissortofproof is anew experience should ﬁnd itpainful andinstructive. *Functorites willnotice thatTheorems land4saythatthere is,uptonatural equiv- alence, aunique functor from thecategory ofC'°° manifolds and C°° maps tothe category ofbundles andbundle maps which isnaturally equivalent to(£",o1d f*)on Euclidean spaces, andtotherestriction ofthefunctor onopen submanifolds. 90 Chapter 3 Let(x,U)beacoordinate system onM.Then wehave thefollowing string ofequivalences. Two ofthem, which aredenoted bythesame symbol 2,are theequivalences mentioned incondition (3).Letaxdenote thecomposition ax1--(t"|x(U))o 1:Ox*0(:)"‘. (TM)1U <__i TU-__’-"=“_> T(x(U)) _-F4» (T]l€")|x(U) .‘l|3‘ﬂ> s"(lR“)lx(U) \ (Ix _ I/V Similarly, using equivalence 1:’forT’,wecandeﬁne ﬁx. 11*’ Xi.- E’ n ’”(T’M)lU <—~—-'~'" T’U —-——> T(x(U)) —-——> (TR )lX(U) I‘x(U) 8n(lRn)|X(U) \ j132:; _7 /l Then l5x"1<><>~'xI (TM)|U —>(T'M)|U isanequivalence, soittakes theﬁbre ofTMover pisomorphically totheﬁbre ofT"M overpforeach pEU.Ourmain taskistoshow thatthisisomorphism between theﬁbres over pisindependent ofthecoordinate system (x,U).This willbedone inthree stages. (I)Suppose VCUisopen andy=X|V- Wewillneed toname alltheinclusion maps i:U—>M iii/—>M jIV—>U kty(V)—> x(U). Tocompare axandozy,consider thefollowing diagram. (TM)|U TU-'E“_> T(x(U)) i (T1Rt")|x(U) 313%. 8"(]R")|x (U) (1) jcQ) jiz.Q) ls...@ la @ la (TMNV<—-~‘3--Tv—'5"—>Tum)—3"'—><TR")1y<1/> ~‘3-mlE"<1R">|y<v> TheYitrzgenl Bundle 91 Each ofthefour squares inthisdiagram commutes. Toseethisforsquare ®, weenlarge it,asshown below. The twotriangles ontheleftcommute bycon- dition (3)forTM, andtheoneontheright commutes because i0j=F. (TM)|U ‘\: Cmqgru TM _ 1;‘K C/TV (TM)|V Square ®commutes because koy=xoj.Square (3)commutes forthe same reason assquare ®; theinclusions x(U) —>IR"and y(V) —>IR"come into play. Square @obviously commutes. Chasing through diagram (1)now shows that thefollowing commutes. (TM)|U -L» £"<R")|x<v) C C <TM)|v —-‘-"”—> e"<1R")|y<v> This means thatforpEV,theisomorphism oz},between theﬁbres over p isthesame asax. Clearly thesame istrue for,6,and ﬂy,since ourproof used only properties (1),(2),and(3),nottheexplicit construction ofTM. Thus ﬂy“! cozy =Bx"! oozxontheﬁbres over p,forevery pEV. (II)Wenow need aLemma which applies toboth TM andT"M_ Again, itwill beproved forTM (where itisactually obvious), using only properties (l),(2), and(3),sothatitisalsotrueforT"M. 92 Chqpter 3 LEMMA. IfACIR"and BCRmareopen, andf:A—>BisC°°, then the following diagram commutes. TA_-E-—>(T]R")|A ’n"’.»;"(n)"|/1 frj f,. TB.__.""-'_> (T1R’")|B ._’fi> .@"'(n)"'|B PROOF. Case1.meisamaj)f;ts"->hmwithf=fonA.Consider the following diagram, where 1':A—>IR"andj:B—>lR’"aretheinclusion maps. (TR)"|/1 ’""’-e"<R")|A % c jc TA_~5i*‘—> Th" ‘" awn") fkj is fl. TB-_’i‘_> Tn"i-~> .=;'"(n'")t Ant(T]R’")|B _’-"_"_>.i.~"'(it"')|B Everything inthisdiagram obviously commutes. This implies that thetwo compositions N n ' TAL->(Tn")|.=1 1a"(lR")|A Lma") °1dI@'"(11t’") and TAisTBi>(T1R’")|B iii .=;*"(1R’")|B L.@'"(n'") areequal and thisproves theLemma inCase l,since themaps “old ft” and “old f,”areequal onA. Case 2.Genera! case. Foreach pEA,wewant toshow that twomaps arethe same ontheﬁbre over p.Now there isamap _)F:IR"—>Rmwith _)F=fonan open setA’,where pEA’CA.Wethen have thefollowing diagram, where every 1:"comes from thefactthatsome setisanopen submanifold ofanother, TheYitrzgent Bundle 93 and2':A’—>Aistheinclusion map. H TAAF~:»~A >(Tll?.")|A l>s”(]R)”|A\\\C \ \'\ 1*) ,(TA)]A’ @ C@ (C / ‘l /c . lt.// -,-V H I (2) Q)TA’ " (TlR")iA’i>s(]l'?.")lA’ @. fli‘ ® Old OlCl ft (1 V ¢'\-I,,, . " (T1R’")lB i 8*"(1l'&"')lB Boxes G), ®, and@obviously commute, and@commutes byCase .7.To seethat square ®(which hasatriangle within it)commutes, weimbed itin alarger diagram, inwhich j:A—>IR"istheinclusion map, and other maps have alsobeen named, foreaseofreference. TR" y K<1») ATA xi-()> (TlR“)|A Kl lﬁ(H) .- ()xx ,,,TA (TR)|A Toprove thatA01',=M0K,itsuﬁices toprove that Uoloi,,,=1)o,u,oK, since uisone-one. Thus itsuﬁices toprove j,.01',=u0,u,0K,which amounts toproving commutativity ofthefollowing diagram. TR” (jofy X A..- ""-' H J‘ TA’———-—--——----+ (TR )|A Since j0iisjusttheinclusion ofA’inIR",thisdoes commute. 94 Chaplet 3 Commutativiry ofdiagram (2)shows that thecomposition ft 2 m r’"lB mmTA—->TB—~>(TlR )|B_-_>e (R)|B coincides, onthesubset (TA)|A’, with thecomposition 2 ,."Afl..,Id;m,.. TA._-_>(TR )|A_ii.._..~; (R)|A_‘i_Le~ (R)|B. andonA’wecanreplace “old )7,”by“old f,,”. Inother words, thetwocom- positions areequal inaneighborhood ofanyp6A,andarethusequal, which proves theLemma. (III)Now suppose (x,U)and(y,V)areanytwocoordinate systems with pE UF1V.Toprove that ,B_,."1 ooz),and,Bx"’ 0oz},induce thesame isomorphism ontheﬁbre ofTM atp,wecanassume without lossofgenerality that U=V, because part (I)applies toxand x|U OV,aswell astoyand y|U F)V. Assuming U=V,wehave thefollowing diagram. Tr-rw>> i»<TR">:x<v> ~51’-‘Q s"<R":»tt-(U) %(3)(TM)iU iiTU (y01"")... old(yOx"‘).,. X ii Tom) i»<TR">|y<v>51’-@> 8"<R":»|yw> The triangle obviously commutes, and therectangle commutes bypart (II). Diagram thus shows that ozy=old(y0x_’),,, oozx. Exactly thesame result holds forT’: By=Old(YQX"’)»= OBx- The desired result ,8)?’ oozy=Bx"! Oozxfollows immediately. Now thatwehave awell-deﬁned bundle map TM—>T’M_ (theunion ofall ,B,,“’ oozx), itisclearly anequivalence eM. The proofthat eNOf,=ft;oeM is leftasamasochistic exercise forthereader. 4* T/1e Itngent Bundle 95 PROBLEMS 1.LetMbeanyset,and{(x,-, U,-)} asequence ofone-one functions x,-:U,-—>IR” with U;CMand x(U,-) open inR",such that each X;QXFII Xr(U=' flUj)—>Xi(UrCU1) iscontinuous. Itwould seem that Mought tohave ametric which makes each U;open andeach x,-ahomeomorphism. Actually, thisisnotquite true: (a)LetM=]RU{>z=}, where >z<¢IR.LetU1=Rand x1:U1—> IRbethe identity, and letU2=R--{0}U{=1<}, with xg: U2—>Rdeﬁned by Xg(£I) =a, a750,=l< )Cg(=l<) =0. Show thatthere isnometric onMoftherequired sort, byshowing thatevery neighborhood of0would have tointersect every neighborhood of»z<_Never- theless, wecanﬁndonMapseudometric p(afunction p:MxM—>IRwith allproperties forametric except thatp(p,q) maybe0forp7':q)such thatp isametric oneach U,-andeach x,-isahomeomorphism: (b)IfACIR”isopen, then there isasequence A1,/lg, A3,...ofopen subsets ofAsuch thatevery open subset ofAisaunion ofcertain A,-’s. (c)There isasequence ofcontinuous functions f-:A—>[0,1],with support f,- CA,which “separates points andclosed sets”: ifCisclosed and pEA-C, then there issome ﬁ-with ﬂ-(p)¢f-(A (WC). Hint: First arrange inasequence allpairs (A,-,/lj)ofpart (b)with C/lj. (d)Letﬂ,j,j=1,2,3, ...besuch asequence foreach open setx,-(U,-). Deﬁne 8:311 M—>[0,1]bl’ “(nun pew 0 P¢Ui- Arrange allg,-,1»inasingle sequence G1,G2,G3,...,letdbeabounded metric onIR,anddeﬁne ponMbygig;(P) PMQ»!(>(p.q)= —-d(Gr(P),Gr(q))-i=1 Show that pistherequired pseudometric. (e)Suppose that forevery p,q EMthere isaU,-and U;with pEU;and qEU;and open sets B,-Cx,-(U,-) and B;Cxj(Uj) sothat pEx,-"’(B,-), qEx,;"’(B,;), andx,-"’(B,-) ﬁx,;_’(B,;) =9.Show that pisactually ametric onM. 96 Chapter 3 2.(a)Suppose (x,U)and (y,V)aretwocoordinate systems, giving risetotwo maps onTM, lxiff ’(U)—> U><]R”, [x,v]q|--> (q,v), ry:n“’(V) —>V><IR“, [y,w]q |—->(q,w). Show that in:r"’(U F)V)thesetsoftheform tx"1(A) forACU><IR"open areexactly thesetsoftheform If!(B)forBCV><IR"open. (b)Show thatifthere isametric onTMsuch thatIx,isahomeomorphism for acollection (xi,U,-)with M=U,U,-,then allIxarehomeomorphisms. (c)Conclude from Problem lthatthere isametric onTMwhich makes each Ix ahomeomorphism. 3.Show that inthedeﬁnition ofanequivalence itsuﬁices toassume that the map E1—>E2iscontinuous. (Toprove theinverse continuous, notethatlocally itisjust amap UxR“—>U><IR“). 4.Show that inthedeﬁnition ofabundle map, continuity offIB1—>B2 follows automatically from continuity off:E1—>E2. 5.Aweak equivalence between two bundles over thesame base space Bis abundle map ()7,f)where _)Fisanisomorphism oneach ﬁbre, and fisa homeomorphism ofBonto itself. Find twoinequivalent, butweakly equivalent, bundles over thefollowing base spaces: (i)thedisjoint union oftwocircles, (ii)aﬁgure eight (>6 , (iii)thetorus. 6.Given abundle map (f,f),show that _)F=g0Iiwhere gandharecontin- uous maps such that htakes ﬁbres linearly toﬁbres, while gisanisomorphism oncach fibre. 7.(a)Show that foranybundle rt:E—>B,themap s:B—>Ewith s(p) theOvector ofn"1(p)isasection. (b)Show that ann-plane bundle Eistrivial ifand only ifthere arensections s1,...,.s',, which areeverywhere linearly independent, i.e.,s1(p),...,s,,( p)E n"1(p)arelinearly independent forallpEB. (c)Show thatlocally every n-plane bundle hasnlinearly independent sections. 8.(a)Check that*;;isanequivalence relation onthesetofpairs (x,v). (b)Check thatthedeﬁnition off,isindependent ofthecoordinate systems x andywhich areused. (c)Check theremaining details inTheorem l. The Itngenl Bundle 97 9.(a)Show that thecorrespondence between TM and equivalence classes of curves under which [x,11],,corresponds tothe1;:equivalence class ofx"’o3/, for3/acurve inIR“with 3/’(O) =v,makes f..correspond tofﬂ. (b)Show that under thecorrespondence [x,a],, t->Z,a='8/3x='|p, themap f, canbedeﬁned by [ft(~‘3)](e) =~‘3(e0f)- 10.IfVisaﬁnite dimensional vector space over IR,deﬁne aC°°structure onV andahomeomorphism from V><VtoTVwhich isindependent ofchoice of bases. Asinthecase ofIR”,forv,wEVwewilldenote byvwEVwthevector corresponding to(w,v). 11.Ifg:IR—>IRisC°°show that gov)=2(0)+e’(0)x+xzhtx) forsome C°°function h:IR—>IR. 12.(a)LetF),bethesetofall C°°functions f:M—>IRwith f(p) =O,and letE:3+],—>IRbealinear operator with €(fg) =0forallf,gEFp.Show that Ehasaunique extension toaderivation. (b)LetWbethevector subspace ofE,generated byallproducts fgforf,gE J“?],.Show thatthevector space ofallderivations atpisisomorphic tothedual space (J'T,,/ W)*. (c)Since (J"5],/ W)*hasdimension n=dimension ofM,thesame must betrue of3"],/W. Ifxisacoordinate system with x(p) =O,show that xl+W,..., x"+Wisabasis for.F,,/W(useLemma 2).The situation isquite different for C1functions, asthenextproblem shows. 13.(a)LetVbethevector space ofall C1functions f:IR—>IRwith f(O) =O, andletWbethesubspace generated byallproducts. Show thatlimf(x)/x2 exists forallfEW. “T0 (b)ForO<s< l,Iet x’+" x30 nu){OH0 Show thatallf,areinV,andthatthey represent linearly independent elements ofV/W. (c)Conclude that(V/W)"‘hasdimension cc=2‘. 14.Iff:M—>Nand f,istheOmap oneach ﬁbre, then fisconstant on each component ofM. 98 Chapter 3 15.(a)Amap ftM—>Nisanimmersion ifand only iff,isone-one on each ﬁbre ofTM, More generally, therank offatpEMistherank ofthe linear transformation ft:Mp—>Nf(,,,). (b)Iff0g=f,where gisadiffeomorphism, then therank offogata equals therank offatg(a). (Compare with Problem 2-33(d).) 16.(a)IfMisamanifold-with-boundary, thetangent bundle TM isdeﬁned exactly asforM;elements ofMpare*5»equivalence classes ofpairs (x,v). Although xtakes aneighborhood ofpE8Monto II-ll“,rather than IR“,the vectors vstillrunthrough IR",soMpstillhastangent vectors “pointing inall directions”. IfpE3Mandx:U—>II-ll”isacoordinate system around p,then M x,,."’(IR""1,,(,,)) CMPisasubspace. Show thatthissubspace does notdepend onthechoice ofx;infact,itisi,,.(8M),,, where 1':8M—>Mistheinclusion. (b)LetaEIR”"1 ><{O}CII-ll". Atangent vector inII-Il”,, issaid topoint “in- ward” if,under theidentiﬁcation ofTII-II" with 8"(II-ll"), thevector is(a,v)where v">0.Avector vEMpwhich isnotin1',(BM),,issaid topoint “inward” if n Q inward aTmoutward x,.(v) EIHl",,(,,) points inward. Show thatthisdeﬁnition does notdepend on thecoordinate system x. (c)Show that ifMhasanorientation ,u.,then 3Mhasaunique orientation Zip.such that [v;,...,v,,_1] =(3,u.),, ifand only if[w,i,.v;,. ..,i,,,v,,_1] =,u,,for every outward pointing wEMp. (d)Ifptistheusual orientation ofIHI",show thatEly.is(—l)" times theusual orientation ofIR""1 =till-ll". (Thereason forthischoice willbecome clear in Chapter 8.) (e)Suppose weareinthesetup ofProblem 2-l4. Deﬁne gt8M><[0,l)—> 3N><[0,l)byg(p,t) =(f(p),t). Show thatTPisobtained from TMUTN The Thngent Bundle 99 byidentifying vE(8M),, with (p*‘).g,a,(v)a(aN),,,,,. (f)IfMand Nhave orientations ,u,and vand ft(8M,8,t.t) —>(8N,8v) is orientation-reversing, show that Phasanorientation which agrees with ,u.andu onMCPandNCP. (g)Suppose MisS2with twoholes cutout,andNis[0,1] xS’.Letfbe adiffeomorphism from MtoNwhich isorientation preserving ononecopy ofS1andorientation reversing ontheother. What istheresulting manifold P? 17.Show that TIP2 ishomeomorphic tothespace obtained from T(.S‘2,t') by identifying (p,v)E(S2,t'),, with (-—p, -v)E(.S‘2,i)_.,,. 18.Although there isnoeverywhere non-zero vector ﬁeld onS2,there isone on5'2-{(0,0,1)},which isdiﬁfeomorphic toIR2.Show thatsuch avector ﬁeld canbepicked sothatnear (O,0,l)thevector ﬁeldlooks likethefollowing picture (a“magnetic dipole”): /Y.‘¢- --. 7I ,r »- I-s I I , l 7 \ I <i‘c-A it \ '\2 l I I r ‘M 1 19.Suppose wehave a“multiplication” map (a,b):—->a-bfrom IR"><IR"toIR" that makes IR"into a(non-associative) division algebra. That is, (Q1 +ﬂ2)'b=01 *b+-02 ﬂ'(b1+1’J'2)=0-b1+ﬂ'b2 rI.(a-b)= (la)-b=a-(lb) fO1";i. EIR at(l,O,...,O) =0 andthere arenozero divisors: a,b;éO=> ab;é0. 100 Chapter 3 (For example, forn=l,wecanuseordinary multiplication, and forn=2 wecanuse“complex multiplication”, (a,b)-(c,d) =(ac-—bd,ad +bc).) Let e1,...,e,,bethestandard basis ofIR". Every point in.S‘""’ isa-e1foraunique aEIR". Ifasé0,then a-e1,...,a~e,,arelinearly independent. Ifp=a~e1 E.S"""1, then theprojection ofa-e2,...,a -enon(.S‘""1,t'),, elinearly independent. Multiplication byaiscontinuous. T.S‘“‘1 istrivial. TIP”""1 istrivial. The tangent bundles TS3 and TS7 areboth trivial. Multiplications with therequired properties onIR“and IR8areprovided bythe“quaternions” and “Cayley numbers”, respectively; thequaternions arenotcommutative andthe Cayley numbers arenoteven associative. Itisaclassical theorem that the reals, complexes, and quaternions aretheonly associative examples. Fora simple proof, seeR.S. Palais, TheClarsgﬁeation ofRealDivision Algebras, Amer. Math. Monthly 75(1968), 366-368. _].F.Adams hasproved, using methods of algebraic topology, thatn=1,2,4,or8. [Incidentally non~existence ofzero divisors immediately implies thatfora75O there issome bwith ab=(l,O,...,O) and b’with b’a=(l,0,...,O). Ifthe multiplication isassociative itfollows easily thatb=b’,sothatwealways have multiplicative inverses. Conversely, thiscondition implies thatthere arenozero divisors ifthemultiplication isassociative; otherwise itsuﬁices toassume the existence ofa unique bwith a~b=b-a=(l,O, ...,0).]/‘R/‘R/1%”/‘R/1%.:>.e..&"[email protected]. 20.(a)Consider thespace obtained from [0,1] xIR"byidentifying (O,v)with (1,Tv), where T:IR"—>IR"isavector space isomorphism. Show thatthiscan bemade intothetotal space ofavector bundle over S1(ageneralized Mobius strip). (b)Show that theresulting bundle isorientable ifand only ifTisorientation preserving. 21.Show thatforpES2,thevector ppEIR3,,isnotini,.(.S‘2,,) byshowing that theinner product (p,c’(O)) =Oforallcurves cwith c(O)=pand|c(t)| =l forallt.(Recall that (f,e>’(¢) =(f’(1)‘,e(f)>+ (f(().e’(t)‘>. where ‘denotes thetranspose; seeCalculus onManifttldr, pg.23.) 22.LetMbeaC°°manifold. Suppose that(TM)|A istrivial whenever ACM ishomeomorphic toS1. Show that Misorientable. Hint: Anarccfrom The Itngent Bundle 101 pgEMtopEMiscontained insome such Aso(TM)|c istrivial. Thus one can“transport” theorientation ofM,,,, toM,,. Itmust bechecked that thisis independent ofthechoice ofc.First consider pairs c,c’which meet onlyatpg andp.The general, possibly quite messy, case canbetreated bybreaking upc intosmall pieces contained incoordinate neighborhoods. Remark: Using results from theAddendum toChapter 9,together with Prob- lem29,wecanconclude thataneighborhood ofsome S1CMisnon-orientable ifMisnon-orientable. The next twoproblems deal with important constructions associated with vector bundles. 23.(a)Supposeé -=71':E—>Xisabundle andf:Y—>Xisacontinu- ousmap. LetE’CYxEbethesetofall(y,e) with f(y) =n(e), deﬁne rt’:E’—>Ybyn’(y,e) =y,anddeﬁne _)FIE’—>Eby_)F(]/,6) =e.Avector space structure canbedeﬁned on n’"‘o> =tote):AEr"‘</om byusing thevector space structure onrt"(f(y)). Show that rt’:E'—>Yisa bundle, and(_)F,f)abundle map which isanisomorphism oneach ﬁbre. This bundle isdenoted byf*(E), andiscalled thebundle induced (from E)byf. (b)Suppose wehave another bundle E”=rt”; E”—>Yand abundle map (_)F,f)from E”toEwhich isanisomorphism oneach ﬁbre. Show that E”2 E’=f*(E). Hint: Map 6EE”to(rr”(e),f(e)) EE’. (<1)IferZ—>Y,then(f<>e)*(E)1.e*(f*(E))-(d)IfACXandi:A—>Xistheinclusion map, then i"‘(.§) 2:E|A. (e)If5isorientable, then f*(.~§') isalsoorientable. (f)Give anexample where Eisnon—orientable, butf*(E)isorientable. (g)LetE=rt:E—>Bbeavector bundle. Since rt:E—>Bisacontinuous map from aspace tothebase space Bof5,thesymbol rr*(.§) makes sense. Show thatifEisnotorientable, then n'*(E) isnotorientable. 24.(a)Given ann—plane bundle E=rt:E—>Bandanm-plane bundle 17= n’:E’—>B,letE”CE><E’bethesetofallpairs (e,e’) with n(e) =n’(e’). Letn”(e,e’) =n(e) =rr’(e’). Show that rt”: E”—>Bisan(rt+m)—plane bundle. Itiscalled theWhitney sum E®17of5and17;theﬁbre ofE®17over p isthedirect sum n"’(p) EBn"'1(p). (blIffiY—>B,Show thatf*(€®'7)1f*(E) ®f*('7)- 102 Chapter 3 riwHriw *&»-09¢Given bundles E;=rt,-IE;—>B,-,deﬁne rt:E1XE2—>B1XB2by e;,e2) =(rt;(e;),n2(e2)). Show thatthisisabundle E1><E2over B1><B2. IfA:B—>BxBisthe“diagonal map”, A(x) =(x,x), show thatE®17Z A(E><'7)‘ (e)IfEandnareorientable, show thatE®17isorientable. (f)IfEisorientable, and I7isnon—orientable, show that E®17isalso non~ orientable. (g)Deﬁne a“natural” orientation onV®Vforanyvector space V,and use thistoshow thatE®Eisalways orientable. (h)IfXisa“ﬁgure eight” (c.f.Problem 5),ﬁnd twonon-orientable l—plane bundles Eand17over Xsuch thatE®17isalsonon-orientable. 25. (a)IfrriE—>MisaC°° vector bundle, then rt...hasmaximal rank at rach point, andeach ﬁbre 71""(p) isaC°°submanifold ofE. (b)The0—section ofEisasubmanifold, carried diffeomorphically onto Bbyrt. 26.(a)IfMandNareC°°manifolds, andJIM[orme] IM><N —>M[orN] istheprojection onM[orN],then T(M ><N)2n'M*(TM) ®n;v*(TN). (b)IfMandNareorientable, then M><Nisorientable. ()IfM><Nisorientable, then both MandNareorientable. (“J 27.Show thatthejacobian matrix ofy,,.0(x,,.)" isoftheform Dry’OX“ 0 ( X D;'J*"°X'“)i This shows that themanifold TM isalutqys orientable, i.e.,thebundle T(TM) is orientable. (Here isamore conceptual formulation: forvETM, theorientation for(TM), canbedeﬁned as [ 3 3 3 3 3(xIorr) vi...’ 3(x"0rr),,’3x1,,”H’3x" 5 theform ofy...o(x,..)'"' shows that thisorientation isindependent ofthechoice ofx_)Adifferent proof that TM isorientable isgiven inProblem 29. 28.(a)Let(x,U)beacoordinate system onMwith x(p)=OandletvEMP be2;, a’8/3x’|p .Consider thecurve cinTMdeﬁned by 3 _ P The Thngenl Bundle 103 Show that dc 8 :;;<°>=an (b)Find acurve whose tangent vector at0is8/8(x" on)|U. 29.This problem requires some familiarity with thenotion ofexact sequences - f e .(cfChapter ll).Asequence ofbundle maps E1——>E2——>E3with f=g= identity ofBisexact ifateach ﬁbre itisexact asasequence ofvector space maps. (a)IfE=rt:E—>BisaC°° vector bundle, show that there isanexact sequence O—>2r*(E) —>TE—>n'*(TB) —>O. Hint: (l)Anelement ofthetotal space ofn*(E) isapair ofpoints inthesame ﬁbre, which determines atangent vector oftheﬁbre. (2)Map XE(TE), to (6,rt,X). (b)If0>E1 >E2 >E3 >0isexact, then each bundle E1isorientable if theother twoare. (c)T(TM) isalways orientable. (d)Ifrt:E—>Misnotorientable, then themanifold Eisnotorientable. (This iswhy theproof that theMobius strip isanon-orientable manifold issosimilar totheproof that theMobius bundle over S1isnotorientable.) The next twoProblems contain more information about thegroups intro- duced inProblem 2-33. Inaddition tobeing used inProblem 32,thisinforma- tionwillallbeimportant inChapter IO. 30.(a)LetpgE.S"""1 bethepoint (O,...,O, 1).Forrt32deﬁne ftSO(n) —> .S"""1 byf(A) =A(p(1). Show that fiscontinuous and open. Show that f'"l(p(1) ishomeomorphic toSO(rt —1),andthen show that f"’(p) ishome- omorphic toSO(n —l)forallpE.S‘""1. (b)SO(l) isapoint, soitisconnected. Using part (a),andinduction onrt, prove thatSO(rt) isconnected forallrt?_l. (c)Show that O(n) hasexactly twocomponents. 31. (a)IfT:IR"—>IR“isalinear transformation, T*1 IR"—>IR",theadjoint ofT,isdeﬁned by(T*v, w)=(v,Tw) (foreach v,themap wi——>(v,Tw) is linear, soitisw:->(T*v, w)foraunique T"‘v). IfAisthematrix ofTwith respect totheusual basis, show thatthematrix ofT"‘isthetranspose A‘. 104 Chapter 3 (b)Alinear transformation TIIR"—>IR“isself-adjoint ifT=T*, sothat (Tv,w) =(v,Tw) forallv,w EIR".IfAisthematrix ofTwith respect tothe standard basis, then Tisself-adjoint ifandonly ifAissymmetric, A‘=A.It isastandard theorem that asymmetric Acanbewritten asCDC "1forsome diagonal matrix D(forananalytic proof, seeCalculus anManiﬁtlds, pg.122). Show thatCcanbechosen orthogonal, byshowing thateigenvectors fordistinct eigenvalues areorthogonal. (c)Aself-adjoint T(orthecorresponding symmetric A)iscalled positive semi- definite if(Tv, v)3OforallvEIR",andpositive deﬁnite if(Tu, v)>0forall vgéO.Show that apositive deﬁnite Aisnon-singular. Hint: UsetheSchwarz inequality. (d)Show thatA‘-Aisalways positive semi-deﬁnite. (e)Show that apositive semi-deﬁnite Acanbewritten asA=B2forsome B. (Remember that Aissymmetric.) (f)Show thatevery AEGL(n,IR) canbewritten uniquely asA=A1-A;where A1EO(rt) andA2ispositive deﬁnite. Hint: Consider A‘-A,andusepart(e). (g)The matrices A1andA2arecontinuous functions ofA.Hint: IfAl”) —>A andAV‘) =A(")1 -A992, then some subsequencc of{A(")1} converges. (li)GL(n,IR) ishomeomorphic toO(n) XR"(“+’)/2 andhasexactly twocom- ponents, {A:detA >O}and{A:detA <0}.(Notice thatthisalsogives us another way ofﬁnding thedimension ofO(n).) 32.Two continuous functions jg,f1:X—>Yarecalled homotopic ifthere is acontinuous function H:XX[0,l]—>Ysuch that f,-(X)=1'](X,I') Ii:-'0,l. The functions H1:X—>Ydeﬁned byH,(x) =H(x,t) may bethought ofas a path offunctions from H11=fgtoH1=f1.The map Hiscalled ahomotopy between foandf1. The notation f:(X,A)->(Y,B),forACXand BCY,means that ftX—> Yandf(A) CB.Wecallﬂ1,f1: (X,A) —>(Y,B)homotopic(as maps from (X,A)to(Y,B))ifthere isanHasabove such that each H1:(X,A)—> (Y,B). (a)IfA:[0,1]—>GL(n,IR) iscontinuous andH:IR”><[0,1] —>IR“isdeﬁned by H(x,t) =A(r)(x),show thatHiscontinuous, sothatH11andH1arehomotopic asmaps from (IR”,IR” —~{O})to(IR",IR" —{O}). Conclude thatanon-singular linear transformation TI(IR”,IR” -—{O}) —>(IR”,IR” --{O}) with detT>Ois homotopic totheidentity map. (b)Suppose f:IR"—>IR"isC°°andf(0) =0,while f(IR" -{O})CIR"- IfDf(0) isnon-singular, show that f:(IR",IR" —-{O}) —>(IR",IR" --{O}~@/Pﬁ-A.....£?.0'}- The Yzmgent Bundle 105 homotopic toDf(0): (]R",]R" -—{O})—>(]R",]R” -{O}). Hint: Define H(x,I) = f(tx) forO<t5land H(x,0) =Df(O)(x). Toprove continuity atpoints (x,O),useLemma 2. (c)LetUbeaneighborhood of0EIR“and fIU-->R"ahomeomorphism with f(O) =O.LetB,CVbetheopen ballwith center 0andradius r,andlet I1:R"—>B,bethehomeomorphism h(x) = arctan |x|)x;rr then f0121 (]R",]R" —{O}) —>(]R",]R“ --{O}). Wewillsaythat fisorientation preserving at0iffohishomotopic tothe identity map I:(lR",lR“ -{O}) —>(]R",lR” -{O}). Check that thisdoes not depend onthechoice ofB,CV. (d)Forp ER“, letTp:]R“ —>R"beTp(q) =p+q. Iff:U-->Visa homeomorphism, where U,VCIR"areopen, wewillsaythat fisorientation preserving atpifT__)-(P) Of0Tpisorientation preserving atO.Show that ifM isorientable, thenthere isacollection C’ofcharts whose domains cover Msuch thatforevery (x,U)and(y,V)inC’,themap yox"1 isorientation preserving atx(p) forallpEU('1V. (e)Notice that thecondition ony0x"!inpart (d)makes sense even ifyox”! isnotdifferentiable. Thus, ifMisany(notnecessarily differentiable) manifold, wecandeﬁne Mtobeorientable ifthere isacollection C3ofhomeomorphisms x:U—>IR"whose domains cover M,such that C’satisﬁes thecondition in part (d).Toprove that thisdeﬁnition agrees with theoldoneweneed afact from algebraic topology: Iff:IR”—>R”isahomeomorphism with f(0)=0 and T:R"—>R“isT(x1,...,x“) =(x1,...,x”"1,»-x“), then precisely one offand T0fisorientation preserving atO.Assuming thisresult, show that ifMhassuch acollection Gofhomeomorphisms, then foranyC°°structure onMthetangent bundle TMisorientable. 33.LetM”CRNbeaC°° n-dimensional submanifold. Byachord ofMwe mean apoint ofRNofthe form p--qforp,q 6M. (a)Prove thatifN>2n+1,then there isavector vESN"'1such that |-|(')nochord ofMisparallel tov, iinotanent laneMcontains v. g P P Hint: Consider certain maps from appropriate open subsets ofM><Mand TMto5”"1. 106 Chapter 3 (b)LetlRN"1 CRNbethesubspace perpendicular tov,andrt:RN—>lRN"1 thecorresponding projection. Show that rr|M isaone-one immersion. In particular, ifMiscompact, then zr|M isanimbedding. (c)Every compact C°°n-dimensional manifold canbeimbedded in1R2""" . Note: This istheeasy case ofWhitney’s classical theorem, which gives the same result even fornon-compact manifolds (H.‘Whitney, Dzﬂerentiabte mangfrtds, Ann. ofMath. 37(1935), 645-680). Proofs may befound inAuslander and MacKenzie, Introduction toDtﬂerentiabte ./Wangfblds and Sternberg, Lectures on ferential Geometry. InMunkres, Elementary Diﬂnezztial Y5j)0l0gy, there isadifferent sortofargument toprove thatanot-necessarily—compact n-manifold Mcanbe imbedded insome RN(infact, with N=(n+l)2). Then wemay show that M imbeds in]R2"+1 using essentially theargument above, together with theexis- tence ofaproper map f:M—>IR,given byProblem '2-30 (compare Guillemin midPollack, Dﬂezentiat Yiipology). Amuch harder 1-esult ofWhitney shows that M"canactually beimbedded inR2"(H.Whitney, Theseb’—z'ntersectz'0n.r qfasmooth 2z—man§ﬁ1ldz'n 2n—space, Ann. ofMath. 45(1944), 220-24 6). CHAPTER4 TENSORS All theconstructions onvector bundles carried outinthischapter have a common feature. Ineach case, wereplace each ﬁbre rr"l(p) bysome other vector space, andthen ﬁtallthese newvector spaces together toform a new vector bundle over thesame base space. The simplest case arises when wereplace each ﬁbre Vbyitsdual space V*. Recall that V*denotes thevector space ofalllinear functions it:V—>IR.If fIV—>Wisalinear transformation, then there isalinear transformation f*: W*—>V*deﬁned by (f*l)(v) =l(fv)- Itisclear thatifIv:V—>Vistheidentity, then 1;/*istheidentity map ofV* and ifg:U—>V,then (fog)"‘ =g*0f*. These simple remarks already show that f*isanisomorphism ifftV—>Wis,for(f'"l 0f)* =1;/*and (f<=f"‘)* =1»/*.The dimension ofV*isthesame asthat ofV,forﬁnite dimensional V.In fact, ifvl,._.,v,, isabasis forV,then theelements v*,-EV*,deﬁned by 11*,-(11,-) =..-5}, areeasily checked tobeabasis forV*.The linear function 11*;depends onthe entire setv1,...,v,,, notjustonvialone, andtheisomorphism from VtoV* obtained bysending v,-to11*;isnotindependent ofthechoice ofbasis (consider what happens ifv1isreplaced by2111). Ontheother hand, ifuEV,wecandeﬁne v**EV**=(V*)* unambigu- ously by v**()t) =Mu) forevery AEV*. Ifv**(l.) =0forevery ll.6V*,then §\.(v) =0forallItEV*,which implies that IO? 10s ctape¢4 v=0.Thus themap v|—>v**isanisomorphism from VtoV"‘*. Itiscalled thenatural isomorphism from VtoV"‘*. (Problem 6gives aprecise meaning totheword “natural”, formulated only after theterm hadlongbeen inuse.Once themeaning ismade precise, wecan prove that there isnonatural isomorphism from VtoV*.) Now let3,’=rr:E—>Bbeanyvector bundle. Let E’=Urn-'1*,peB anddeﬁne thefunction rt’:E’—>Btotake each [rr_'(p)]* top.IfUCB, andtIrt"(U)—>U><R"isatrivialization, then wecandeﬁne afunction 1’;1-H-'(u) _>U><(n")* intheobvious way: since themap Irestricted toaﬁbre, orrr"(p) —>{P}><R", isanisomorphism, itgives usanisomorphism <:,,*>"':tn-'1* —>{1>}><<R">*. Wecanmake rt’:E’—>Bintoavector bundle, thedual bundle 5*ofQ’,by requiring that allsuch I’belocal trivializations. (We ﬁrstpick anisomorphism from (lR”)"' toR”,once andforall.) Atﬁrst itmight appear that 11-'*23;,since each rr_'( p)isisomorphic to rr’“' (p). However, thisistrue merely because thetwovector spaces have the same dimension. The lackofanatural isomorphism from VtoV*prevents usfrom constructing anequivalence between §*and£1.Actually, wewillsee later that in“most” cases 35*isequivalent to3;;forthepresent, readers may ponder thisquestion forthemselves. Incontrast, thebundle §'**=(t‘§*)* is atwqys equivalent to§.Weconstruct theequivalence bymapping theﬁbre V of£5over ptotheﬁbre V"‘*ofE**over pbythenatural isomorphism. Ifyou Tensor; 109 think about how 1‘;*isconstructed, itwillappear obvious thatthismap isindeed anequivalence. Even ifEcanbepictured geometrically (e.g., if5isTM), there isseldom a geometric picture for§*.Rather, §*operates on§:Ifsisasection of3;‘andcr isasection oft,=*,then wecandeﬁne afunction from BtoRby s(P)Grr"(P) PHg(p)(S(p)) o(P)e11"‘(P)=Ir"(P)*- This function willbedenoted simply byo(s). When thisconstruction isapplied tothetangent bundle TMofM,there- sulting bundle, denoted byT*M, iscalled thecotangent bundle ofM;theﬁbre ofT*M over pis(M,,)*. Like TM, thecotangent bundle T*M isactually a C°°vector bundle: since twotrivializations x,,,andy,,ofTM areC°°—related, thesame isclearly trueforx,’andy,.’(infact, y,,'0(X,,")_' =ya0(x,,,)"). Wecanthusdeﬁne C°°,aswellascontinuous, sections ofT*M. IfwisaC°° section ofT*M and XisaC°° vector ﬁeld, then w(X) istheC°°function P'—>w(P)(X(P))- Iff:M—>RisaC°° function, then aC°° section dfofT*M can be deﬁned by df(P)(X) =X(f) F01‘/Y EMP- The section dfiscalled thedifferential off.Suppose, inparticular, that Xis do/dt|,,,, where c(t0) =p.Recall that Q_cidz," *dz This means that df(§ )=c,(% )0) d =Em(f°¢') =(for-m) orW IIO C/zopter 4 Adopting theelliptical notations g for gr, d%ff§£2 for g'(t), thisequation takes theniceform dc_d(f(c(r))) If(x,U)isacoordinate system, then thedxfaresections ofT*M over U. Applying thedeﬁnition, weseethat . 3 . P Thus dx'(p), ...,dx”(p) isjust thebasis ofMp* dual tothebasis 3/Bx‘ Ip,..., 8/3x”|_,, ofMp. This means thatevery section wcanbeexpressed uniquely onUas I’! w(P)=Zwi-(P)dx"(P),l'=| forcertain functions w,-onU.The section wiscontinuous orC°°ifandonly if thefunctions w,-are. Wecanalsowrite H w= 2w,-dxi, i=1 ifwedeﬁne sums ofsections andproducts offunctions andsections inthe obvious way (“pointwise” addition andmultiplication). Thesection elfmust have some suchexpression. Infact,weobtain aclassical formula: Tertrors 111 1.THEOREM. If(x,U)isacoordinate system andfisaC°°function, then onUwehave df=Z:a—’_f,dx‘._3a1..-=1 PROOF. IfXpeMpis It ,-8 XP=Z“ W1':-"I P then at=Xp<x‘)=di-"<Xt>. Thus df(P)(XP) =Xpm=Z@‘§7{.[=1 =Z,a—,{,dx"<X.,>. ~:~{:1 Classical differential geometers (and classical analysts) didnothesitate totalk about “inﬁnitely small” changes dxlofthecoordinates xi,justasLeibnitz had. Noonewanted toadmit that thiswasnonsense, because true results were ob- tained when these inﬁnitely small quantities were divided intoeach other (pro- vided onediditintheright way). Eventually itwas realized that theclosest one cancome todescribing an inﬁnitely small change istodescribe adirection inwhich thischange issupposed tooccur, i.e.,atangent vector, Since dfissupposed tobetheinﬁnitesimal change offunder aninﬁnitesimal change ofthepoint, dfmust beafunction ofthischange, which means that dfshould beafunction ontangent vectors. The dxlthemselves then metamorphosed into functions, and itbecame clear thattheymust bedistinguished from thetangent vectors 8/8x". Once thisrealization came, itwasonly amatter ofmaking newdeﬁnitions, which preserved theoldnotation, and waiting foreverybody tocatch up. In short, allclassical notions involving inﬁnitely small quantities became functions ontangent vectors, likedf,except forquotients ofinﬁnitely small quantities, which became tangent vectors, likedc/dz. Looking back atthcclassical works from ourmodern vantage point, onecan usually seethat, nomatter how obscurely expressed, thispoint ofview wasin 112 Chapter 4 some sense theonealways taken byclassical geometers. Infact, thedifferential dfwasusually introduced inthefollowing way: CLASSICAL FORMULATION Letfbeafunction ofthex',...,x“, sayf=f(x',...,x").MODERN FORMULATION Letfbeafunction onM,and xa coordinate system (sothat f=fox forsome function fonR",namely f=fOxii). Letxibefunctions ofr,sayxi= x'(t). Then fbecomes afunction OTF,f(f)=f(X'(I),---,X”(I))-LetCIR—>Mbeacurve. Then foc:lR—> R,where foﬁe(t),= j(x's¢(i),...,x"s¢(1)). Wenow have a_iyd_x"dz_,=,andz' (The classical notation, which suppresses thecurve c,isstillused byphysicists, asweshall point out once again inChapter 7.)Wenow have (f=>@)’(r) =ZD.-f<x<c<r)))-o-*1» cm)l'=I =a—f-(cm) -ofQcm) I,__=IBx or d<f<c<r)))_ "af ,dx"<c<:))dz _g@(C(i)) dz Multiplying bydtgives n i {=1 (This equation signiﬁes thattrue results areobtained bydividing by dragain, nomatter what t/zefunctioru i h x’(I)are. Itistheclosest approac inclassical analysis totherealiza- tion ofdfasafunction ontangen vectors.)IConiseciuently, dc_“af ,-dc Since every tangent vector atc(t)is oftheform dc/dt,wehave H I f=l Yénsors 113 Inpreparation forourreading ofGauss and Riemann, wewillcontinually examine theclassical way ofexpressing allconcepts which weintroduce. After awhile, the“translation” ofclassical terminology becomes only alittle more difﬁcult than thetranslation oftheGerman inwhich itwaswritten. Recall that iff:M—>NisC°°, then there isamap f,,:TM —>TN; foreach peM,wehave amap )’,,,,,: Mp —>Nf(,,). Since f,,,,isalinear transformation between twovector spaces, itgives risetoamap Nf(.v)* _*MP*' Strict notational propriety would dictate thatthismap bedenoted by(f*_,,)*, buteveryone denotes itsimply by f;INfrpf‘ —>Mf- Notice that wecannot putallj,,*together toobtain abundle map from T*N toT*M; infact, thesame grENmay bef(pg)formore than onep,-EM, andthere isnoreason why f,,_,,, should equal f,,.,,,. Ontheother hand, wecan dosomething with thecotangent bundle thatwecould notdowith thetangent bundle. Suppose wisasection ofT*N. Then wecandeﬁne asection r]ofT*M asfollows: 7l(p) : 0ftp: i.e., ﬂ(P)(Xp) =w(f(P))(f*pXp) forXpEMp' (The complex symbolism tends tohide thesimple idea: tooperate onavector, wepush itover toNbyf,,.,andthen operate onitbyw.)This section 17is denoted, naturally enough, byf*w.There isnocorresponding wayoftrans- ferring avector ﬁeld XonMover toavector ﬁeld onN. Despite these differences, wecansay,roughly, thatamap f:M—>Npro- duces amap J’...going inthesame direction onthetangent bundle andamap f*going intheopposite direction onthecotangent bundle. Nowadays such situations arealways distinguished bycalling thethings which gointhesame direction “covariant” and thethings which gointheopposite direction “con- travariant”. Classical terminology used these same words, anditjusthappens tohave reversed this: avector ﬁeldiscalled acontravariant vector ﬁeld, while asection ofT*M iscalled acovariant vector ﬁeId. And noonehashad the gallorauthority toreverse terminology sosanctiﬁed byyears ofusage. So it’sV61’)! easy toremember which kind ofvector ﬁeld iscovariant, andwhich contravariant—it’s justtheopposite ofwhat itlogically ought tobe. 114 Chapter 4 The rationale behind theclassical terminology canbeseen byconsidering coordinate systems xonR"which arelinear transformations. Inthiscase, if x(v,-) =E3,then x(a'v| +~--+ a"v,,) =(a',...,a"), sothexcoordinate system isjustan“oblique Cartesian coordinate system”. 3,11’ --—"IIr ""-I I, 1 (Q3 I U2 “ £1121 vi __________ Ifx’isanother such coordinate system, then x’?=EL, a,~,-x" forcertain a,-J-. ._ -' 5Clearly a,-J-3;”/Bx,so ."*1 <*> »"’=Z?—1x’; II'M] thiscanheseendirectly from thefactthatthematrix (3x'j /Bx’) istheconstant lnatrix D(x' 0.\""') =x’0x'"l. Comparing with ) 4"1'i:ax“d* (=1==1= x= i. X, ml3x‘ from Theorem l,weseethatthediﬁerentials dxi“change inthesame way” as tliccoordinates xi,hence they are“covariant”. Consequently anycombination H w=23(1),-dx“ Fm] isalsocalled “covariant”. Notice thatifwealsohave H‘ w=Zto’;dx”, I'm] Yimors 115 then wecanexpress thew',-interms ofthe0);.Substituting " I .f__ ax 1"dl —-; 1xJ J% into theﬁrstexpression forwandcomparing coefﬁcients with thesecond, we ﬁndthat “ Bx‘I Q; ..,,_‘—Z..,,_ax,,..1"-1"-=1 Ontheother hand, given twoexpressions i:a1_é_€C_IT =ﬁg/i% fml rm] foravector ﬁeld, thefunctions admust satisfy H -3)" GU. :Z151‘-Lag.3. fr-=1 A These expressions canalways beremembered bynoting that indices which are summed over always appear once “above” andonce “below”. (Coordinate func- tions xl,...,x"used tobedenoted byx1,...,x,,.This suggested subscripts w,- forcovariaut vector fields andsuperscripts LI“forcoiitravariant vector ﬁelds. Af- terthiswasﬁrmly established, theindices onthe.x’swere shifted upstairs again tomake thesummation convention work out.) Covariant andcontravariant vector ﬁelds, i.e.,sections ofT*M andTM, respectively, arealsocalled covariant andcontravariant tensors (ortensor ﬁelds) oforder 1,which isawarning that worse things aretocome. Webegin with some worse algebra. IfV1,...,Vmarevector spaces, afunction T:V1><---><V,,,—>liR ismultilinear if Ul_> T(Ul>--->Uk--l>U>Uk+l»--->Um) islinear foreach choice ofv|,...,vk..|,v;¢+|,...,v,,,. The setofallsuch T isclearly avector space. IfV1,...,V”,=V,thisvector space willbedenoted 116 Chapter 4 byT’"(V). Notice that T'(V) =V*. Iff:V—>Wisalinear transfor- mation, then there isalinear transformation f*:T”‘(W) —>T"‘(V), deﬁned completely analogously tothecasem=I: f*T(v1,---.vm) =T(f(vl)>--->f(Um))- ForTET"‘(V), andSeT'(V) wecandeﬁne the“tensor product” T®SE f,~k+i(V) by T®S(Ul:"'>Uk>Uk+l>- -'1-viii-I-ll) ZT(v|9"'>vk) 'S(Uk+l:---:Uk+1)' Ofcourse, T®S isnotS®T. Ontheother hand, (S®T)®U =S®(T®U), sowecandeﬁne :1-fold tensor products unambiguously; thistensor product operation isitself multilinear, (S1+S2)®T=S1®T+S2®T,etc. In particular, ifv|,...,v,, isabasis forVand11*],...,v*,, isthedual basis for V*=T'(V), then theelements v*,-,®---®v*;,_ 15s,,...,i,,gn areeasily seen tobeabasis forT“(V), which thus hasdimension nk. Wecanusethisnew algebraic construction toobtain anew bundle from any vector bundle §=rt:E—>B.Welet E’=UT“(rr"'(p)), _pEB and let rt’:E"—> Btake 3'""‘(rr'“'1(p)) top. IfUCBand r:rr“l(U)—> U><1x" isatrivialization, then theisomorphisms Ia:rr"'(p) —>{P}><R" yield isomorphisms (rp*)"I 5'”"(rr"(p)) —>{P}><9""(1R”)- Ifwechoose anisomorphism T“QR") —>113"“once andforall,these maps can beputtogether togive amap 2":;rr"'"1(U)—> U><lR"“. Yfznsors 117 Wemake rt’:E"—>Binto avector bundle T“(15)byrequiring that allsuch I’ helocal trivializations. The bundle §*isthespecial case k=1. Forthecase ofTM, thebundle T“(TM) iscalled thebundle ofcovariant tensors oforder k,andasection iscalled acovariant tensor ﬁeId oforder k.If (x,U)isacoordinate system, sothat dx’<p).....dx" isabasis for(M,a)*, then thek-fold tensor products dx"(p)®---®dX“(P) eT"(Mp) 151'],---Jr 5" areabasis forT"‘(M_,,). Thus, onUevery covariant tensor ﬁeld Aoforder k canbewritten /1tp)= ZA1....i,,.<p)dx‘*<p)®---sdxrtpi.I],...,.l’j,- orsimply 1-" -A=Z: A;,___,,_.dxf1®---®dxi*, it :3£ where dx“ ®---®dx"*'-' now denotes asection ofTi‘(TM ).Ifwealsohave A=ZA’i.....-,_.dx”"®---®dx-"'~.l'],...,lk then _it lkA, _ A Bx Bx 0t(...Ot;; — Z l(...i;,- ax*"‘"‘,a} --.?xfak f(,.",ik (theproducts arejust ordinary products offunctions). Toderive thisequation, wejustuseequation (=+==1=)onpage ll4,andmultilinearity of®.The section A iscontinuous orC°°ifandonly ifthefunctions A,=,___,-,, are. Acovariant tensor ﬁeldAoforder kcanjustbethought ofasanoperation A onkvector ﬁelds X1,...,Xkwhich yields afunction: i1'<X1....,Xi) =/1(p)(X1(p). ...>Xk(P))- Notice that Aismultilinear ontheset'VofC°°vector ﬁelds: .i"(X,,...,X,-+X',,...X,,)=.i'(X,,...,X,,...,X,,)+Z(X,,...,X’,,...Xk) ,&"(X,,...,.iX,,...X,.) =aE(X,,...,X,.). 118 Chapter 4 Moreover, because Aisdeﬁned “pointwise”, itisactually linear avertheC°° fmzctions 37;i.e.,iffisC°°, then A(X|,...,fX,-,...,X;,)_fA(X|,...,X,-,...,Xk), forwehave JRX1.---.fX.-.....Xi> =A<.v><X1 (P),---=f(P)Xi'(P)> ---=Xk(P)) =f(p)A(p)(X1(p),- ...X1(p),. ..>Xk(P)) _ 'A(X1:" ->Xf:' -'>Xk)(p)' Weareﬁnally ready foranother theorem, onethat isused over andover. 2.THEOREM. If .A»:'V><---x'V—>J'7MM ktimes islinear over 37,then there isaunique tensor ﬁeld Awith A»=A. PROOF. Note ﬁrstthatifveMPisanytangent vector, then there isavector ﬁeldXe'Vwith X(p)=v.Infact,if(x,U)isacoordinate system and v=gai% then wecandeﬁne n .8 X: fgﬂiﬁ OI] U 0 outside U, where each ainow denotes aconstant function and fisaC°°function with f(p) =1andsupportf CU. Now ifv|,...,vk EMpareextended tovector ﬁelds X;,...,Xk E'Vwe clearly must deﬁne A(11)(v1,-- -,v:<) =<>4>(X1,---,X1<)(P)- The problem istoprove that thisiswell—deﬁned: IfX;(p)=Y,-(p) foreach 1', weclaim that A(Xl:- - =Aiyls -'-1 Yérzsors 119 (The map A“lives atpoints”, tousetheinterminology.) Forsimplicity, take the case k=I(thegeneral case isexactly analogous). The proof that .>%(X )(p)= .;%(Y)(p) when X(p) =Y(p) isintwosteps. (l)Suppose ﬁrstthat X=Yinaneighborhood Uofp.LetfbeaC°° function with f(p)=1andsupport fCU.Then fX=fY,so f=>4>(/Y) =¢“~(fX)= =>4>(fY) =f¢‘~(Y); evaluating atpgives =>‘»(X)(P) ==>‘¥(Y)(P)- (2)Toprove theresult, itobviously sufﬁces toshow that .>%(X)(p) =0if X(p)=0.Let(x,U) beacoordinate system around p,sothat onUwe canwrite ".3 .X=Zb=F wherea1lb'(p)=0. i=-I Ifgis1inaneighborhood Vofp,andsupport gCU,then n .8 n 8 1'-=1 1‘-1"-=1 isawell—deﬁned C°°vector ﬁeld onallofMwhich equals XonV,sothat i =>‘~(X)(1>) =<>4»(Y)(p). by(I)- Now wtritm =Zen») iA (P)i=1 O00.0 =0,since11-’(p)= Because ofTheorem 2,wewillnever distinguish between thetensor ﬁeld A andtheoperation /T,norwillweusethesymbol Aanylonger. Note that Theorem 2applies, inparticular, tothecase k=1,where T'“(TM) =T*M, thecotangent bundle: afunction from 'V—>37which islinear over Fcomes from acovariant vector ﬁeldcu._]ustaswith covariant vector ﬁelds, aC°°map fIM—>Ngives amap f*taking covariant tensor ﬁelds Aoforder konN tocovariant tensor ﬁelds f*Aoforder konM: f*»4(P)(X1,,.---XI<,,) =/1(f(P))(f»=X1,,,--~,fiX1<,,)- 120 Chapter 4 Moreover, ifAand Barecovariant tensor ﬁelds oforders kand1,respectively, then wecandeﬁne anew covariant tensor ﬁeld A®Boforder k+1: (A®B)(p) =A(p) ®B(p) (operating onMpx ><Mp k+1times). Although covariant tensor ﬁelds willbeourmain concern, ifonly forthe sake ofcompleteness weshould deﬁne contravariant tensor ﬁelds. Recall that acontravariant vector ﬁeld isasection XofTM. Soeach XPeM,,.Now an element vofavector space Vcanbethought ofasalinear function v:V*—>R; wejustdeﬁne v(}t) tobeMu). Acontravariant tensor ﬁeld oforder kisjust asection Aofthebundle T"‘(T*M); thus, each A(p)isak-linear function onM,,*. Wecould alsousethenotation ?‘},(TM), ifweuseTk(V) todenote all k-linear functions onV*.Inlocal coordinates wecanwrite .. B BAU’): A11---Jr-(p)m ®...®W J-i:'"IJk P (remember thateach B/Bx!‘ |,,operates onM,,,*), orsimply ..3 B A= AJ""J"i. i.. jZ, Bxll ® ®BxlkI:-"1 )'\‘ Ifwehave another such expression, --B B “=42. '4'"lu---tfk then weeasily compute that __3x»'ﬁ1 axrﬁrrﬁ _ y...;.i____ i41'“Uzi .41wax,‘ am. |,...3k Acontravariant tensor ﬁeld Aoforder kcanbeconsidered asanoperator A taking kcovariant vector ﬁelds w;,...,wkintoafunction: /Tim],---=wk)(P) =A(p)(w1(p)..-..wi(P))- Naturally, there isananalogue ofTheorem 2,proved exactly thesame way, that allows ustodispense with thenotation A,andtoidentify contravariant tensor ﬁelds oforder kwith operators onkcovariant vector ﬁelds that arelinear over theC°°functions 37. Ténsars 121 Finally, weareready tointroduce “mixed” tensor ﬁelds. Tomake theintro- duction lesspainful, weconsider aspecial case ﬁrst. IfVisavector space, let T,‘(V)denote allbilinear functions T:VxV*—>R. Avector bundle F,’=rt:E—>Bgives risetoavector bundle T,‘(if),obtained by replacing each ﬁbre rr_'(p)byT,1(rr_'(p)). Inparticular, sections ofZ1(TM ) arecalled tensor ﬁelds, covariant oforder Iandcontravariant oforder 1. There areallsorts ofalgebraic tricks onecanplaywith T,‘(V);although they should bekept toaminimum, certain ones arequite important. LetEnd(V) denote thevector space ofalllinear transformations T:V—>V(“endomor- phisms” ofV).Notice thateach SGEnd(V) gives risetoabilinear SET,‘(V), §;V><1/*_>ix, bytheformula (*) $(v=X)=3»($(v))- Moreover, thecorrespondence S|—>Sfrom End(V) toZ‘(V)islinear andone- 0116,for§=oimplies that).(S(v)) =0forall2.,whichimplies thatso)=0, forallv.Since both End(V)and T]'(V) have dimension n2,thismap isan isomorphism. The inverse, however, isnotsoeasy todescribe. Given S,for each vthevector S(v)eVismerely determined bydescribing theaction ofait onitaccording to(=1=).Itisnothard tocheck thatthisisomorphism ofEnd(V) and ﬁ'(V) makes theidentity map 1:V—>VinEnd(V) correspond tothe “evaluation” map 4;v><1/*_>ninT,'(V) given by e(v,}t) =}t(v). Generally speaking, ourisomorphism canbeused totransfer anyoperation from End(V)toT,‘(V). Inparticular, given abilinear T:V><V*—>R, wecantake thetrace ofthecorresponding S:V—>V;thisnumber iscalled thecontraction ofT.Ifv|,.. .,v,,isabasis ofVand T=Z: Ti)-U*l' ®Uh ilj 122 C/zapter 4 then wecanﬁnd thematrix A=(a,-J-) ofS,deﬁned by fl‘ SW1) =ZajiUj> jzl interms oftheT,j;infact, ﬁn=v*;'(S(v1)) =T(vr>v*;) =7}’.- Thus n - contraction ofT=ET,‘. Fm] (The term “contraction” comes from thefactthatthenumber ofindices iscon- tracted from 2to0bysetting theupper andlower indices equal andsumming.) These identiﬁcations andoperations canbecarried out,ﬁbre byﬁbre, inany ﬁbre bundle T,‘(35).Thus, asection AofT,‘(35)canjustaswell beconsidered asasection ofthebundle End(¢‘,=), obtained byreplacing each ﬁbre rr“‘(p)by End(rr_‘ (p)). Inthiscase, each A(p)isanendomorphism ofrr_1(p).More- over, each section Agives risetoafunction (contraction ofA):B—>R deﬁned by p|—>contraction ofA(p) _ =trace A(p) ifwe consider A(p) eEnd(.1r_I(p)). Inparticular, given atensor ﬁeld A,covariant oforder 1and contravariant oforder 1,which isasection ofT,‘(TM ),wecanconsider each A(p)asan endomorphism ofMp,andweobtain afunction “contraction ofA”.Ifina coordinate system A= dx' ®F thenfl (contraction ofA)=ZA:-. {=1 The general notion ofamixed tensor ﬁeld isastraightforward generalization. Deﬁne 17}‘“(V) tobethesetofall(k+1)-linear T:]/®-~-®I(><1/*®---®Vj-> lit. lctimes ttimes Tensor; 123 Every bundle 1Egives risetoabundle T}"‘(§). Sections of3T}“(TM) arecalled tensor fields, covariant oforder kandcontravariant oforder 1,orsimply oftype (if),anabbreviation thatalsosaves everybody embarrassment about theuseof thewords “covariant” and“contravariant”. Locally, atensor ﬁeld Aoftype canbeexpressed as — J“"J‘ fl f M M A—_Z_ At1...r';_.dx ®®dx':®,x,-, ®eax,-,.t_(,...,t;,- J1,---:1! and if - - 3 3A’=iZ;=‘1'§.'...if dx“1rs---®dx"*<<2»55®---®.[,..., {,- J|1---1.” then . . fl axik ax-"BI axfﬁfrﬁ _ Jl---J! ax W _____ (*) Aoei...ozi,, _i A1';...t;,- ax,-on axyqk axj, ax], ' ¢,...,r;,- in.---Jr Classical differential geometry books areﬁlled with monstrosities likethis equation. Infact, theclassical deﬁnition ofatensor ﬁeld is:anassignment ofn"""" functions toevery coordinate system sothat(=1=)holds between then"""' functions assigned toanytwocoordinate systems xandx’.(l)Oreven, “aset of12"” functions which changes according to(=i=)”. Consequently, inclassical differential geometry, allimportant tensors areactually deﬁned bydeﬁning the functions Ag,interms ofthecoordinate system x,andthen checking that(=1=) holds. Here isanimportant example. Inevery classical differential geometry book, onewillﬁndthefollowing assertion: “The Kronecker delta 5,?‘isatensor.” In other words, itisasserted thatifonechooses thesame :12functions 5,1‘foreach coordinate system, then (=1=)holds, i.e., -Bx"Bx’55_ i. 5“— Bx*'°¢ Bxi’ ,1 thisiscertainly true, for Z5,ax‘ax?_ Bx”-B 50,, M Bx°‘Bxl ,2,Bx°‘Bx’ 124 C/zapter 4 From ourpoint ofview, what thisequation shows isthat - B_ J2iA_Z:5,-dx®axJ. 1,1 isacertain tensor ﬁeld, independent ofthechoice ofthecoordinate system x Toidentify themysterious map A(p): Mp><M,,* —>R, weconsider vEM,,andAeM,,*with theexpressions H 8 H v=Za“aia ,)t=Z:b5dx5(p); oc=l x P ﬁn] then -. B/1(P)(v.A) =Z6?drip)®Htum1,; P -.“ a a “= Bfd '(p)( a°‘—O, (bdx‘5(p)) 5: xO; Bx,,Bx!,,,2;5 =Z5,.jafb,- i,j =i: Gib; iml =).(v). Thus A(p) isjusttheevaluation map MP><M,,*—>R;considered asanendo- morphism ofM,,,itisjusttheidentity map. The contraction ofatensor isdefined, classically, inasimilar manner. Given atensor, i.e.,acollection offunctions A,oneforeach coordinate system, sat- isfying -Bx" Bx'5'l3_ 1?? Aa—'ZJ':Ai Bx*'°‘ Bx?’ Ténsars 125 wenote that iA’:=i(Z4.%,’-‘.§§)o:=l 0z=l t,j .“ 'J =g;/1,! 2;i,;:‘c—: =Z445;atH =£3,411, iv-lQ sothat thissum isawell—deﬁned function. This calculation tends toobscure theonepartwhich isreally necessarywveriﬁcation ofthefactthatthetrace of alinear transformation, deﬁned asthesum ofthediagonal entries ofitsmatrix, isindependent ofthebasis with respect towhich thematrix iswritten. Incidentally, atensor oftype canbecontracted with respect toanypair ofupper andlower indices. Forexample, thefunctions N B53=Z*“°iii3§Otml “transform correctly” ifthe do.Ifweconsider each A(p) eT},3(M,,), then wearetaking B(p) E‘T}2(M,,) tobe B(p)(v|,v2,)t|,}t2) =contraction of:(v,).) |—>A(p)(v,v|,v2,).|,)t2,)t). Wliile acontravariant vector ﬁeld isclassically asetofnfunctions which “transforms inacertain way”, avector atasingle point pisclassically justan assignment ofnnumbers a‘,...,a”toeach coordinate system x,such thatthe numbers a"‘,. ..,a”‘assigned tox’satisfy .“."1'an_Z:aIai(p)_ —_ Bxirm] This isprecisety thedeﬁnition weadopted when wedeﬁned tangent vectors as equivalence classes [x,a],,. The revolution inthemodern approach isthat the setofallvectors ismade into abundle, sothat vector ﬁelds canbedeﬁned as sections, rather than asequivalence classes ofsetsoffunctions, andthatallother 126 Chapter 4 types oftensors areconstructed from thisbundle. The tangent bundle itself was almost avictim oftheexcesses ofrevolutionary zeal. Foralong time, theparty lineheldthatTMmust bedeﬁned either asderivations, orasequivalence classes ofcurves; thereturn totheolddeﬁnition wasinﬂuenced bythe“functorial” point ofview ofTheorems 3-1and3-4. The modern revolt against theclassical point ofview hasbeen socomplete incertain quarters thatsome mathematicians willgiveathree page proof that avoids coordinates inpreference toathree lineproof thatusesthem. Wewon’t goquite that far,butwewillgive an“invariant” deﬁnition (one that does not useacoordinate system) ofanytensors thataredeﬁned. Unlike the“Kronecker delta” andcontractions, such invariant deﬁnitions areusually notsoeasy to come by.Asweshall see,invariant deﬁnitions ofalltheimportant tensors in differential geometry aremade bymeans ofTheorem 2.Weseldom deﬁne A(p) directly; instead wedeﬁne afunction Aonvector ﬁelds, which miraculously turns outtobelinear over theC°° functions 37,and hence must come from some A.Attheappropriate time wewilldiscuss whether ornotthisisallabig cheat. Teasers 127 PROBLEMS 1.Letf:M"—>N”', and suppose that (x,U) and (y,V)arecoordinate systems around pandf(p),respectively. (a)Ifg:N—>R,then atsef) _”' as 30/"<=>f)ax,(P) ay,(f(p)) -ﬁrm. (Proposition 2-3isthespecial casef=identity.) (b)Show that 3 m30”‘°f) 3 x")_Z Bx? (P)I ’P j=| f(P) and, more generally, express jI..(Z§'m, a"B/Bx‘|p) interms oftheB/By1'|,,. (c)Show that (f*dy")(r) =ZaLif)(P) -dx’(r)-fml ax (d)Express flZ4,-....,~..dr"'®---®dy”')I J-|e--will interms ofthedxi. 2.Iff,g: M—>NareC°°, show that dtfs) =fds+ stif- 3.Letf:M—>RheC°°. ForveMp,show that fitvl=df(”)f(p) GRic»)- 4.(a)Show thatifthe ordered bases 11],...,1),,andw1,. ..,w,,forVareequally oriented, thenthesame istrueofthe bases 11*],...,v*,,and10*],...,w*,,forV*. (b)Show thatabundle Eisorientable ifandonlyif£*isorientable. 128 C/zapter 4 5.The following statements andproblems arealltaken from Eisenhart’s clas- sical work Riemannian Geometry». Ineach case, check them, using theclassical methods, andthen translate theproblem andsolution intomodern terms. An “invariant” isjust a(well—deﬁned) function. Remember that thesummation convention isalways used, sokin; means EL, }t";.t,-. Hints and answers are given attheend, after (xiii). (i)Ifthequantity kin; isaninvariant andeither iforpt;arethecomponents ofanarbitrary [covariant orcontravariant] vector ﬁeld, theother setsarecom- ponents ofavector ﬁeld. (ii)If).,,;*' arethecomponents ofnvector ﬁelds [inann-manifold], where i fori=l,...,1:indicates thecomponent andcaforor=l,...,rtthevector, and these vectors areindependent, thatis,det(}t,,,;") 760,then anyvector-ﬁeld Uis expressible intheform 2.‘=a“2.,,,*, where thea’sareinvariants. (iii)If;.t,-arethecomponents ofagiven vector—ﬁeld, anyvector—ﬁeld A5satisfy- ingit‘11.;=0isexpressible linearly interms of12—Iindependent vector ﬁelds la)’.foroz=l,...,n—1which satisfy theequation. (iv)Ifaij=a/"iforthecomponents ofatensor field inonecoordinate system, then a"1=a2’forthecoordinates inanyother coordinate system. (v)Ifavandbi!‘arecomponents ofatensor ﬁeld, soareaij+Zr”.Ifail‘and bk;arecomponents ofatensor ﬁeld, soareaijbki. (vi)Ifa,-J-}t")H' isaninvariant forifanarbitrary vector, then a,-;+a,1;arethe components ofatensor; inparticular, ifa,-)~}t'}J =0,then a,-;+a,-,-=0. (vii) Ifa,-J-)t"}tj =0forallvectors idsuch that kin; =0,where pt;isagiven covariant vector, ifviisdeﬁned [c.f.(iii)]byarjlafvt =0,or=1,...,1:—Iand pt,-vi #0,andbydeﬁnition a,-J-vi=a)~ v'p.;=r, then (Hg;—-%;.t,-cry-)§‘¢‘,*j =0issatisﬁed byevery vector ﬁeld 3,",andconsequently 1 ﬁr;+ﬂji='1'j(lli<Fj +llj<F1')~ (viii) Ifa,,arethecomponents ofatensor andbandcareinvariants, show thatifban+can=0,then either b=—canda,,_issymmetric, orb=cand a,-Sisskew-symmetric. (ix)Bydeﬁnition therank ofatensor ofthesecond order a5)istherank of thematrix (erg)-). Show that therank isinvariant under alltransformations of coordinates. Ténsars 129 (x)Show that therank ofthetensor ofcomponents ail?)-, where a,-and by arethecomponents oftwovectors, isone; show that forthesymmetric tensor Gib) -l—ﬂjb; tl‘l(-3 I'2lI1l< lSYWO. (xi)Show that thetensor equation ai;A;=alt), where orisaninvariant, can bewritten intheform (ai;—a5i,-)}t,- =0.Show also that a’)=5*’)-oi, ifthe equation istohold foranarbitrary vector X,-. (xii)Ifa",~).,- =alt)holds forallvectors A;such thatttilf =0,where [Liisa given vector, then ai)=0:5") +tr)-;.t". (xiii) If 0 ifj,,,=j5forsomecz;é)3ori,,,=i5forsomea;é)3 6j,___j,,_ OI-If {J-ls---ij-P}T£{2.l>'--sip} """" 1 ifj],...,j,, isaneven permutation of2'1,...,1’), ——1 ifj|,...,j,,isanoddpermutation of1'1,...,z',, then <i,i,1_:'_',.f’ arethecomponents ofatensor inallcoordinate systems. HINTS AND ANSWERS. (i)toisdetermined ifw(X) isknown forallX,andviceversa. (iii)Given w[with w(p)#0forallp],there areeverywhere linearly indepen- dent vector ﬁelds X1,...,X,,_1 which span kerw ateach point. (This istrue only locally Forexample, onS2><Rthere isantosuch thatkerw(p,I)consists ofvectors tangent toS2><{t}.) (vi)ForT:V><V—>R,letT'(v, w)=T(w, v).Then T+T’isdetermined by S(v) =T(v,v).For,T(v+w,v+w)=T(v,v)+T(v,w)+T(w, v)+T(w, w). Similarly, T(v, v)=0forallvimplies that T+T’=0. (vii)Given to[with w(p) 950forallp],choose Ycomplementary tokerw at allpoints. Ifo'(Z) =T(Y, Z),then T(Z, Z)=w(Z)o'(Z)/w(Y) forailvector ﬁelds Z. (ix)T:V><V—>Rcorresponds toT:V—>V*[where T(v)(w) =T(v,w)]. The rank ofTmay bedeﬁned astherank ofT(consider thematrix ofTwith respect tobases 111,...,11,,and 11*],...,v*,,). (xii)LetV=M,,*. IfT:V—>VandpteV*andT(v) =cwforallvel£CI‘[L, there isaycomplementary toker/.tsuch that T(v) =av+;i(v)y forallv. (Begin bychoosing yocomplementary tokerp.andwriting vuniquely asv@+cy@ for‘U9ekerpt.) 130 Chapter 4 (xiii)Deﬁne 5:_V><---><V><V*><---xVi—>R ,0times ,0times by5(1)], ...,Up,A],...,AP) =dCt(}tj ('Uj)). 6.(a)Let1';/:V—>V**bethe“natural isomorphism” z'V(v)(}t) =)t(v). Show that foranylinear transformation f:V—>W,thefollowing diagram com- mutes: _ . VM, 1/** fl14i *4:WL W ’h)Show thatthere donotexist isomorphisms 1';/:V->V*such thatthefol- lowing diagram always commutes. V {V V: fl_itW ii, W* Hint: There does noteven exist anisomorphism 2':R—>ll?’which makes the diagram commute foralllinear f:R—>R. 7.Acovarzantfunctor from (ﬁnite dimensional) vector spaces tovector spaces isa function Fwhich assigns toevery vector space Vavector space F(V)andtoev- crylinear transformation f:V—>Walinear transformation F(f):F(V)—> F(W), such that F(l;/) =l}=‘(V) and F(gof) =F(g) OF(v). (a)The“identity functor”, F(V)=V,F(f)=fisafunctor. (b)The “double dual functor”, F(V)=V**, F(f)=f**isafunctor. (c)The“Tafunctor”, F(V) =T}<(V) =T"‘(V*), F(f)(T)(3~1»---,3~i<)=T(l1°f>---tlksf) isafunctor. (d)IfFisanyfunctor and f:V—>Wisanisomorphism, then F(f)isan isomorphism. Acontravan'antfw1ct0r isdeﬁned similarly except that F(f): F(W) —>F(V)and F(gof)=F(f)oF(g). Functors ofmore than oneargument, covariant in some andcontravariant inothers, may alsobedeﬁned. (e)The “dual functor”, F(V)=V*,F(f)=f*isacontravariant functor. (f)The “Tkfunctor”, F(V) =T"‘(V), F(f) =f*isacontravariant functor. Ténsors 131 8.(a)LetHom(V, W)denote alllinear transformations from VtoW.Choos- ingabasis forVand W,wecanidentify Hom(V, W)with them><nmatrices, andconsequently giveitthemetric ofll-"km". Show thatadifferent choice of bases leads toahomeomorphic metric onI-Iom(V, W). (b)Afunctor Fgives amap from Hom(V, W)toHom(F(V), F(W)). Call F continuous ifthismap isalways continuous [using themetric inpart (a)]. Show thatif§=rt:E—>Bisanyvector bundle, andFiscontinuous, then there is abundle F(§) =Ir’:E’—>Bforwhich Tr"_](p) =F(rr'"1(p)), and such that toevery trivialization I:rr_1(U)—> U><R" corresponds atrivialization I’::rr’_'(U) —>U><FUR"). (c)The functor St"}(V) =T‘(V"‘) =V**iscontinuous. (The bundle T1(TM) is justacase oftheconstruction in(b).) (d)Deﬁne acontinuous contravariant functor F,andshow how toconstruct a bundle F(!‘;'). (e)The functor F(V)=V*iscontinuous. (The bundle T*M isaspecial case oftheconstruction in(d).) Generally, thesame construction canbeused when Fisafunctor ofseveral arguments. The bundles ﬁk(M) areallspecial cases. Seethenext twoproblems forother examples, aswell asanexample ofafunctor which isnotcontinuous. 9.(a)LetFbeafunctor from V",theclass of:2-dimensional vector spaces, toVi‘.Given AeGL(n,lR) wecanconsider itasamap A:R”—>R”.Then F(A): F(lR") —>FUR"). Choose, once andforall,anisomorphism FUR") —> Rk. Then F(A)canbeconsidered asamap 11(A): Rk—>Rk. Show that 12:GL(n, R)—>GL(k, R)isahomomorphism. (b)How does thehomomorphism 11depend ontheinitial choice oftheisomor- phismF(R")->Rl‘? (c)Letv=(v1,...,v,,) and W=(101,. ..,w,,) beordered bases ofVand let e=(e1,...,en)bethestandard basis ofR".Ife—>vdenotes theisomorphism taking e,-tov,-,show that thefollowing diagram commutes RN 8—>V V Al/—>W 2 Rn I32 Chapter 4 where A=(a,-J-) isdeﬁned by U);=Z:ﬂj;'UJ'. 1'“! After identifying FUR") with 113;‘,thismeans that Rk F(€ —>V) h(A) l W) Ric alsocommutes. This suggests away ofproving thefollowing. THEOREM. Ifh:GL(n,R) —>GL(k,lR) isanyl‘l0m0m0rpl‘liSm, there isafunctor F;,:V"—>Vksuch thatthehomomorphism deﬁned inpart (a)isequal to/1. (d)Forq,q’eRf‘,deﬁne (mt)~(w.<1’) ifq=/'1(A)q" where w,-=Z;-‘=1 aj,-vj. Show that~isanequivalence relation, andthatevery equivalence class contains exactly oneelement (v,q)foragiven v. Wewilldenote theequivalence class of(v,g) by[v,q]. (c)Show that theoperations ["=9'1]+[Y>q2l=[Y>9'1+ 9'2] Q'[Y=9'l =["=99'] arewell-deﬁned operations making thesetofallequivalence classes into a k—dimensional vector space F;,(V). (f)IfV,W eV"and f:V—>W,choose ordered bases v,w, deﬁne Aby f(v,-) =237:] aj,-wj, anddeﬁne Fn(f)[",9'l =[“’,/1(A)(9')l- Show thatthisisawell—deﬁned linear transformation, thatF;,isafunctor, and that F;,(A) =I:(A) when weidentify F;,(lR") with El‘by[e,q] |—>g. (g)Letoz:R—>Rbeanon-continuous homomorphism (compare page 380), andlet11:GL(n,lR) —>GL(1,]R.) =Rbeh(A) =a(det A).Then F;,:V”—>V‘ isanon-continuous functor. YE’:2s'0rs 133 10.Inclassical tensor analysis there are,inaddition tomixed tensor ﬁelds, other “quantities” which aredeﬁned assetsoffunctions which transform according toyetother rules. These newrules areoftheform , Bx“A=Aoperated onbyf1F . Forexample, assignments ofasingle function atoeach coordinate system x such thatthefunction a’assigned tox’satisﬁes Eixia’=det( .)-a 3x” arecalled (even) scalar densities; assignments forwhich a’=det(Lid)Bx” arecalled oddscalar densities. The Theorem inProblem 9allows ustocon- struct abundle whose sections correspond tothese classical entities (later we willhave amore illuminating way): (a)Let/2:GL(n,lR) —>GL(l,lR) takeAintomultiplication bydetA. LetF;, bethefunctor given bytheTheorem, andconsider the1-dimensional bundle F;,(TM) obtained byreplacing each ﬁbre Mpwith F;,(Mp). If(x,U)isa coordinate system, then 3 !"' )>]]€Fh(MP) P P isnon-zero, soevery section onUcanbeexpressed asa-axforaunique function a.Ifx’isanother coordinate system anda-ax=a’-ax», show that a'=det -a. Bx” (b)If,instead, 1'1takes Aintomultiplication by|detA|, show thatthecorre- sponding equation is ‘ a’=det(LxBx” (c)Forthis11,show thatanon-zero element ofF;,(V) determines anorientation forV.Conclude thatthebundle ofoddscalar densities isnottrivial ifMisnot orientable. 134 C/rapier‘ 4 (<1)VVecanidentify 3t'}"(lR") withn~"“"’bytaking . .. . . it-+1e*,-,®---®e*,-,_®e,-, ®---®e,-, |—>(z1,...,1k,]1,...,];)‘h basis vector oflR” . Recallthatitf;v->v,wedeﬁne53%;); 3t‘}"‘(V)_>St‘}"‘(V)by 1;"<f)<T)tv1.....vi..>~1.....>v>= Ttftvm.--.ftvi.).>~. @f.---.ii<>f)- Given AGGL(n,llR), wecan consider itasamap A:R”—>R”. Then 3t‘}"‘(A): F}"‘(]R") —>3t":,"‘(lR") determines anelement ﬁ"(A) ofGL(n""*“',]R). Let/2:GL(n,lR) —>GL(n"+",li?.) bedeﬁned by /z(A) =(det A)wfi}k(A) waninteger. Thebundle F(TM)iscalled thebundle of(even) relative tensors oftype andweight w.Fork=1=0weobtain thebundle of(even) relative scalars of weight w[the(even) scalar densities arethe(even) relative scalars ofweight 1]. lf(detA)“ isreplaced byldetA|‘“(wanyrealnumber), weobtain thebundle ofoddrelative tensors oftype andweight w.Show thatthetransformation lawforthecomponents ofsections ofthese bundles is A.-6|.--ﬁt _dc,31"‘ wZA1‘:---1': 3"’-'...... ax“axi§'..... axiﬁia|...ak "" axyj I'|...i,q- ax.r0t| ax/12‘); ax)‘ ax)‘, ;I f_|,... ';,- Jl:---:J.~' (orthesame formula with det(3x"/3x”_) replaced by|det(3x"/Bx”-)|). (e)Deﬁne +1 iffl,...,r',,isaneven permutation of1,...,n .9,-|___,ﬂ = -1 if1'1,...,1},isanodd permutation of1,...,n 0 iffO,=f5 forsomea 755. Show thatthere isacovariant relative tensor ofweight -1with these compo- nents inevery coordinate system. Also show that sf‘"J"=.9,-|___,-,, arethecom- ponents inevery coordinate system ofacertain contravariant relative tensor ofweight 1.(SeeProblem 7-l2forageometric interpretation ofthese relative tensors.) GHAPTER 5 VECTOR FIELDS AND DIFFERENTIAL EQUATIONS We return toamore detailed study ofthetangent bundle TM, and its sections, i.e.,vector ﬁelds. LetXbeavector ﬁeld deﬁned inaneigh- borhood ofpGM.Wewould liketoknow ifthere isacurve pt(—e,s)—>M through pwhose tangent vectors coincide with X,thatis,acurve pwith -Z, o--?> »<><°>=P _//' '*-“.__’.__',X> d dp X P \ *i 2* = ( - .______* Phi.) Pt” -err-\. QP Since thisalocal question, wewish tointroduce acoordinate system (x,U) around pandtransfer thevector ﬁeld Xtox(U) CR”.Recall that, ingeneral, a,..X does notmake sense forC°°functions atM—>N.However, ifatisa diﬂeomorpliism, then wedeﬁne (o:,,.X)q =o:,,(X,,,-|(q)) [i.e., =a,m_|(q)(XO,-|(q))]. Itisnothard tocheck (Problem l)thato:,,X isC°°ono.'(M). Inparticular, we have avector ﬁeld x,,X onx(U) CR".There isafunction f:x(U) —>R" with (-x#X)q =f(q)q QRnqw i.e.,(x,,.X)q has“components” f1(q),. ..,f"(q). Gonsider thecurve c=xop. The condition dpI—X(P(5)) means that d p...(5=X<p<r>>; 136 Chapter 5 hence d d IfI-'=xtpvk I)=35*(Xi/7(0)) =(x=|=X)x(p(r)) =(x*X)c(r)- Ifweusec"(t) todenote theordinary derivative ofthelR”—valued function c, then thisequation ﬁnally becomes simply 6'0)'=f(C(I))- This isasimple example ofadifferential equation forafunction c:R—>ll-R”, which may alsobeconsidered asasystem of21differential equations forthe functions cf, c“(t)=ff(c'(t),...,c"(t)) t'=1,...,n. Wealsowant the“initial conditions” 03(0)=15(19)- Solving adifferential equation used tobedescribed as“integrating” the equation (the process isintegration when theequation hasthespecial form c"(t) =f(I)forf:R—>R,aform towhich ourparticular equations never reduce); solutions were consequently called “integrals” oftheequation. Partof thisterminology isstillpreserved. Acurve p:(——s,.9)—>Mwith 10(0)=P d d-if=Xtpto) iscalled anintegral curve forXwith initial condition p(0) =p.Similar ter- minology isapplied, ofcourse, tothediﬁferential equations oneobtains upon introducing acoordinate system. Forquite some time, wewillwork entirely in Euclidean space, andforawhile x,y,etc., willdenote points ofR”.IfUCR" isopen andf:U—>R”,then acurve c:(-8,s)—>Mwith c(0) =x xEU 6(1)=f(c(r)) iscalled anintegral curve forfwith initial condition c(0)=x. Vector Fields andDﬁerential Eqzmtiorzs 137 Before stating themain theorem about theexistence anduniqueness ofsuch integral curves, weconsider some special cases. Theequation foracurve cwith range R, ea)=-tr.-(1)12, which would bewritten classically interms ofafunction y:R—>Ras dy 2 2'; _Ty 1- isthespecial case f(a)=—a2. The standard method ofsolving thisequation istowrite dla =dx -1’ dy 1—=x+CY 1y=?~x+C Thus thecuwes I “‘)=i+_caresupposed tobesolutions. This canbechecked directly ifyoudon’t believe theabove manipulations. (They really domake sense; theequation inquestion asserts that y’=f0y,so (Iyy’1 -—0 - = -f 5 (F°J’)"=1 F0/(I))=I+Chence, ifF’=1/f, then forsome C.)Toobtain theinitial conditions c(0)=a,wemust take 1 c(t)=inI+1/a This works inallcases except a=0.lnthiscase, thecorrect solution is c(t)=0 forallI 13s Chapter 5 I (which wemissed bydividing byy).Interms ofvector ﬁelds, thecurves care theintegral curves of dA/(C1) =-—-£12-‘Eu W 4---- .11..~—4-ow —~ 1. 41-1. 4-.4 --C1 We ~ _l _l 0 1 12 4 4 2 Notice thatnointegral curve, except c(I)=-0,canbedeﬁned forallI,even though Xisdeﬁned onallofR.Itmight bethought thatthissomehow reﬂects thefactthatX(0)=0,butthishasnothing todowith thecase. Fora>0,the curve c(I)=1/(I+1/a) isdeﬁned foralllarge I,andasI—>ooitapproaches, butnever reaches, 0.Ontheother hand, asI—>-1/Iithecurve escapes to inﬁnity because thevector ﬁeld getsbigtoofast. This willcontinue tobetrue even ifwemodify thevector ﬁeld near 0sothatitisnever 0. Another phenomenon isillustrated bytheequation c’<o=not”. written classically as dy 2/3 dxTy' There aretwodifferent solutions with theinitial condition c(0)=0,namely (l) c(I)=0 forallI, (2) c(I)=T29 forallI. Inthiscase, thefunction f,given byf(a)=I12/3, isnotdifferentiable. Unique- nesswillalways beinsured when f:U—>R"isC1,butitcanalsobeobtained with arather lessstringent condition. Wesaythatthefunction fsatisﬁes a Lipschitz condition onUifthere issome Ksuch that lf(X)—-f(y)|.sK|>~‘—-yl f<>ra1IX,yeU- Notice thatf(a)==I121’3isnotLipschitz; infact, there isnoKwith lftx) ""f(0)| .5K|>~'| forxnear 0,since xz/3 2” --— =x""'/3—> :l:oo asx—> 0”. f(x)=-"XA. lizctor Fields andDgﬁfeiential Equations 139 ALipschitz function isclearly continuous, butnotnecessarily differentiable (for example, f(.>c) =|x|). Ontheother hand, aC1function islocally Lipschitz, thatis,itsatisﬁes aLipschitz condition inaneighborhood ofeach point—this follows from Lemma 2-5. ALipschitz function isalso clearly bounded onany bounded set. The basic existence and uniqueness theorem fordifferential equations de- pends onasimple lemma about complete metric spaces. l.THEOREM (THE CONTRACTION LEMMA). Let(M,p)beanon- empty complete metric space, andletf:M—>Mbea“contraction”, that is, suppose there issome C<1such that otftx), ft)-')) 5Cots.)/) forallmyGM- Then there isaunique xeMsuch thatf(x) =x(thefunction fhasaunique “ﬁxed point”). PROOF. Notice thatfisclearly continuous. Letxo eManddeﬁne asequence {x,,} inductively by x"'i'l =f(xR)a i.e., X~+1=f”(I0)=f°f°"'°f(I0)- ntimes Then aneasy induction argument shows that Pixnixn-t-I) $CnP(x0:xl)- Thus p(rn,x.,+i) sptxi.-,xn+1) +---+p(x..+i-1 .rn+.t) s<6"+---+C“*""')[email protected])~ Since C<1,thesum Z20 C"converges, soC”+- --+C”‘*"l"'l —>Oasn—>00. Thus thesequence {x,,} isCauchy, sothere issome xwith x=limx,,. -ﬂ—'¥OO Continuity offthen shows that Q§.O f(x) -=limf(x,,) -=limx,,+| rx.F|'—')OO II—')OO 140 Chapter 5 Wearegoing toapply theContraction Lemma tocertain spaces offunctions. Recall that if(M,p)isametric space and Xiscompact, then thesetofall continuous functions f:X—>Misametric space ifwedeﬁne themetric O’by vtﬂg) ==SupatftX),stX))-xeX IfMisbounded, then wedonoteven need Xtobecompact. Moreover, ifM iscomplete, then thenew metric space isalsocomplete; thisisbasically justthe theorem that theuniform limit ofcontinuous functions iscontinuous, plus the factthateach xlirréo f,(x) exists since Miscomplete. Inparticular, ifMisa compact subset ofR",then thesetofallcontinuous functions f:X—>Mis complete with themetric <Itf,s) =llf~ell, where llfll=$l1P|f(X)|-xeX Our basic strategy insolving differential equations willbetoreplace differen- tiable functions andderivatives bycontinuous functions andintegrals. IfUC R"andfIU—>R"iscontinuous, then acontinuous function (II(--b,b) —>U, deﬁned onsome interval around 0,clearly satisﬁes I Mr)-=fmo)U 01(0) =x ifitsatisﬁes theintegral equation I (2) av)--=x+(ftwtu))du. where theintegral ofanR”—valued function isdeﬁned byintegrating each com- ponent function separately. Conversely, ifatsatisﬁes (l),then orisdifferentiable, hence continuous; thus oz’==-foatiscontinuous, so a(I)-x=oI(I)-—a(0) =fia"(u) du=[If(oI(I;)) du. 0 0 Fortheproof ofthebasic theorem, weneed only onesimple estimate. Ifa continuous function f:[o,b] —>R”satisﬁes |fI5K,then f(u)du 5K(b-o). Toprove this, wenote that itistrue forconstant functions, hence forstep functions, andthus forcontinuous functions, which areuniform limits on[0,b] ofstepfunctions. Hzctor FieZa’s andDgﬁferential Equations 141 2.THEOREM. Let f:U—>R“beany function, where UCR”isopen. Letx0EUandleto>Obeanumber such that theclosed ball B;,,(x0), of radius 2oandcenter x0,iscontained inU.Suppose that )|f|sLonFates) ()lftr)-f(y)|5KIX-1/Iformye§2e(X0)- Choose b>0sothat (l55-Q/L ()b<1/K. Then foreach xeE,,(x0) there isaunique ax:(—-b,b) —>Usuch that <Ix'tI) =ftorx(1)) oI,,(0) =x.I\3'TZ." >-PUD PROOF Choose xeBa(xo), which willbeﬁxed fortheremainder oftheproof. Let M={continuous oz:(-—b,b) —>§;;,,(x0)}. Then Misacomplete metric space. Foreach aseM,deﬁne acurve So:on trbib) by I So:(I)=x-t-A f(tx(u))du (theintegral exists since fiscontinuous onB2a(x0)). The cuwe Satisclearly continuous. Moreover, foranyIG(-—b,b) wehave |-Mr)-x| =ftwtu>>du <bL by 5o byn""'-wr"‘-w()3\--|éﬂéﬂ Since |x-—x0|5o,itfollows that |S0z(I)-—x0|<2o,forallIE(-—b,b), so (=t=) SO!(I) EB29 (X0) CB-29 (X0) f0I‘I G Thus S:M—>M. Now suppose oz,)3eM.Then use-son=sgp£ft<1tu)) —ftfit-11))at <1>1<suplate)-aw bye)~b<uMisacontraction. Hence Shasaunique ﬁxed point: There isaunique oi:(--b,b) —>B20060) with I Ct(I) =x+‘/(I) f(o:(u))du. This, alas, isnotquite what thetheorem states. Having used theelegant Con- traction Lemma, wepayforitbyﬁnishing offwith aﬁnicky detail: The map atistheunique )3:(-—b,b) —>Usatisfying I no=X+[0ftﬁondu- Reoson: Weclaim that anysuch )3actually liesin§;a(A'0), infact, inB;,,(x0). Consider ﬁrst numbers I>0.Wehave already seen (statement (=t=))that for eachI with05I <b, (=t==t=) ﬂ(t)=x+ftf(ﬁ(u)) du isinB2_,,(x0) [theopenball] 0 provided that )3(u) GB;,,(x0) foralluwith 05u<I, socertainly if )3(u) EBg,,(x0) foralluwith 05It5I. Wecannowuseasimple least upper bound argument. Let A={It05I<17and)3(u)€B;,,(x0)for05u<I}. Letat=supA. Suppose or<b.Weclearly have ,B(u) eB2,,(x0) for05u<oz. Soﬁ(a) eB;,,(xq), by(=t=*). This clearly implies that)3(a+s)EBga(XQ) for sufﬁciently small s>0,which contradicts thefactthat or=supA.Soitmust bethat supA ==b.Asimilar argument works for--b<I50. Tosum up,theunique fixed point axofthemap Sistheunique curve with thedesired properties. '1' Vector Fields andD§*jfe?'ential Equations 143 Notice thatsolutions ofthedifferential equation 05(1)=f(°l(f)) remain solutions under additive changes ofparameter; thatis,if - t3(1)=¢¥(‘0+1). then f3'(f)=<¥'(I0+1)=f(°¢(l0 +I))=f(t3(I))- This remark allows ustoextend theuniqueness partofTheorem 2. 3.THEOREM. Suppose f:U—>R"islocally Lipschitz, thatis,around each point there isaballonwhich fsatisﬁes condition (2)ofTheorem 2forsome K (and hence alsocondition (l)forsome L).LetxeUandleton,a;betwomaps onsome open interval Iwith a1(I),a;(I) CUand on-’(l)==f(0u(I)) ,__12 01,-(0)=x T’' Then oz]=(Z2onI. PROOF. Suppose oz](I0)=O!2(I()) forsome I0EI.Ifwedeﬁne ﬁst!) =<Ii(lo+1), then thefunctions )3;satisfy thesame differential equation, )3,-’(I) =f()9,-(I)), andhave thesame initial condition )3;(0)=oz](I0)=--tx;(I0) EU.Hence 51(I)= ,8;(I) forsufﬁciently small I,byTheorem l.Thus theset {Ie110-'1(I)"-'= 0120)} isopen. Itisclearly alsoclosed andnon-empty, soitequals I.+9 Wenowrevert tothesituation inTheorem 2.Wewillwrite ozx(I)asa(I,x), sothatwehave amap Oil('-O,b) XBa(JtI0) —>U satisfying Q-§(0, X)=X d ;1";tY(5ix)= f(°l(iiX)) [i.e., D1oz(I,x) =f(oz(I,x)), butwewillfrequently use8/BI ord/dt inthis 144 Chapter 5 discussion]. This map oriscalled alocal ﬂow forfin(-—b,b) ><B,,(x0). To picture thismap oz,thebestwecandoistodraw theimages oftheintegral B24(X0) It curves ax.Ify=ax(I0), then theintegral curve axwith theinitial condition a,,(0) =xdiffers from theintegral curve aywith initial condition Q‘.iy(0) =y onlybyachange ofparameter, sothetwoimages overlap. Foreach ﬁxed x,the mapI |—>a(I,x) for~—b<I<bliesalongpart ofthecurve through x.Onthe other hand, ifweﬁxI,then themap x|—>Cd(I,)C) gives theresult ofpushing each xalong theintegral curve through it,foratime interval ofI.Tofocus attention onthismap, wedenote itby¢,»: ¢,(x) =ci(I,x) [-==oIx(I)]. This map qt),isalways continuous. Infact, thewhole fiow oriscontinuous (asa function ofboth Iandx): 4.THEOREM. Iff:U—>R”islocally Lipschitz, then theflow oz:(-—b,b) ><B,,(.x0) —>U given byTheorem 2iscontinuous. PROOF. Letusdenote themap Sdeﬁned intheproof ofTheorem 2byS1, toindicate explicitly theroleofx.Then lltlx"-Syﬂfxll =llsxax "-Syﬂfxll ==IX"-yi- l/lzctor Fields andDgﬁizrential Equations 145 Recall that Ilsa"S13"SbK||°t *"I3||- lfSifdenotes then-fold iterate ofSy,then ||<>a-S;ax||5ma-Sycix||+||-mi-Siva||+---+||$,’§"'<rx -S;ax||l K K"-‘ - --— -. s<1+1> ++05) iny|.4,___,K|x yl Recall alsothatinTheorem Itheﬁxed point 0a,,ofSyisthelimit ofS,’,fo: for anyoz.Hence 0a,.=limS,’§0z,,, soweobtainH-¥OO l O!'"'Ci <i" I*"- . ||XJ»’ll_,_,K| y| Since ||ot,,'"'-Ofyll =sup|ot(I,x)--ot(I, y)|,thiscertainly proves continuity ofoz.+I+ I Ifadditional conditions areplaced upon themap f,then further smoothness conditions canbeproved foroz.Infact, Ff:U—>R"isCk,thentfzeﬂowoti (——b,b) ><B,,(x0) —>UisalsoCk. Unfortunately, thisisavery hard theorem. Aclean exposition oftheclas- sical proof isgiven inLangs Introduction toDﬁerentiable Maniﬁllds (2nd ed.), and arecently discovered proof canbefound inLang, Real andFunctional Analysis (3rded.), pp.37l-379. Inorder toread thishigh—powered proof, youmust ﬁrst learn theelements ofBanach spaces, including theHahn-Banach theorem, and thenreadabout differential calculus inBanach spaces, including theinverse and implicit function theorems (Real andFuncIionaZAnalysis, pp.360—365), butthisis probably easier than reading theclassical proof (and, besides, when you’re ﬁn- ished you’ll alsoknow about Banach spaces, anddifferential calculus inBanach spaces). Wewilljustaccept thisfact. Notice thatthemaps qt),areconsequently C°° iffisC°°. Since themap oz:(-b,b) ><B,,,(x0) —>U satisﬁes ot(0,x) =x,wehave <13{0}><Fa/2(X0) —>Ea/2050) CBa(X0)- 146 Chapter 5 Continuity ofatand compactness of{0}><Ba/2(x0) imply that there issome 6>Osuch that C5:(""3>5) XBa/2 (350) _*Ba(x0)- l B21: (X0) U Ba(X0) [IfxGBa/2(x0), then theintegral curve with initial condition xstays inBa(x0) for|I|<8.] Soif|s|<.9,andxeBa/;_(x0), then thepoint O!(S,X) EB,,,(x0), sowecan alsodeﬁne 3/(I) =O!(I,O!(S,)C)) |I|<8. This satisﬁes 1/(I)=ft:/(1)) 3/(0) =oI(s,x). Wehave alsonoted that ).‘3(I) =Ci(S+I,x), deﬁned for|s+I| <8, satisﬁes 5'0)=f(l3(I)) )3(0) =ot(s,x). Consequently, ﬁ(I) ---:a(I,a(s,x)) for|I|<s.Inother words, if|s|,|I|, |s+I| <8,then oI(I,ot(s,x)) ---:ot(s+I,x). Ifwenow let¢,:Ba/;(x0) —>R"be¢,(x) =ot(I,x) forxeBa/g()C()), wecan say: if|s|,|I|,|s+I|<eandx,¢>,(x) eBa/;(x0), then ¢i(¢:(Il) =¢>i+:(X)- Roughly speaking, ¢I~+-S =¢>;oqt),=<1’),0<,t>,.This shows, inparticular, that for |s|<8each ¢_,-isadiffeomorphism, with inverse ti),-"I =¢>_._,. Everything we have said, since itislocal, canberesaid, without requiring anymore proof, on amanifold. léctor Fields andDﬁrential Equations 147 5.THEOREM. LetXbeaC°°vector ﬁeld onM,andletpEM.Then there isanopen setVcontaining pand ans>0,such that there isaunique col- lection ofdiffeomorphisms <,t>,»:V—>¢,(V) CMfor|I|<swith thefollowing properties: ()¢:(—e,1:)>< V—>M,deﬁned by¢(I,p)=<,t>,(p), isC°°. (2)If|s|,|I|,|s+I|<e,andq,,(q) eV,then\--| ¢'s+r(9') -'=‘Rs°¢'r(9')- (3)IfqeV,then Xqisthetangent vector atI=0ofthecurvet |—>¢>,(q). The examples given previously show thatwecannot expect ¢,tobedeﬁned forallI,oronallofM.Inonecase however, thiscanbeattained. The support ofavector ﬁeld Xisjust theclosure of{pEM:X,,75O}. 6.THEOREM. IfXhascompact support (inparticular, ifMiscompact), then there arediffeomorphisms <,t>,:M—>MforallIeRwith properties (l), (9),(3)- PROOF. Cover support Xbyaﬁnite number ofopen setsV1,...,V,,given by Theorem 5with corresponding .91,...,e,,anddiffeomorphisms of.Lete= min(s|,...,1:,,). Notice thatbyuniqueness, ¢f(q) =¢,’(q)forqEV,-F1V,-.So Wecandeﬁne _ 3 ' .¢t(q) :{¢r(§l) if9'GV: q ifgtatsupport X. Clearly qt):(—s,s) ><M—>MisC°°, and<,t>,.(., --=qt),o<,t>_, if|I|,|s|, |I+s|<8, andeach ¢,isadiffeomorphism. Todeﬁne qt),for|I|3.9,write I=k(s/2) +1- with kaninteger, and |1'|<s/2. Let ¢u (pg/2 0---o(pg/2 0(pr [¢E/2 iterated lflIlI‘l1CS:| f0l‘kZ0 I <,t>.._,/2 o---o<,t>..,,/2 oqt), [¢..,.;/2 iterated -ktimes] fork<0. Itiseasytocheck thatthisisthedesired {¢v,}. '1' 14s Chapter 5 The unique collection {¢,} given byTheorem 6,ormore precisely, themap I|—><,t>,from Rtothegroup ofalldiffeomorphisms ofM,iscalled a1-parameter group ofdiffeomorphisms, andissaid tobegenerated byX.Inthelocal case of Theorem 5,weobtain a“local 1-parameter group oflocal diffeomorphisms”. The vector ﬁeld Xissometimes called the“inﬁnitesimal generator” of{¢,} (vector ﬁelds used tobecalled “inﬁnitesimal transformations”). Condition (3)inTheorem 5canberephrased interms oftheaction ofXq onaC°°function f:M—>R.Recall that dc dcI 5,-to=% -=(faor)- Thus, tosaythatX4isthetangent vector atI=0ofthecurve t|—>¢,(q) amounts tosaying that (Xfm) :Xqf_,,i_)n0ft¢atq)f)—f(4)_ This equation willheused very frequently. The ﬁrstuseistoderive acorollary ofTheorem 5which allows ustosimplify many calculations involving vector ﬁelds, andwhich alsohasimportant theoretical uses. 7.THEOREM. LetXbeaC°°vector ﬁeld onMwith X(p) 750.Then there isacoordinate system (x,U)around psuch that 3XW OI] PROOF. Itiseasy toseethatwecanassume M=R"(with thestandard coor- dinate system I',...,I", say), andp=0eR".Moreover, wecanassume that X(0)=3/BI‘ |0.The idea oftheproof isthatinaneighborhood of0there isa unique integral curve through each point (0,a2,...,a”);ifqliesontheintegral __ _ 4-,’?ii __,_,-f T“""*-——— (0,02 ""'-\-._,_____‘ "'\-uq,-‘-__i ""'\-1-__ _5,; ~ ’/,./"' ________.____________._\\)\ Vector Fields andDgﬁhrmztial Equations 149 curve through thispoint, wewilluse1:12,...,a"asthelastn—1coordinates ofq andthetime interval ittakes thecurve togettoqastheﬁrst coordinate. To dothis,letXgenerate qt),andconsider themap Xdeﬁned onaneighborhood of0inR"by ;((a',...,a") =<,i>,,1(0,a2,...,a"). Wecompute thatfora=(a',. ..,a"), xe(% )(f)==% (fox) =[33,,f;;f<><<a' +/1.-22.....a">>-ftxte))] =)i_{n,%if(¢a1+h(0, at.....-=r">)-ftxto))] -)3,%if(¢1=(X(“)))— ftxt==1))] =tXf)txte))- Moreover, for1'>1wecanatleast compute xe(%0)tf)=-5?-gotfox) =,{1I}},%[ftxt°,---./=,---,0)) -ft0)] =liml[f(0,...,h,...,0)-f(0)]h—+0 /1 ._3f TBI? Since X(0)=3/BI‘ |0byassumption, thisshows that ;(,,,0=Iisnon-singular. Hence x=;(""may beused asacoordinate system inaneighborhood of0. This isthedesired coordinate system, foritiseasy toseethattheequation X,,(3/31') =X0X,which wehave justproved, isequivalent toX=3/3x‘. +I* The second useoftheequation tXf)te)=liml[f(¢>a(P)) -mm l1—>0 /2 ismore comprehensive. The factthatXfcanbedeﬁned totally interms of theClllT€0I'l10I‘])l1iSI'I1S ¢,=,suggests thatanaction ofXonother objects canbe 150 Chapter 5 obtained inasimilar way. Toemphasize thefundamental similarity ofthese notions, weﬁrstintroduce thenotation Lxf f0!‘ Xf. WecallLXfthe(Lie) derivative offwith respect toX;itisanother function, whose value atpisdenoted variously by(L,rf)(p) =LXf(p) ==(Xf)(p) = Xp(f). Now ifwisaC°° covariant vector ﬁeld, wedeﬁne anew covariant vector ﬁeld, theLiederivative oftowith respect toX,by .l (Lxw)(p)=A111},;[(¢i*w)(p) -w(p)]- This istlielimit ofcertain members ofMp*. Recall that ifXPEMp, then (¢i*w)(p)(Xp) =w(¢h(P))(¢h*XP)- Afairly easy direct argument (Problem 8)shows thatthislimit always exists, andthattlienewly deﬁned covariant vector ﬁeld LXwisC°°,butwewillsoon compute thisvector ﬁeld explicitly inacoordinate system, andthese facts will then beobvious. lfYisanother vector ﬁeld, wecandeﬁne theLiederivative ofYwith respect toX, _ l (LXY) 2 Ely}? ""(¢'i1*Y)P]- Thevector ﬁeld¢;,,,,Y appearing here isaspecial caseofthevector ﬁeld oz,,,Y deﬁned atthebeginning ofthechapter, foroz:M—>Nadiffeomorphism andY avector ﬁeld onM.Thus (¢>;,,,.Y)P=¢;,,,,(Y¢__h(p)) isobtained byevaluating Y 3*¢i»"l (P)=¢-4,(p),andthen moving itback topby¢;,*. /P integral curve I*—>MP) Y¢'-!:(P)ofXthrough p Q5-hip)(¢'Ii=r Y)p Thedeﬁnition ofLXYcanbemade tolookmore closely analogous toLgf andLxw inthefollowing way. Ifoz:M—>Nisadiffeomorphism andYisa l/E6101‘ Fields" andDéﬁizrential Equations 151 vector ﬁeld ontherange N,then avector ﬁeld a'*Y onMcanbedeﬁned by (°¢*Y).v =(°!"])*(Y=1(P))- Ofcourse, oz*(Y) isjust(a""),,.Y. Now notice that _1 .Y—(<;i>;,*Y) ,1 ,§1_1g;;[<¢,.*Y)p -YP]=pgP ”~go-,;[Yp -(¢'--k*Y)p] =gin,%tYp~(¢>k*Y)p] =<Lm- Nevertheless, wewillstick totheoriginal (equivalent) deﬁnition. Vilenow wish tocompute LXwand LXYinacoordinate System. The cal- culation ismade aloteasier byﬁrst observing 8.PROPOSITION. IfL);Y;and L);w;exist fori=1,2, then (I)Lx(Yi +Y2)=LXY1 +LxY2, (2)Lx(wi +012) =Lxwi +Lxw2- IfLXY and Lxw exist, then ()LXfY=Xf'Y+f'LXY> (Lxf- =Xf'w+f-Lxw ) w . Finally, ifw(Y) denotes thefunction p|—>w(p)(Y_,,) and LXwandLXYexist, then (5)Lx(w(Y)) =(Lxw)(Y) +w(LxY)->-PKUO PROOF. (l)and(2)aretrivial. The remaining equations areallproved bythe same trick, theoneused inﬁnding (fg)" (x).Wewilldonumber (3)here. <L.t»fY> -limlimo ""(¢h*fY)] p'_I1->0}? P P _ l =,II1i% ,?[f(p)Y,0 T¢'Il=i=(fY)¢_._;,(p)] =31%%tfYp -f(¢--h(P))¢h=i=Y¢.._;,(p)] . l "=,€1_I;%f(p)E[Yp -¢'h*Y¢_;,(p)l +iii-fa ¢,,*y¢__h(p)_ 152 C/zapter 5 The ﬁrstlimit isclearly f(p)-LXY(p). Inthesecond limit, theterm inbrackets approaches liml 2Xﬂp)’ k—+0 -—k while aneasy argument shows that¢;,,,,Y¢,__,,(p) —>Yp.'1' ‘Wearenow ready tocompute LXinterms ofacoordinate system (x,U) onM.Suppose X=XL, a"3/Bx‘. Weﬁrstcompute LX(dxi).Recall (Prob- lem4-l)thatiff:M—>Nandyisacoordinate system onN,then f*(dyl) =ﬁ dxj. jzl xi Wecanapply thisto<;'>;,*, where yisx.Then LX(Mir)=335;,%t<¢».*>dx‘ -dr’(p)] =,l,1_,n},5Z:ﬁzi;-?Q(1>)dX"(p) -'dX‘(p)] -\-_..1: Q: Now thecoefﬁcient ofdxj(p)is -13(Xl°¢t) r__- l3(Ii°¢h) 3(Xi°¢'0) i!lL“@zlW*“‘1"l1lL‘%,;;l ax:”‘P>c at-1‘ml 3 - : : (*)=g ,l,1__)I'%%[(X °¢>1i) -(X°¢>0)l P {this step willbejustiﬁed inamoment} 3 - Ba‘ ='3;PX(X') ='3F(P)- Tojustify (=i=)wenote that themap A(fz,q) =xi(¢»;,(q)) isC°° from R><M toR;thus 32A/Ezlhaxj =32A/Bx]. Biz,which iswhat theinterchange oflimits amounts to. Itnowfollows that,, .. B' . Lxdx' =g5%dx’. Wecould nowuse(2)and(4)ofProposition 8tocompute LXwingeneral, but l/E0102" Fields andDtyifﬁfllttll Eqztatzbns 153 wearereally interested incomputing LXY.Tocompute LX(3/ Bxl)wecould imitate thecalculations ofLXdxi; butthere would beacomplication, because ¢;,,,,onvector ﬁelds involves onemore composition than ¢;,*oncovariant vector ﬁelds. The trick needed todeal with thiscomplication hasalready been used to prove (3),(4),and(5)ofProposition 8,andwecannow use(5)togettheanswer immediately: i i 3 :' 3 :' 3OZLX 51-'-ZLX [dx =(LX SO _ - 8 3’dx' ‘-==—-éi;xj xj thus, 3 HBa‘. 3 L15?"-- Using (3)weobtain . .3 .8 . 3 1"‘? =L J-"i. J "i. LX(I7axj) X17 ax,-l-17 LX(axj) n - n - ,8b1a ,a@=a “Q”wt?-gb inw- Sumniing over jandthen interchanging fandjinthesecond double sum we obtain "",abi ,a@i a ",a ”,a L"‘Y=§(§“ w""’ Hg" W’Y=,§"w~ This somewhat complicated expression immediately leads toamuch simpler coordinate—ﬁ'ee expression forLXY.Iff:M—>NisaC°°function, then Yf isafunction, soXYf=X(Yf)makes sense. Clearly "la “-if .-war .--W I J] :15: 8 _= 3x1 __ Bx3x1 3x131 Thesecond partial derivatives which arise here cancel those intheexpression forY(Xf), andweﬁnd that LXY =XY—YX, alsodenoted by[X,Y]. 154 C/zapter .5 Often, [X,Y](which iscalled the“bracket” ofXand Y)isjust deﬁned as XY—YX; note that thismeans lX>YlP(f) =Xplyfl _YP(Xf)- Astraightforward veriﬁcation shows that [X,Ylptfg) =f(P)[X, Ylplg) +g(P)[X, Ylptf), sothat[X,Y]Pisaderivation atp,andcantherefore beconsidered asamember ofMp. Wearenow inavery strange situation. Two vector ﬁelds LXY and[X,Y] have both been deﬁned independently ofanycoordinate system, butthey have been proved equal using acoordinate system. This sort ofthing irks some people tonoend. Fortunately, inthiscase thecoordinate-free proof isshort, though hardly obvious. lnChapter 3weproved alemma which forthespecial caseofRsaysthata C°°function f:(—s,£) —>Rwith f(O)=0canbewritten f(I)=Est!) foraC°°function g:(—s,s) —>Rwith g(0) =f*'(0), namely 1 g(t)=-A f’(st)ds. This hasanimmediate generalization. 9.LEMMA. lff:(-2,:-:) ><M—>RisC°°and f(0,p) =0forallpEM, then there isaC°°function g:(—s,s) ><M—>Rwith ftw) =Istw) aa—{(0,P) ==s(0,p)~ PROOF. Deﬁne 'ago,P)=(ﬁrstP)d3-~=~ Véctor Fzslds andD§ﬂE’r6?ztz'al Equatioizs 155 l0.THEOREM. IfXand YareC°°vector ﬁelds, then LX1’=[X,Y]. PROOF. Letf:M—>RbeC°°. LetXgenerate <,t>,,|I|<2.ByLemma 9 there isafamily ofC°°functions gronMsuch that f°¢1=f+‘gr 8'0=Xf- Then (¢'h=rY)p(f) ='—“¢h*(Y¢-J';(_P))(.f) =Y¢_,,(p)(f O¢h) ='—"Y¢_i.u>)(f +/18/1), SO 330%tYp—(¢h=i=Y)Pl(f) =gs,,-';t<Yf> —(Yf)(¢--h(P))l —,]i_I%(Y£,’h)(-t(P)) =(LXrm»)~<Yg@> =Xpiyf)—Yp<Xf>-~=~ The equality LXY =[X,Y]=XY—YXreveals certain facts about LXY which arebynomeans obvious from thedeﬁnition. Clearly so Consequently, LXY =—LyX, soLXX =0. Since weobviously have LX(aY1+bY;) =HLX Y1+bLX Y2,itfollows imme- diately that Lisalsolinear with respect toX: L4,;/,+g,_;/21’ =HLXI Y-l—bLX2Y. Finally, astraightforward calculation proves the‘jacobi identity . [X,[Y,Z]] +[Z,[X,Y]] +[Y,[Z,X]] =0. This equation iscapable oftwointerpretations interms ofLiederivatives: (lLx[Y=Zl =[LxY,Zl+ [Y,LxZl, (b)asoperators onC°°functions, wehave L[X,y] =LX0Ly —LyOLX (wllitill might bewritten 21$[LX, Ly]].N 156 Chapter 5 Finally, note thatLXYislinear over constants only, notover theC°°functions 5*“. Infact, Proposition 8,orasimple calculation using thedeﬁnition of[X,Y], shows that [fX,gY] =fg[X,Y]+ f(Xg)Y -—g(Yf)X- Thus, thebracket operation [,]isnotatensor—that is,[X,Y]], does notde- pend only onX],and Y],(which isnotsurprising—what can one dototwo vectors inavector space except take linear combinations ofthemi’), butonthe vector ﬁelds XandY.Inparticular, even ifX],=0,itdoes notnecessarily follow that[X,Y]],=—_~0—in theformula [X,Ylp(f) =Xp(Yf) "*Ypt/Yf) theﬁrstterm X],(Yf)iszero, butthesecond may notbe,forXfmay have a non-zero derivative intheY],direction even though (Xf)(p)20. The bracket [X,Y],although notatensor, pops upinthedeﬁnition ofprac- tically allother tensors, forreasons thatwillbecome more andmore apparent. Before procecding toexamine itsgeometric interpretation, wewillendeavor tobecome more atease with theLiederivative bytaking time outtoprove directly from thedeﬁnition ofLXYtwofacts which areobvious from thedeﬁ- nition of[X,Y]. (l)LXX=0. lfXgenerates qty,itcertainly sufﬁces tosliow that (¢;],,X)],=X],forall/1. Recall that (¢>;,,,X)], =<]i>;,,,X,],__,,(],). Now X,],__],(],) isjust thetangent vector at Xv P 'Y¢'--hip) ¢'--ll time t=-—/itothecurve t|—>gt),(p),andthus thetangent vector, attimer =0, tothecurve I/(Z) 2¢'I--li(p)- Thus ¢v;,,,X],__,,(],)isthetangent vector, attime I20,tothecurve ¢l1° I/(I) "Z¢l:(¢I--it =¢'I Butthistangent vector isjust X],. l/Ector ﬁelds‘ and])§*jTei"en tzialEquations 157 (2)IfX],and Y],areboth 0,then LXY(p) =0. Since X],=0,theunique integral curve cwith c(0) =panddc/dz=X(c(I)) issimply c(z)=p(anintegral curve starting atpcannever getaway; conversely, ofcourse, anintegral curve starting atsome other point cannever gettop). Then Y],=0and (¢h*Y)p :¢/1* Y¢_,_;, (p)=¢'h>i= Yp=¢l:=r0 =0: soLXY(p)=0. Todevelop aninterpretation of[X,Y]weﬁrstprove twolemmas. ll.LEMMA. Letoz:M—>Nbeadiffeomorphism andXavector ﬁeld onM which generates {¢,}. Then a:,,X generates {ozoqt‘),ocz'"'}. PROOF. Wehave (QY»=X)q(f) 2l0‘5*Xa"‘l(q)l(f) =Xa""l(q)(f Oa) =,]i_I:10%l(f°a)(¢li(Ci“l(9'))) —(f=w)tw"'<q>)1 =]i_q,%[f(°l0¢,.Oor‘an—f(q)]-~:+ 12.COROLLARY. Ifoe: M->M,thena,,X ==Xifand onlyif¢>,ooz =o:o¢, forallt. l3.LEMMA. LetXgenerate {¢,} and Ygenerate {gm}. Then [X,Y]—-=0if andonly ifqt),otbs=1,0,oqt),foralls,I. PROOF. If¢;01,0,=1,0],oqt),forallS,then ¢:*Y =YbyCorollary l2.Ifthis istrue forallt,then clearly LXY ==0. Conversely, suppose that [X,Y]=0,sothat . _ l (=l=] 0=]1_m)Z[Y,] -—(<,t>;,,,Y)],] forallq. Given pEM,consider thecurve CI(--6, 6)—>M],given by Z (¢]*Y)p. 158 Chapter 5 Forthederivative, c"(r), ofthismap intothevector space M],wehave do=33;],,';tc<r+/1)~cm] =,§i_,11;,;;t<¢[,+;.,.Y),, -<¢,.r),1 _1=},1_,"},;l¢r*(¢>h»=Y)¢_..tp> "'¢-Y¢_..l .1=¢I*{,]1_,"},;l(¢h*Y)¢-.itp> "'Ya-1uni} =¢r,(0) 11sing(*) with9'=¢’-.=(P) =.0. Consequently c(I)=c(0), so¢,,,Y =Y.ByCorollary l2,¢>,-0tbs=1,0,0qt),for alls,t.'1' Wehave already shown thatifX(p)¢0,then there isacoordinate system x with X=3/3x‘. IfYisanother vector ﬁeld, everywhere linearly independent ofX,then wemight expect toﬁnd acoordinate system with 3 3 ‘*> X-555 Y='w¢—2'However, ashort calculation immediately gives theresult 3 3_0 3x‘’F —’ sothere isnohope ofﬁnding acoordinate system satisfying (*)unless [X,Y]=0. The remarkable fact isthat thecondition [X,Y]=0issqﬂicient, aswell as necessary, fortheexistence ofthedesired coordinate system. 14.THEOREM. IfX1,...,X],arelinearly independent C°°vector ﬁelds in aneighborhood ofp,and[X,,,,X],] =0for15o:,ﬁ 5k,then there isa coordinate system (x,U)around psuch that 3Xa: OI'l U, (IT-.l,...,l€. PROOF Asintheproof ofTheorem 7,wecanassume that M=R",that p=0,and, byalinear change ofcoordinates, that 3 Ci-'=l,...,/{. 0 Wctoz‘ Fields andDgjrerential Equations" 159 IfXC]generates {¢}"}, deﬁne Xby ;((a',...,a") =(pg,(¢ﬁ,(. ..(¢:]‘].(0,.. .,O,a"“*",...,a")) ...)). Asintheproof ofTheorem 7,wecancompute that 3 ,1 mi =,...,k 3 X(0) at],0or 1 it»>13:" 30 min Qlzk-l-l,...,H. Thus x=;(""'canbeused asacoordinate system inaneighborhood ofp=0. Moreover,just asbefore weseethat 3X] Nothing said sofaruses thehypothesis [X,,,,X];] -==0.Tomake useofit, we appeal toLemma 13;itshows that foreach cabetween 1andk,themap Xcan alsobewritten X(a1,...,a”) =¢§a(¢>,:](...(0,...,0,a"‘H,...,a")...)), andourprevious argument then shows that X],= '1' Wethus seethat thebracket [X,Y]measures, insome sense, theextent to which theintegral curves ofXand Ycanbeused toform the"coordinate lines” ofacoordinate system. There isamore complicated, more difiicult to prove, andlessimportant result, which makes thisassertion much more precise. IfXand Yaretwovector ﬁelds inaneighborhood ofp,then forsufiiciently smal; hwecan (1)follow theintegral curve ofX Y through pfortime fr; (2)starting from that point, follow the integral curve ofYfortime I2; (3)then follow theintegral curve ofX X backwards fortime h; p (4)tlien follow theintegral curve ofY backwards fortime /1. YX 160 chum 5 Ifthere happens tobeacoordinate system xwith x(p) =0and 3 3Xx? m——- 3x1’ Y 3x2’ then these steps take ustopoints with coordinates M-"NF.-"W" -/~,_/:3(haoaos so) (/I,/1,0 (0,/1,0, ,0) (0,0,0,...,0), E sothatthis“parallelogram” isalways closed. Even when Xand Yare(linearly independent) vector ﬁelds with [X,Y];é0,theparallelogram is“closed up toﬁrstorder”. The meaning ofthisphrase [anextension oftheterminology “c=3/uptoﬁrst order at0”,which means that c’(0)=3/(0)] isthefollowing. Lett-(ft) bethepoint which step (4)ends upat, ‘c \ ¢‘(/1)=it-/I(¢-h(Wt(¢h(OD)- Then thecuwe cistheconstant curve puptoﬁrstorder, thatis, 15.PROPOSITION. C10)=0. PROOF. Ifwedeﬁne 011(I,/1)=¢i(¢t(P)) 420»/1)=¢>-1(W(¢1»(P))) ‘I39,/1)=W-r(¢-h('#;i(¢h (P)))), then c(t)=tx3(t,t). Moreover, Q3ll 0!2(0=5)=0!1(1=I) (bl a3(0:t):a.-7-(tar) l/Ector Fields andD§*ji'ei'ential Equations 161 andforanyC°°function f:M—>R, (C) mYfoCi| (<1) @ =-—Xfotxg (e) = --Yfoozg while (I) Ow. /1)=Xft<><1<0./1))- Consequently, repeated useofthechain rulegives (f°CH0) =D1(f °<¥3)(9=0) +D2(f °<¥3)(9=9) =D1(f°0!3)(9=0) +[D1(f °<¥2)(9»9) +D2(f° <12)(9, 9)] “Sing (bl =Ditf °<¥3)(9=0) +D1(f°<¥2)(9,0) +[D1(f0cz|)(0,0) +D;z(f o0z|)(0,0)] using (a). Thus, (c),(d),(e),and(f)give .00:0tf0¢')’(0)=~Yf(p) -Xftr) +Yftr)+Xftr) = Whenever wehave acurve c:(--s,.<:) —>Mwith c(0) =pand c’(0) =0E M],,wecandeﬁne anewvector c”(0) ordzc/dt2|0 by c”w)tf> =(fQc>”<0>~ Asimple calculation shows, using theassumption c’(0) =0,thatthisoperator c”(0) isaderivation, c”(0) eM],. (Amore general construction ispresented inProb- leml7.)Itturns outthatforthecurve cdeﬁned previously, thebracket [X,Y]], isrelated tothis“second order” derivation. Until wegettoLiegroups itwill notbeclear howanyone ever thought ofthenexttheorem. Theproof, which ends thechapter, butcaneasily beskipped, isanhorrendous, butclever, cal- culation. Itisfollowed byanaddendum containing some additional important points about differential equations which areused later, andasecond addendum concerning linearly independent vector ﬁelds indimension 2. 162 C/zaater 5 16.THEOREM. mo)=2[X,Y]],. PROOF. Using thenotation ofthe previous proof, since (foc)(I)=(foa3)(t,t) wehave (*)(f0¢)”(°) =131,1(f0<¥s)(0,0) +292,1(f°¢¥s)(0>0) +D2.2(f Q<Ia)(°i9)- Now r""""qas(1) D1,:(f°0t3)(9,0) =D1('—Yf °0t3)(9,0) byll =YYf(P) by Wealso have (2)2D2,1(f °0l3)(0>9) =2D!(“Yf °Q3) by(Pl =2[D1(Yf 0O!2)(0,0) +D2(Yf 00l2)(0,0)] by()andthechain rule =2XYf(r) ~2Dz(Yf<>ou)(0,0) by x2XYf(p) -—2[D1(Yf 0a|)(0,0) +D2(Yf 0c¢])(O, 0)] by()and thechain rule =2XYf(P) "2YYf(P) '"'3XYf(P) bylland(ll23¢" Q3 C‘) Since (b)gives D2(f °0ls)(0,$) =D1(f 00l2)(S,-Y) +D2(f °°t2)(S,S), wehave (3)D2.2(f° 0¢s)(9,9) ==D1,1(f 0O12)(9,9)+2132,: (f0012)(9,0) +D2,2(f °<¥2)(0=0) ZD|(--Xfoczg)(0,0) +2D;_>(-'Xf ocz;_)(0,0) +D2,2(f °0l2)(0»0) by(d) ='-XXf(P) ""2[D1(Xf<>tI1)(0=0) +D2(Xf 00.'1)(0,9)l +D;,;(f 0a;)(0,0) by(d)andthechain rule =XXf(r) -—3YXf(P) -2XXf(r) +D2,2(f °<¥2)(9>0) bl’(C)and(ll l/Ector Fielris‘ andD§*jTerentz'al Equations Finally, from D2(f °<¥2)(0,~Y) =D1(f°<¥1)(S,S) +D2(f °<¥2)($>~Y) (from (all wehave (4) D2,2(f °0i2)(0,0) =131,1(f°¢¥1)(0,0) +2D2,1(f °0l1)(0>9) +D2,2(f °<¥1)(9>9) =YYf(r) +2XYf(r) +XXf(r) by(c)and Substituting (l)-(4) in(=|=)yields thetheorem. '1' 164 Chapter 5 ADDENDUM 1 DIFFERENTIAL EQUATIONS Although wehave always solved diﬁerential equations 3-,;a(t,x) =f(a(t,x)) with theinitial condition a(0,x) =x, wecouldjust aswellhave required, forsome to,that a(t0,x) =x. Toprove this, onecanreplace 0bytoeverywhere intheproof ofTheorem 2, orelsejustreplace orbyI|—>a(t—r0,x). Another omission inourtreatment ofdifferential equations ismore glaring: thedifferential equations a"(t) =f(o:(t)) donoteven include simple equations ofthe form o:"(t) =g(t), letalone equations likeo:"(t) =to:(t). Ingeneral, we would liketosolve equations 3 5~<I(I,x) =f(I,w(I,X)) <X(0,x) ==-"X, where f:(—c,c) xU—>R".One waytodothisistoreplace f(a(t,x)) by f(t,oz(t,.r)) wherever itoccurs intheproof. There isalsoaclever trick. Deﬁne fr(—c,c') ><U—>R"'H by _ f(S,X) =(1,f(»Y,X))- Then there isaflow (6z',6z2) ==ti:(-—b,b) ><W—>R><R"with §<itw.x) =-=ft<i<r.s,x)) oi(0,s,x) =(s,x). Fortheﬁrstcomponent function 6:‘thismeans that 3-1-50: (t,s,x) =1 &'(O,s,x)=--s; hector Fields‘ andDﬁerential Equatz'0n.s' 165 thus c'é'(I,s,x) =3-t-I. Forthesecond component (E2wehave 'a%&2(t:Ssx) =f(&(t>Ssx)) =f(¢i1(I,S,X),<32(I,S,x)) =f(s+ t,6e2(r,s,x)). Then ﬁ(t,x) =ts’-’(r,0,x) isthedesired flowwith grtr.X)-=me(rm) ,8(0,x) =x. Ofcourse, wecould alsohave arranged forB(t0,x) ==x(byﬁrstﬁnding 6: with6z(t0,s,x) =(s,x), notbyconsidering thecurve r|—>ﬁ(t--t0,x)). Finally, consider thespecial case ofalinear differential equation Oftf)=s(t)~<I(t), where gisann><nmatrix-valued function on(a,b). Inthiscase ftwt) ='-"st!)-X- Ifcisanyn><n(constant) matrix, then tc~@¢)’(I)=c-av)=st!)-me) soc-ozisalsoasolution ofthesame diﬁferential equation. This remark allows us toprove animportant property oflinear differential equations, distinguishing them from general differential equations tx"(£) =f(t,cz(t)), which may have solutions deﬁned only onasmall time interval, even iff:(a,b) ><R"->R" isC°°. l7.PROPOSITION. Ifgisacontinuous n><nmatrix-valued function on (a,b), then thesolutions oftheequation 0/<1)=em-wit) canallbedeﬁned on(a,b). 166 Chapter 5 PROOF. Notice that continuity ofgimplies that f(t,x) =g(t) -xislocally Lipschitz. SoforanytoE(a,b)wecansolve theequation, with anygiven initial condition, inaneighborhood ofto.Extend itasfaraspossible. Iftheextended solution orisnotdeﬁned forallIwith to51<1l1,let1|betheleast upper bound ofthesetoft’sforwhich itisdeﬁned. Pick )9with ]3'(t) =g(t) -]6(t) fort near I1 591) 5'59~ Then ]6(t*) 750for1*<£1close enough toI].Hence there iscwith tr-rite)=wt")- Byuniqueness, c-,8coincides with ozontheinterval where they aredeﬁned. lliI.1$czma heextended ast£1asc-,acontradiction. Similarl ormust beY P Y, deﬁned forallIwith a<t5to.'1' l/eater Fieltls andDg']§?2:erztz'al Equations 167 ADDENDUM 2 PARAMETER CURVES INTWO DIMENSIONS Iff:U—>Misanimmersion from anopen setUCR"into ann—dimen- sional manifold M,thecurve I|—>fl[a1,...,a,-__1,t,a,-,,.1,...,a,,) iscalled a parameter curve inthe1'‘hdirection. Given nvector ﬁelds X1,...,X,,defincd inaneighborhood ofpEMandlinearly independent atp,weknow thatthcre isusually noimmersion fiU—>Mwith pGf(U), whose parameter curves intheithdirection aretheintegral curves oftheX,-—for wemight nothave [X,-,X]-] =0.However, wemight hope toﬁndanimmersion fforwhich the parameter curves intheithdirection liealong theintegral curves oftheX]-,but have di1"ferent parameterizations. Asimple example (Problem 20)shows that even thismodest hope cannot befulﬁlled indimension 3. Ontheother hand, inthespecial case ofdimension 2,such animbedding canbefound: l8.PROPOSITION. LetX|,X; belinearly independent vector ﬁelds ina neighborhood ofapoint pina2-dimensional manifold M.Then there is animbedding f:U—>M,where UCR2isopen andpef(U), whose id‘ parameter lines liealong theintegral curves ofX,-. PROOF. Wecanassume that p==0ER2,and that X]-(0) =(e,-)0. Every point qinasufﬁciently small neighborhood of0isonaunique integral curve OfX1through apoint (0,x2(q))—we proved precisely thisfactinTheorem 7. Similarly, qisonaunique integral curve ofX;through apoint (x'(q),0). -Y2(<1)ll q IT it‘(<1) The map q|—>(xl(q),.\'2(q)) isC°°, with _]acobian equal toIat0(these facts alsofollow from theproof ofTheorem 7).Itsinverse, inasufliciently small neighborhood of0,istherequired diffeomorphism. '2' 168 Chapter 5 Vilecan always compose fwith amap oftheform (x,y)|—>(a(x), B(y)) fordiﬁfeomorphisms orand BofR,which gives usconsiderable ﬂexibility If, forexample, CCR2isthegraph ofamonotone function g,then themap C . (x,y)|—>(x,g(y)) takes thediagonal {(x,x)}toC.Moreover, foranyparticular parameterization c=(c;,c'2): R—>R2ofC,wecanfurther arrange that (‘(1') maps to(e(t),c(r)), bycomposing with (x,y)l—>(c]""(x), y).Consequently, wecanstate l9.PROPOSITION. LetX;,X; belinearly independent vector ﬁelds ina neighborhood ofapoint Pina2—dimensional manifold M,and letcbea curve inMwith c(0)=pandc’(z) never amultiple ofX;orX2.Then there isanimbedding f:U—>M,where UCR2isopen andpGf(U), whose ilh parameter lines liealong theintegral curves ofX]-,andforwhich f(t,I)=c(r). l/E0102‘ Fields and1)§ﬁ§;>:'e?ztz'al Equalions 169 PROBLEMS 1.(a)Ifozi M—> Nis C°°, then 05*:TM—> TNisC°°. (b)Ifoz:M—>Nisadiffeomorphism, and XisaC°°vector ﬁeld onM,then o:,,,X isaC°°vector ﬁeld onN. (c)If0::IR—>Ris01(2) =:3,then there isaC°°vector ﬁeld XonRsuch that a:,,.X isnotaC°°vector ﬁeld. 2.Find anowhere 0vector ﬁeldon1Rsuch thatallintegral curves canbedeﬁned onlyonsome interval around 0. 3.Find anexample ofacomplete metric space (M,p)andafunction f:M—> Msuch thatp(f(x),f(y)) <1p(x,y) forallx,y eM,butfhasnoﬁxed point. 4.Letf:(~—c,c) xU><V—>R"beC°°,where U,VCR”areopen, andlet (xo,yo)EU><V,Prove thatthere isaneighborhood Wof(xo,yo)andanum- berb>0such that foreach (x,y) EWthere isaunique cc=a(x,y): (--b,b) —> Uwith cz'(r) EVforIE(--b,b) and 0¢”(f) =f(r,0¢(I)=0¢'(f)) 01(0) =x (1/(0) =y. Moreover, ifwewrite a(x,,.)(t) =a(r,x,y), then oz:(--b,b) ><W—>UisC°°. Hint: Consider thesystem ofequations 0/(I)=5(1) ﬁW)=fmwvLMO) 5.Wesometimes have tosolve equations “depending onparameters”, 3 (*) 5<r(r.y.>~') =f(I,ynrtr. y.X)) M0.y.I)=1', where f:(--c,c) ><VxU—>R",foropen UCIR"andVCRm,andweare solving fora(_,,,x): (--Z7, b)—>Uforeach initial condition xand “parameter” y. Forexample, theequation oft!)=wt!) 01(0)==X, 170 C/zapter 5 with solution cx(r) =X83”, issuch acase. (a)Deﬁne _ f:(—-c,c)><V><U—>lR”‘><lR" by _ ftrmr) =(0,f(r.r.X))- If(6z‘,6z2) =6::(-b,b) ><W—>Rm><IR"isaﬂowforfinaneighborhood of (y0,x0), sothat a_ _-_a,v¢(r.y.X) f(r.<>1(r.y,>v)) &(0$ y! A’) == (J?! x)! show thatwecanwrite '50,)/»>~') 7'(y’a(!:y»x)) forsome oz,andconclude thatcasatisﬁes (=|=). (b)Show thatequations oftheform 3(=l==!=) 5oz(r,x) =f(z,x,a(r, x)) o:(0,x)=x canbereduced toequations oftheform (=t=)(and thus toequations %<1(r.x)= f(<1(r,x)), ultimately). [VVhen oneproves that aCkfunction f:U—>R"hasaCkﬂow oz:(-b,b) ><W—>U,thehard part istoprove that iffisC‘, then oris diﬁlzreiitiable with respect tothearguments inW,andthatifthederivative with respect tothese arguments isdenoted byD201, then (***) D;D;o:(r,.r) =D;f(a(r,x)) -D;o:(r,x) (aresult which follows directly from theoriginal equation D;a(:,x) =f(a(z,.\')) iffisC2,since D;D2 =DZD1). Since (=t==i==t=) isanequation forD20: ofthe form (am), itfollows that D20: isdiHCI‘€l1[i&blC ifDgf isC‘,i.e.,iffisC2. Diﬁerentialuility ofclass Ckisthen proved similarly, byinduction.] l/Ector Fields and1)zﬁérerztz'al Erjuaiiorzs 171 6.(a)Consider alinear diﬁerential equation 05(1) =3'(I)°((f), where g:IR—>R,sothatwearesolving forareal-valued function ca.Show thatallsolutions aremultiples of av) 2efg(r)dr where fg(r) dzdenotes some function Gwith G'(r) =g(one canobtain all positive multiples simply bychanging G).The remainder ofthisproblem inves- tigates theextent towhich similar results hold forasystem oflinear diﬁerential equations. (b)LetA=(a,-J-) beann><nmatrix, andlet|A|denote themaximum ofall |ﬂgj|. Sl'l0W ll‘l3.l IA+Bl5.I/4|+ IBI IABI sﬂl/1! -IBI- (c)Conclude that theinﬁnite series ofn><nmatrices A A2 A3 A4 converges absolutely [inthesense that the(i,j)‘h entry ofthepartial sums converge absolutely foreach (i,j)]anduniformly inanybounded set. (d)Show that exp(TAT"l) =T(expA)T"'}. (e)IfAB=BA, then exp(A +B)=(exp A)(exp B). §“;t4_+_W_(§£)(§§5)+Rp=0 _- p=O p=0 NHint: Write andshow that IRNI —>0asN—>00. (Y)(exp A)(exp --A) ==I,soexpAisalways invertible. (g)The map exp, considered asamap exp: lR"2—>lR“2, isclearly diﬁerentiable (itiseven analytic). Show that ¢><p’<@> =B(=¢><p<@>~B)- (Notice thatfor|A|,theusual norm ofAGR"2,wehave |A|5|A|5n|A|.) 1'72 Chapter 5 (h)Use thelimit established inpart (g)toshow that exp’(A)(B) =exp(A) -B ifAB=.-BA. (i)LetA:R—>lR"2bediﬁerentiable, andlet 3(1)=-'@><P(A(I))- IfB’(r) denotes thematrix whose entries arethederivatives oftheentries ofB, show that B10=/11:)-¢><P(/10)). provided thatA(r)A'(r) 2A’(r)A(r). (This isclearly true ifA(s)A(r) 2A(t)A(s) foralls,r.) (j)Show thatthelinear differential equation oft!)=gt!)-01(1) hasthesolution I o:(r)=exp(ig(s)ds) provided thatg(s)g(r) =g(t)g(S) foralls,r.(This certainly happens when g(t) isaconstant matrix A,soevery system oflinear equations with constant co- eﬁicients canbesolved explicitly—the exponential of[Jg(s) ds=-.zAcanbe found byputting Ain_]ordan canonical form.) 7.Check that ifthecoordinate system xisx=)("'i, forX:R"—>M,then X=3/3.1" isequivalent to;(,,,(3/31') =X0X. 8.(a)LetMand NbeC°°manifolds. ForaC°°function f:M><N—>R andgreN,letf(-,q)denote thefunction from MtoRdeﬁned by PI—>f(19,17)- If(x,U)isacoordinate system onM,show thatthefunction 3f/Bxi,deﬁned by 5-,=(P,q) ="-§"'(P), isaC°°function onM><N. (b)If<35:(——s,e) ><M—>Misal—parameter group ofdiffeomorphisms, show thatforevery C°°function f:M—>R,thelimit ;li_I;T:)'fi1'lf(¢h(P))'" ma] litter Fields andDﬂrrezztial Equatzi0rz.s‘ 173 exists, anddeﬁnes aC°°function onM. (c)If¢*: (—£,£) ><TM —>TM isdeﬁned by ¢*(t= U)=¢r=|=(v), show that¢,..isC°°,andconclude thatforevery C°°vector ﬁeld Xandcovari- antvector ﬁeld 0)onM,thelimit ggno';;l(¢h*w)(-Yp) -we->1 exists anddeﬁnes aC°°function onM. (d)Treat LXY similarly. 9.Give theargument toshow that ¢;,,,.Y¢_,,(_,,; —>Ypintheproof ofProposi- tion 8. 10.(a)Prove that L,-((f-oa)=Xf-cv+f-L,-(co LXlfv(Y)l =(LXw)(Y) +¢v(LXY)- (b)How would Proposition 8have tobechanged ifwehaddeﬁned (LXY)(p) as _1 ,ig_1_~,_no;.;t<¢t..Y>,., —11,1? 11.(a)Show that ¢*(df)(Y) =Y(f°¢)- (b)Using (a),show directly from thedeﬁnition ofLythat forYGMp, lLX df(P)l(Yp) zYp(LXf), and conclude that LXdf=d(L,vf). The formula forLXdxi,derived inthetext, isjustaspecial case derived inan unnecessarily clumsy way. Inthenext part wegetamuch simpler proof that L,-(Y =[X,Y],using thetechnique which appeared intheproof ofProposi- tion l5. (c)LetXand Ybevector ﬁelds onM,and frM—>IRaC°°function. IfX generates {¢,}, deﬁne oz(t,h) ==Y¢_,(,,)(f 0¢;,). 1'74 Chapter 5 Show that D;0z(O,O) =—Xp(Yf) D20:'(O, 0)= Conclude thatforc(h)=o1(h,h)wehave —~C"(0) =LxY(P)(f) =[X,Y]p(f)- 12.Check theJacobi identity. 13.OnR3letX,Y,Zbethevector ﬁelds 3 3X=_--23y ‘V32: 3 3Y=—.'Z— '— 3x+A32 3 3Z=—————.y3x x3y (a)Show that themap aX+bY +cZ v—>(a,b,c) eR3 isanisomorphism (from acertain setofvector ﬁelds toR3)andthat [U,V]|—> thecross-product oftheimages ofUand V. (b)Show thattheﬁowofaX+bY+cZisarotation ofR3about some axis through O. 14.IfAisatensor ﬁeld oftype onNand¢:M—>Nisadiffeomorpliism, wedeﬁne ¢*A onMasfollows. Ifv|,...,vkGMp, and M,...,K;GMp*, then l¢*A(P)](v| ,---Mt,M,---,1!) =A<¢u»))t¢.v.....,¢.vi. t¢">*i......t¢*'>*m. (a)Check that under theidentiﬁcation ofavector ﬁeld [orcovariant vector ﬁeld] with atcnsor ﬁeld oftype [ortype thisagrees with ourold¢*Y. (b)Ifthevector ﬁeld XonMgenerates {¢>,}, andAisatensor ﬁeld oftype onM,wedeﬁne (LX/Dip) =,p;n0;1;t<¢,.*A> -Arm]. l'Z'ct0r Fietdr and1)gﬂ.3:re2z!z'al Equrztimts 175 Show that LX(/1 -l"B)=LXA -l"LXB Lxl/1®B)=(LX/1) ®3+Aone (sothat Lx(f/1) =X(f)/4 +f-LXA)> inparticular). (c)Show that LX,.;.X2A =LX,/1 +LX2/1. Hint: Wealready know thatitistrue forAoftype (E),(ii), (d)Let C:rm/>—> I/:;'tv> beanycontraction (CT)(v1,---,v1<-|,l|,---Jr-1) =contraction of (v,k) |—>T(v|,...,v,,,_.|,v,v,,,_;.|,...,vk_|,k|,...,}\5_|,l,l5+|,...,)\;__;). Show that LX(CA) =C(L,vA). (e)Noting thatA(X| ,...,Xk,w|,...,w;)canbeobtained byapplying contrac- tions repeatedly toA®X|®---®X;,®w|®---® 0);,use(d)toshow that LX(A(X;,...,X;<,w|,...,w;)) =(LX/1)(X|,...,Xk,(z)1,...,(z),~') k +Z:A(X;,...,L,vX,-,...,X;,,w|,...,w;) i=1 I +Z:A(X1,...,X;,,w;,...,LXw;,...,w;). t'=I . . n I n (f)IfAhascomponents A,il‘_'_'_',ﬁ inacoordinate system xandX=Z0'3/3x’, i=1 show thatthecoordinates ofLXAaregiven by :1 J’!---1'! k rz - J1---J! _ 'It---11¢ J1---J -11.! +1---J! (LXA)i1---is _Ea‘? _2Z:Ail---1'1.-G U 3,1-1' 1'=| cz=| j=| I rz '-- 30’+2 :A!""." .. ._,-_1]..-Ja_]llQv_|_]-.Jk ax;a, otwl :'=l 176 Chapter 5 15.LetDbeanoperator taking theC°° functions Fto37,and theC°° vector ﬁelds '17toV,such that D:37—>Fand D:'17—>'17arelinear over IRand D(fY)=f-DY+Df-Y. (a)Show that Dhasaunique extension toanoperator taking tensor ﬁelds of type tothemselves, such that Dislinear over IR (D(A®B)=DA®B+A®DB (foranycontraction C,DC=CD. t_>oy~Q’I"'_.‘-u_t/‘-u_/‘-u_/ Ifwetake Df=Xf and DY =L,-(Y, then thisunique extension isLy. (la)LetAbeatensor ﬁeld oftype sothat vgecanconsider A(p) GEnd(_/lép); then A(X) isavector ﬁeld foreach vector eldX.Show that ifwedene D,;f =O,DAX ==A(X), then DAhasauni-que extension satisfying (l),(2), and (3). (c)Show that (DAw)(1>)=—A(1>)*(w(P))- (d)Show that LIX=fLx —DX®df- Hint: Check thisforfunctions andvector ﬁelds ﬁrst. (e)IfTisoftype show that I’! H N (DAr)jj'.-=Zr,j”.4f, +Z:r,j".4,{ -ZT3343. aml 0:=| ct‘-=-I Generalize totensors oftype 16.(a)Letf:IR—>IRsatisfy f"(O) =O.Define g(t‘) = fort‘ 3O.Show that theright-hand derivative I .(/)— (0) f”(0) so ’1.g2~(Use Taylor’s Theorem.) (I3)Given ctIR—>Mwith c'(O) =OEMp, deﬁne 7/(I) = forIZO. Show thatthetangent vector c”(0) deﬁned byc”(O)(f) _-=(f0c)”(O) canalso bedescribed byc”(O) =2)/"(0). Vector Fields and1)§ﬂi*:'er2tz'a[ Eqzratiurzs“ 1'77 17-(a)Letf:M—>IRhave pasacritical point, sothat ft),=O.Given vectors Xp’Y),eMp, choose vector ﬁelds X,Ywith X},=X),and Y),=Yp. Deﬁne f..t/Y...Y,»=Xptrfa. Using thefactthat [X,Y]_,,(f) =0,show that f,.,,(X_,,, Yp)issymmetric, and conclude thatitiswell—deﬁned. (b)Show that rz rz g8 rz __32]- f i" , bl i. )3 Gib)"-?T(p). fml P 3x1P wig, 3a3x1 (c)The rank of(32f/3x"3x1' (p))isindependent ofthecoordinate system. (d)Letf:M—>Nhave pasacritical point. ForXp,Y),eMandg:N—>IR deﬁne ~~ f|=*(X: Y)(g) =XP(Y(g ° Show that fr-*5 MPXMP_*Nftp) isawell-deﬁned bilinear map. (e)Ifc:IR—>Mhas0asacritical point, show that takes (1g,lg) tothetangent vector c”(O) deﬁned byc”(O)(f) =(f0c)”(O). 18.Letcbethecurve ofTheorems I5and I6.Ifxisacoordinate system around pwith x(p) =O,and rz _8 Z (iii? :| ‘D 3x,, show that f@UD=aWL+MFL where 0(t2) denotes afunction such that 1m0Myﬁ=ur—>0 19.(a)IfMiscompact andOisaregular value off:M—>IR,then there is aneighborhood UofOeIRsuch thatf"(U) isdiffeomorphic tof'"'(0) ><U, 178 Chapter 5 byadiffeomorphism ¢>:f"(O) ><U—>f"(U) with f(¢(p,t)) =I.Hint: Use Theorem 7and apartition ofunity toconstruct avector ﬁeld Xona neighborhood off‘i(O) such thatf,,X =d/dt. (b)More generally, ifMiscompact andqeNisaregular value off:M—> N,then there isaneighborhood Uofqand adiffeomorphism ¢:f‘! (q)XU—> f“(U)withf(¢(P=q’)) =q’- (c)Itfollows from (b)thatifallpoints ofNareregular values, then f_l(q1) andf"(q2) arediffeomorphic forqhq; sufﬁciently close. Iffisonto N,does itfollow that Misdiffeomorphic tof"(q) xN? 20.InIR3,letYand Zbeunit vector ﬁelds always pointing along they-and :—axes, respectively, andletXwillbeavector ﬁeld oneofwhose integral curves isthex-axis, while certain other integral curves areparabolas intheplanes y=constant, asshown intheﬁrstpartoftheﬁgure below. Using thesecond part oftheﬁgure, show thatProposition I8does nothold indimension 3. Z ,P\ J, a I +~+ r—~> >> Jt _ / 0 CHAPTER 6 INTEGRAL MANIFOLDS PROLOGUE Amathe1natician’s reputation rests on Beauty istheﬁrsttest: there isno thenumber ofbadproofs hehasgiven. permanent place intheworld for [Pioneer work isclumsy] ugly mathematics. A.S.Besicovitch, quoted inE.Littlewood, G.H.Handy, AM'at/zernaiiciarfs /ldzlrcellany AMal/zernalicianir /1ﬂ0[0g_’)‘ Intheprevious chapter, wehave seen that theintegral curves ofavector ﬁeld onamanifold Mmay bedeﬁnable only forsome small time interval, even though thevector ﬁeld isC°°onallofM. W'ewill now vary ourquestion alittle, sothat global results canbeobtained. Instead ofavector ﬁeld, sup- pose thatforeach pGMwehave al—dimensional subspace APCMp. The function Aiscalled aI-dimensional distribution (thiskind ofdistribution has nothing whatsoever todowith thedistributions ofanalysis, which include such things asthe“ti-function”). Then Aisspanned byavector ﬁeld tocalty; that is, wecanchoose (inmany possible ways) avector ﬁeld Xsuch that O#XqGAq forallqinsome open setaround p.WecallAaC°°distribution ifsuch a vector ﬁeld Xcan bechosen tobeC°° inaneighborhood ofeach point. ForaI-dimensional distribution thenotion ofanintegral curve makes no sense, butwedeﬁne a(1-dimensional) submanifold NofMtobeanintegral manifold ofAiffor every [JGNwehave f,,.(N_,,) =AP where iiN—>M istheinclusion map. Foragiven pGM,wecan always ﬁnd anintegral manifold NofaC°° distribution Awith pGN;wejust choose avector ﬁeld Xwith O79XqGAq forqinaneighborhood of[2,ﬁndanintegral curve c‘ofXwith initial condition 0(0) =p,andthen forget about theparameterization ofc,bydeﬁning Ntobe {c(I)}. This argument actually shows that forevery pGMthere isacoordinate system (x,U)such that foreach ﬁxed setofnumbers az,...,a", theset {I1EUIrztq)=H2.---.x"(q) =r1”} 179 180 Chapter 5 isanintegral manifold ofAonU,and that these aretheonly integral manifolds inU. This isstillalocal result, butbecause wearedealing with submanifolds, rather than curves with aparticular parameterization, wecanjoin overlapping integral submanifolds together. The entire manifold Mcanbewritten asa disjoint union ofconnected integral submanifolds ofA,which locally look like --i-i-.--.-"- (rather than like .--.._..--i._..._. -___.i,,,_ ' icated. Forexample, there isadistribution mfolds alllook likethedense l—clin1ensionalorsometlnng even more compl ) onthetorus whose integral ma'. submanifolcl pictured inChapter 2.Ontheother hancl, there isadistribution onthetorus which hasonecompact connected integral manifold, and allother integral manifolds non-compact. Ithappens thattheintegral manifolds ofthese twodistributions arealsotheintegral curves forcertain vector ﬁelds, butonthe 1zztq.<,tral Jl-fanphtdr 131 Mobius strip there isadistribution which isspanned byavector ﬁeld only locall )4 ,ﬂ|||t|IlIII IIlIIIlIlIlIl](|. \'\leareleaving outthedetails involved inﬁtting together these local integral manifolds because wewilleventually dothisover again inthehigher dimen- sional case. Forthemoment wewillinvestigate higher dimensional cases only locally. Alt'~dimensional distribution onMisafunction p|—>AP,where APCMP isak—dimensional subspace ofMP. ForanypGMthere isaneighborhood U andkvector ﬁelds X,,. ..,Xk such that X|(q),. ..,X;P(q) areabasis forAP, foreach qGU.WecallAaC°°distribution ifitispossible tochoose C°° vector ﬁelds X1,...,XPwith thisproperty, inaneighborhood ofeach point p. A(lc-dimensional) submanifold NofMiscalled anintegral manifold ofAif forevery pGNwehave t'...(NP) =AP where 1':N—>M istheinclusion map. Although tlicdeﬁnitions given sofaralllook thesame asthel-dimensional case, theresults will look very different. Ingeneral, integral manifolds donot exirl, even locally. Asthesimplest example, consider the2-dimensional distribution AinIR3for which AP--:A(P,P,P; isspanned by -2- +bi and 3- .3xP 3::P 3}‘P Thus 3 3 3AP={r—-- +s—-- +bt'—_ :r,sGIR}. 3:.P 3yP 3;P Ifwe identify TIR3 with IR3><IR3,then APconsists ofall(r,s,br)P. Thus AP may bepictured astheplane with theequation 2-c=b(x—-a). 182 Chapter 5 Theﬁgure below shows APforpoints [2=(a,b,O). Theplane A(,,,,;,,c; through (a,b,c-) isjust parallel totheonethrough (a,b,O). _ __i /.///4,//;/ 4// yaw ya\\\*:\ \ Ifyoucanpicturc thisdistribution, youcanprobably seethat ithasnointegral manifolds; aproof can beQven asfollows. Suppose there were anintegral manifold NofAwith OeN.Theintersection ofNand{|[0,y,z)} would bea curve yinthe(y,:)-plane tlirougli Owhose tangent vectors would have tolie intheintersection ofA(@,,.,,; andthe(y,2)-plane. The only such vectors have third component O,so7/must bethey-axis. Now consider, foreach ﬁxed yg, theintersection NO{(x,_vg, 2)}.This willbeacurve intheplane 1[(x,y@,z)} through (O,y@,O), with alltangent vectors having slope yo,soitmust bethe linc {[X, yo,y0x)}. Our intcgral manifold would have tolook likethefollowing picture. Butthissubmanifold cloes notwork. Forexample, itstangent space at (I,O,O)contains vectors with third component non-zero. // Inlegml ﬂ4an§@Zds 133 Toseeingreater detail what ishappening here, consider thesomewhat more general case where A(,,,;,,,,) =APis 3 3 3AP :{T5'; p'l' S5; P+[?'_f((I,b)‘l'Sg((I,b):| g PIi',S E geometrically, APistheplane with theequation ZT“Czf(a:b)(x -T(I) T” Asintheﬁrstexample, theplane A(,,,;,,c) through ((1,b,c) willbeparallel to theonethrough (a,b, O),since fandgdepend only onaandb. \'Venow askwhen thedistribution Ahasanintegral manifold Nthrough each point. Since APisnever perpendicular tothe(x,y)-plane, thesubmanifold is given locally asthegraph ofafunction: N={(X,y.-r-J II=01(I,y)}- / Now thetangent space atp==(a,b,01(a, b))isspanned by 3 301 3_.... _ ,5_ 3xP+3x(a )8;: a +31 MiByp 3y(a’ 32 These tangent vectors areinAPifand only if 3 rm»)=-‘~”-(ma,3x 3 gtmn=altamn.J’ Soweneed toﬁnd afunction 0::R2—>IRwith 301 3 l*l E=f= i=8- 184 Chapter 5 Itiswcll-known that thisisnotalways possible. Byusing theequality ofmixed partial derivatives, weﬁnd anecessary condition onfandg: af3g Inourprevious example, 3 f(a1b):=b: ix]: 3y g(a:b):0: ' _"-:0: sothisnecessary condition isnotsatisﬁed. Itisalsowell-known thattheneces— sziry COI1Clili0I1 (=i==e=)issziﬂirient fortheexistence ofthefunction orsatisfying (=i=)in aneighborhood ofanypoint. 0.PROPOSITlON. Iff,g: R2—>IRsatisfy 3f as(**) =5";1 inaneighborhood ofO,and :9eIR,then there isafunction oz,deﬁned ina neighborhood ofOGR2,such that oe(0,0) =:0 30: [=i=) 3x-Tf 30:__ 3yHgi PROOF. Weﬁrstdeﬁne 0z(x,0) sothata(0,0) =:0and tn §~§<>:,o> =./ti-.0); _3“!£*°> A namely, wedeﬁneX 0z(,\',0) =:0+f f(r,0)dr. 0 Integml zléfaizyfolds 185 Then, foreach x,wedeﬁne 0e(x, y)sothat ——x =g.1} lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllnamely, wedeﬁne 1' @1(><,y) --=01(>~',0)+f gt/1',I)dr0 x y =.:@+f f(r,0)dr+~/Q g(x,r)dr. 0 o This construction does notuse(=i=*), andalways provide uswith anorsatisfy— ing(2),30:/3y :g.Weclaim thatif[>i==:=)holds, then also30:/3x =f.Toprove this, consider, foreach ﬁxed x,thefunction yI—>gi-f(x,y) —f(X,y)-It This is0fory=0by(I).Toprove that itequals 0forally,wejust have to show that itsderivative isO.But itsderivative atyis 3101 af a30: af5~5;(A.y) —§};(x,y) -—5; (my)—a—~y(x,y) 3 3 =git-i-,y> -ai;<x.y> by(2) =Oby(ms). '1' Wearenow ready tolook atessentially themost general case ofa2-dimen- sional distribution inR3: 3 3 3 Ap={r5~;p+S53-I-p+[I'f(P)-l'Sg(]J)]$pIIQSGIR}, where f,g: R3—>IR.Suppose that N={(x,y,z) :2=a(x,y)} 186 C/tapter 6 isanintegral manifold ofA.The tangent space ofNatp=(a,b,0e(a,b)) is spanned, once again, by 3 30: 3_ _.._ ,1;_._. 3x,,+3x(a )3z a 301 a_""" ab """' ay, 3y”lag These tangent vectors areinAPifandonly if 30: f(H.b,v1(-4.5)) =gtrub). (*) 301 g(aa biGT0‘: I5;(a: b)‘ Inorder toobtain necessary conditions fortheexistence ofsuch afunction 0:, weagain usetheequality ofmixed partial derivatives. Thus (=t=)andthechain rule imply that 2 ii"-tab) =9-{<a.b.a<a.b>> +aiY~.b,a<~,b>> -aim11>3y3x 3y 32 3y 32 3 3 3 -_9’»<~.b> =5-<~.b.a<~.b>> +-§i<a.b.<i<a.b>> --°i<a,z>>.3.x3y 3x 3; 3y This condition isnotvery useful, since itstillinvolves theunknown function 0:, butwecansubstitute from [>:=)toobtain 3 3 ,,—£(~.b.aw.b>> +,—{<a.b,<i<a.b>> -g<~,b.<it~.b>> =§~'§;w.b.a<a.b>> +%‘§w,b.<i<~.b>> -nu.b.a<~.b>>. Now wearelooking forconditions which willbesatisﬁed byfand gwhen there isanintegral manifold ofAthrough everypoint, which means thatforeach pair(a,b)these equations must holdnomatter what a(a,b)is.Thus weobtain ﬁnally thenecessary condition ifif-95 asl**) 3y+3z g”a>é+az'f' Izzregral Jkfanﬂfnlcir 187 Inthismore general case, thenecessary condition again turns outtobesuf- ﬁcient. Infact, there isnoneed torestrict ourselves toequations forasingle function deﬁned onR2;wecantreat asystem ofpartial differential equations for1:functions onRm(i.e., apartial differential equation forafunction from Rm toR"). Inthefollowing theorem, wewilluseItodenote points inRmand x forpoints inR";soforafunction f:Rm><R"—>Rf‘weuse 3f§ f0r Dff, 3faxi f0!" Dm+f f. 1.THEOREM. LetU><VCRm><R"beopen, where Uisaneighborhood of0eRm,andletfi:U><V—>R"beC°°functions, for2'=l,...,m. Then forevery xGV,there isatmost onefunction oz:W—>V, deﬁned inaneighborhood WofOinRm, satisfying 01(0) =x (ii) go)=f,-(i,a(i)) forallIew. (More precisely, anytwosuch functions 0:1and01;,deﬁned onW1andW2,agree onthecomponent ofW|flW;which contains O.)Moreover, such afunction exists (and isautomatically C°°)insome neighborhood Wifandonly ifthere isaneighborhood of(O,x) eU><Vonwhich 3f‘ aft H3f‘ Hafi ..[*>l=) T';—$+EaT';cﬁk—Zﬁ_)3'k=0 l,_]=I,...,I7’I. k=:1 k=1 PROOF. Uniqueness willbeobvious from theproof ofexistence. Necessity of theconditions (>z==:=)islefttothereader asasimple exercise, andwewillconcern ourselves with proving existence ifthese conditions dohold. The proof willbe likethatofProposition O,with adifferent twist attheend. Vilefirst want todeﬁne o1(t,O, ...,O)sothat 0e(0,0,...,O) =x (1) a£(I,0,...,O)=ﬁ(I,O,...,0,0((I,O,...,O)). 183 Chapter 5 Todothis,weconsider theordinary differential equation 151(9) =X ﬁ1’(r)=f1(r.0..._.0,ﬁ1(r)). This equation hasaunique solution, deﬁned forltl<2;.Deﬁne w(r,0,....0) =1510) lrl<£1. Then (l)holds forlrl<2;. Now foreach ﬁxed I‘with lt‘|<s1,consider theequation =0z(t1,0,...,0) :f2(I1:t=0:---=O!ﬂ2(r))' This hasaunique solution forsufﬁciently small I.Atthispoint thereader must refer back toTheorem 5-2,andverify thefollowing assertion: Ifwechoose 2; suﬁiciently small, then forit‘;<2|thesolutions oftheequations for,6;with theinitial conditions 132(0) =oz(t‘,0, ...,0)willeach bedeﬁned forit}<£2for some £2>0.Wethen deﬁne'¢'1>‘0:>"*1ﬁg\\-n-/\\-u-/ o1(r‘,r,0,...,0)=,62(z) |t'l<s;,lt|<a;. Then 01(0,0,0,...,0)=x 3°’ | | |(2) F0 ,r,0,...,O) =f;(r ,r,0,...,0,a(z ,r,0,...,O)) 1:‘:<£|,lrl <81- W’eclaim that foreach ﬁxed I1with |I|]<2|wealso have, forallIwith |I|<£2, 3°‘1 1 1(3) O=g(t‘)=- F0 ,r,0,...,O)— f|(r,t,0,...,0,0z(t ,r,0,...,O)). Note ﬁrstthat (4) g(0)=0by(1)- Wenowderive anequation forg’(t). Inthefollowing, allexpressions involv~ ingoraretobeevaluated at(I1,t,0,...,0)andallexpressions involving j}are tobeevaluated at(t1,I, 0,...,0,0z(t1, I,0,...,0)). W'ehave 320: afl"af,30:!’ I T; T; g(fl3I23t'1 an,2askail’ Integral Manﬁicir 189 and thus , a30: af, af,,,(5)8(F)=§‘,T(5§)—g2'—l§;é?f2 l3Y(2) 3f; "3f;i30/‘ af, ”afi,, . =w+,§ﬁsF-as-ggnrfe ">’l2)"‘g“““ =%+ [s"(r)+f1f‘] lc-=1 8i af,"aft,, ..-5-25;fa byCl6ﬁn1t10I1, (3) rt if,,,—_§g*(r> byan=11! Now equation (5)isadifferential equation with aunique solution foreach initial condition. The solution with initial condition g(0) =0,given by(4),is clearly g(I) =0forall1'.So(3)istrue. Itisasimple exercise tocontinue thedeﬁnition oforuntil itiseventually deﬁned on(—e1,s;) >< ><(——e,,, en)andsatisﬁes (=i=).*1‘ Theorem lessentially solves forustheproblem ofdeciding which distributions have integral manifolds. Our investigation oftheproblem sofarillustrates one basic factabout theorems indifferential geometry: Many ofthefundamental theorems ofdifferential geometry fallinto oneoftwoclasses. The ﬁrstkind oftheorem saysthatifonehasa certain nice situation (e.g., adistribution with integral submanifolds through every point) then certain other conditions hold; these con- ditions areobtained bysetting mixed partials equal, andarecalled “integrability conditions”. The second kind oftheorem justiﬁes this terminology, byshowing that the“integrability conditions” aresuffi- cient forrecovering thenicesituation. The remaining parts ofourinvestigation, inwhich wewillessentially begin anew, illustrates aneven more important factabout thetheorems ofdifferential geometry: There a|'ealways incredibly concise andelegant ways tostate thein- tegrability conditions, andprove their sufﬁciencyg without ever even mentioning partial derivatives. 190 Chapter 5 LOCAL THEORY Iff:M—>NisaC°°function, and Xand YareC°°vector ﬁelds onM and N,respectively, wesaythat Xand Yaref-related ifﬂ,.,,(X,,) =Yﬂp) for each peM.Ifg:N—>RisaC°°function, then Ymits) =ﬁtpXp(.s) Z OJr): 5° (Yn@f=wnf@m. Conversely, ifthisistrue forallC°°functions g:N—>R,then Xand Yare f-related. Ofcourse, agiven vector ﬁeld Xmay notbef-related toanyvector ﬁeld Y, normust agiven vector ﬁeld Ybef-related toanyvector ﬁeld onM.Inone case, thelatter condition isfulﬁlled: 2.PROPOSITION. Let f:M—>NbeaC°° function such that fisan immersion. IfYisaC°°vector ﬁeld onNwith Yftpi '5fP*(MP)v then there isaunique C°°vector ﬁeld XonMwhich isf-related toY. PROOF. Clearly wemust deﬁne Xptobetheunique element ofMpwith Yfm =f,,...X,,. Toprove thatXisC°°,weuseTheorem 2-l0(2): there are coordinate systems (x,U)around peMand(y,V)around f(p) eNsuch that J’° fo x_I(a1:\"".\an) =(aI‘J"'Jan‘JO$"'$O)' This iseasily seen toimply that fr.)=Thus if '1l_3 K Y=go: $5, where 01'areC°°functions, then X=ir§si=1 where 0150f=135.This implies that thefunctions ,3’.areC°°(Problem 3).*3‘ The most important property off-relatedness forusisthefollowing: 3.PROPOSITION. IfX,-andY,»aref-related, fori =l,2,then [X1,X;] and [Y|,Y2]aref-related. Inregml 1l4an§?J[cis 191 PROOF. Ifg:N—>IRisC°°, then (ll (Ys8)°f=X:(§°f) i=1,2- So {lYi,Yzlgl °f={Y1(Y2£)} °f-{Y2(Y18)} °f =X1([162] Of)—X2([Y1s] <>f) by(l),with greplaced byYgg and Ylg, respectively =X1(X2(£ °f))—Xz(X1(§ 0f)) by(1) =IX1,-1’z](£ °f)-*9 Now consider ak-dimensional distribution A. \'Vewill saythat avector ﬁeld Xbelongs toAifXpEAPforallp.Suppose that Nisanintegral manifold ofA,and I:N—>Mistheinclusion map. IfXand Yaretwovector ﬁelds which belong toA,then forallpENthere areunique Xp,YpGNpsuch that Xp=i*X_,,, Yp=J',,.Yp. Inother words, Xand Xarei-related, and Yand Yarei-related. Proposition 2 shows that Xand YareC°°vector ﬁelds onN,andProposition 3then shows that [X,Y]and[X,Y]are1'-related. Thus i*]:¢17, Z [X, Here [X,Y]], GNp; thistherefore shows that [X,Y]], GAP. Consequently, if there isanintegral manifold ofAthrough every point [1,then [X,Y]alsobelongs 10A. Foramoment look back atthedistribution AinR3given by 8 3 8A,,_{i5A~;p+s5~;p+[:f(p)+sg(p)]$p. .=,selR}. The vector ﬁelds 3 3 X=—--+f— 3). 3‘- ¢- 3 3Y=—~»- — 3y+832 belong toA.Using theformula onpage l56, weseethat 3 3 3 3 3 tX,Y1= (§-£+fa—f-g-55»)-;. This belongs toAonly when theexpression inparentheses is0,which isprecisely thecondition ForAtohave anintegral manifold through every point. 192 C/rapier 5 Ingeneral, Aiscalled integrable if[X,Y]belongs toAwhenever Xand Y belong toA.This condition canbechecked fairly easily: 4.PROPOSITION. IfX1,...,X;, span Ainaneighborhood Uofp,then A isintegrable onUifandonly ifeach [X,-, isalinear combination k tnm=Zqn|2‘=] forC°°functions C3-. PROOF. Such functions clearly exist ifAisintegrable, since [X,-,X;]q GA4, .i1ich isspanned bytheX.,(q). Conversely, suppose such functions exist. IfX and Ybelong toAwecanclearly write k X=Z.r~X.»I2] k Y=Z:3i-Yr I'=] Toprove [X,Y]belongs toA,itobviously suffices totreat each If}-X,-,g,-Xj] separately. Since wehave lfX.gY] =fg[X. Y]+f(Xg)Y —g(Yf)X. clearly [fX,gY] belongs toAifX,Yand [X,Y]do.*Z* Wearenow ready forthemain theorem. Itisequivalent toTheorem l;in fact, Theorem lcan bederived from it[Problem 7).But theproof isquite different. 5.THEOREM (THE FROBENIUS INTEGRABILITY THEOREM; FIRST VERSION). LetAbeaC°° integrable k—climensional distribution onM.Forevery [JEMthere isacoordinate system (x,U)with >~'(1>)=0 x(U) =(—-e,e) >< ><(-—e,e), such that foreach ak+', ...,0"with all[ail<e,theset {QeU1»-*+‘<q>= W‘. x"<q>=asisanintegral manifold ofA. Any connected integral manifold ofArestricted toUiscontained inoneof these sets. Izzlegml Jli[6l?Z§ﬁJM.§ 193 PROOF. Wecan clearly assume that weareinIR",with p:O.Moreover, we can assume that AgClR"0 isspanned by 3 3 8:‘0"“ 8:!‘0 Letrt:IR“—>Rkbeprojection onto theﬁrstkfactors. Then rt.,.:A0—>Rkois anisomorphism. Bycontinuity, rt...isone-one onAqforqnear 0.Sonear O, wecanchoose unique X1(q),---,Xk(€?)€ A1,? sothat 3IF*X,'(q)=i. l'=i,...,k, 33"rrtq) Then thevector ﬁelds X,-(onaneighborhood ofOGIR")and3/3!"(onRk)are rt-related. ByProposition 3, 3 3J'1'#]:A,ia Z =0.rte) But, [X,:,X_;]q EAqbyassumption, and rt...isone-one onAq. So[X;, =O. ByTheorem 5-14, there isacoordinate system xsuch that 3 ./Y;"-=5 1:1,.-.,k. The sets{qGU:xk+'(q) =a"‘+',...,x"(q) =0"}areclearly integral man- ifolds ofA,since their tangent spaces arespanned bythe3/3x" =X;for i=1,...,k. IfNisaconnected integral manifold ofArestrictcd toU,with inclusion map i:N—>U,consider d(x"’ o1')fork+l5m5ii.Forany tangent vector Xq ofNqwehave =0, since i,,.Xq GAq, which isspanned bythe3/3x="|q forj==l,...,k. Thus d(x"‘ o1')=O,which implies that x"'o1'isconstant ontheconnected mani- foldN.*I* 194 C/zapter 6 GLOBAL THEORY Inorder toexpress theglobal results succinctly, weintroduce thefollowing terminology. IfMisaC°° manifold, a(usually disconnected) k-dimensional submani- fold NofMiscalled afoliation ofMifevery point ofMisin[some com- ponent of)N,and ifaround every point [JEMthere isacoordinate system (X,U),with x(U) =(-e,e) >< ><(—e,e), such thatthecomponents ofNHUarethesetsofthe form {qev:x"+'<q>=a"+',...,x"<q> =asrm<8. Each component ofNiscalled afolium orleafofthefoliation N.Notice that twodistinei components ofNOUmight belong tothesame leafofthe foliation. \ Q -. \ \‘ \ \ I I *' I / 1 / / L i W _ 6.THEOREM. LetAbeaC°°k-dimensional integrable distribution onM. Then Misfoliated byanintegral manifold ofA(each component iscalled a maximal integral manifold ofA)."-,_,_.-__-- .-I‘,__’ .-1"’- -"'- ""'-__ - --,_ "'-.|- PROOF. Using Theorem l-2,weseethat wecancover Mbyasequence of coordinate systems (x,-,U,-)satisfying theconditions ofTheorem 5.Forsuch a coordinate system (x,U),letuscalleach set {qGU: xk+1(q) :ak+l’ ‘H, xn(q) zan} aslice ofU. Itispossible forasingle slice SofU,-tointersect U;inmore than oneslice ofUj,asshown below. ButSOU;hasatmost countably many components, Inregral Jlflangfilcir 195 andeach component iscontained inasingle slice ofU;byTheorem 5,soSFlU_; iscontained inatmost countably many slices ofUj. ,,.-.-.t‘-;»‘ﬁ'i—-i=‘;:'->.-._,;,-- =,-; ..=7.;._..;,¢;_._ 3,;.._.51‘.s?-.-9.’; I-..-----'\-—--- .' I---' -: -‘-11* 1§=?.;;'sl.-=.=iI-1::‘.€§'=;-- :~-==:=2~. ..,:-" ..-:;;:*:.‘-‘T'¥£§;- -,1-1:-;.-'"-1-'=:2-.::.:v '-'-Y1-\:§1‘,;--_.::1-:.-:.-:..;-,5-::.3-1-.3? ='*:-.""..::-r.--\-¢-- >-~p- /.'*:-J .-..»- -'.'.’\‘~ -- _-___~_U-\__\,_______ ..,,_..,-_.¢----,._‘_,>.1.. _.-_=-‘.3.,13.§';='-15-ii?ifQildisitl'$2r,5;.;;tj1i-*_1:J(l'f='ll'- .an-.21 ;;_;:<;-€‘:!¢?.|L:1:-.P§§Y‘:F-"' '*‘-}'.';.":'._N;:j-1}. ';‘.i.'.-21_..-{.1--1"-<-'-L ..-:' --"--1' >.--,-_:\1;-3,; ':*~.:=;=§es;=.%r5:-2- 5.=--= ".';.a'=;i\.t<::}-. -'-."-.;>=:~. "'92i§5i~“==*".=JFI3*-‘E '-I*I‘1??= "-s~"‘ =15=1‘-'-=.§;-;=~. 2:452;$ ‘ f§'E".\'..';i!;: ."-1.=- =-=‘;1'-;'.l§;‘_.f'-rkiy. 1551.2&'~i:’>'»;'-;=;»;;ie*=52;=§;r_;-=;;:-z_|}'=§-jig .5;.%i- Y:Ci"-511-E?5%}-‘; -?='-§?'-’==1-ti:= .1515" $ ::'=.‘=‘-t:'l\I;!\-;!=;-; __ ..=:-*:’t'z-=i¢f?I' ‘2»-,, r!-5-0.-1'5;-;~t;=.'-.'-s ’-= -:2"--11.5%‘-'-.1. .-'.=.;';é.-.2"-"..:f.-'=' '1?-."-.'.§.=;¢:='.-.:’f-;--‘ ‘:1.-_-.;f.-= .::;'.;‘-‘-_.1;-'.r_..;.,,_ '-""22;:;:;-.!;'.-:;-.'-"'_ _.3;_,_- Given peM,choose acoordinate system (x0,U0)with peU0,andletS0 betheslice ofU0containing p.Aslice Sofsome U,-willbecalled joined top ifthere isasequence O= i0,i1,...,i; =1’ and corresponding slices S0 =Si{pS§]s' ~-ssif :S with S,-“OS,-a+|;él?l Ql:0,...,!'—l. Since there areatmost countably many such sequences ofslices foreach se- quence z'0,...,1'1,andonly countably many such sequences, there areatmost countably many slices joined top.Using Problem 3-},weseethat theunion ofallsuch slices isasubmanifold ofM.Forq7:5p,thecorresponding union iseither equal to,ortotally disjoint from, theﬁrst union. Consequently, M isfoliated bythedisjoint union ofallsuch submanifolds; thisdisjoint union is clearly anintegral manifold ofA.*3 [Ifweareallowing non-metrizable manifolds, theproof iseven easier, since wedonothave toﬁndacountable number ofcoordinate systems foreach leaf, andcanmerely describe thetopology ofthefoliation asthesmallest onewhich makes each slice anopen set. Inthiscase, however, thediscussion tofollow will notbevalid—in fact, Appendix Adescribes anon-paracompact manifold which isfoliated byalower-dimensional connected submanifold] 196 Chapter 5 Notice thatif(x,U)isacoordinate system ofthesortconsidered intheproof ofthetheorem, then inﬁnitely many slices ofUmay belong tothesame folium. Ital (lﬂﬂﬂﬂKQKQ. However, atmostcouizlably man) slices canbelong tothesame folium; otherwise thisfolium would contain anuncountable disjoint family ofopen sets. This allows ustoapply aproposition from Chapter 2. 7.THEOREM. Let MbeaC°° manifold, and M; afolium ofthefolia- tioncleterniined bysome distribution A.LetPbeanother C°°manifold and ftP—>MaC°°function with f(P) CMt. Then fisC°°considered asa map into M1. PROOF. According toProposition 2-I1,itsuﬁices toshow thatfiscontinuous asamap into M]. Given [JGP,choose acoordinate system (x,U)around f(p)such that theslices {qeU:x"""(q) =ak+', ...,x"(q) =0"} areintegral manifolds ofA.Now fiscontinuous asamap intoM,softakes M1 1nlegml J14aizyblds 197 some neighborhood Wofpinto U;wecanchoose Wtobeconnected. For k+151'5iz,ifwe hadx"(f(p’)) ¢a’.foranyp’eW,then x"0fwould take onallvalues between a"and x"(f(p")), bycontinuity. This would mean that f(W) contained points ofuncountably many slices, contradicting thefactthat ffwl CM1- Consequently, x"(f(p’)) =(Iiforallp’eW.Inother words, f(W) is contained inthesingle sliceofUwhich contains p.This makes itclear thatf iscontinuous asamap intoM1.*1* 198 Clzapler 5 PROBLEMS 1.(a)LetE=IIIE—>Bbeann-plane bundle, andE’=Ir’:E’->Ba k-plane bundle such that E’CE.If1':E’—>Eistheinclusion map, and 1,5»:B—>Btheidentity map, wesaythat if’isasubbundle ofifif(i,I3)isa bundle map. Show that ak-dimensional distribution onMisjust asubbundle ofTM. (b)Forthecase ofC°°bundles 1;‘and8'over aC°°manifold M,deﬁne aC°° subbundle, andshow thatak-dimensional distribution isC°°ifandonly ifitis aC°°subbundle. 2.(a)Intheproof ofTheorem l,check theassertion about choosing atsulﬁ- ciently small. (b)Supply theproof oftheuniqueness part ofthetheorem. 3.(a)Intheproof ofProposition 2,show that 3 3 fi(ax? P)—3-Viftp). (b)Complete theproof ofProposition 2byshowing that if PI -3Y= *—., aw sothat "aX= with 0:5of=15",then thefunctions 5';areC°°. 4.Intheproof ofProposition 4,show thatthefunctions Cf}actually areC°°. 5.LetA1,,.,,A0beintegrable distributions onM,ofdimensions d1,...,d;,, Suppose thatforeach peM, Mp=(A|)_0 €B ®(A;,)p. Show thatthere isacoordinate system (x,U)around each point, such thatA, isspanned by3/3X',...,3/3x"", etc. Integral zlélanyfilclr 199 6.Prove Theorem lfrom Theorem 5,byconsidering thedistribution Ain Rm><R“(with coordinates t,x), deﬁned by m -3 - 3 Ap={Z l'i5I—f Z?‘i_fI:k([)))'a—JCT Il'€lRm}. [=1 P = =1 P+ TM-_f'—"‘~s Notice thateven when the donotdepend onx,sothat theequations areof theform 30: 5(1)=f,-tr). with theintegrability conditions 3f} 31‘;- w=w= wenevertheless work inRmxR“,rather than Rm. This isconnected with the classical technique of“introducing new independent variables”. 7.This problem outlines another method ofproving Theorem l,byreducing thepartial differential equations toordinary equations along lines through the origin. Asimilar technique willbevery important inChapter II.7. (a)Ifwewant 0e(ur) =)3(u,t) forsome functi0n ,6:[0,e) ><W—>V,show that /5must satisfy theequation %i-(1.01) 2211'-;f,-(t.=:,;3(a,:)) j=I ﬁlm)==x- Weknow thatwecansolve such equations (weneed Problem 5-5,since the equation depends onthe“parameter” IeRm). One hastocheck that onee can bepicked which works forallIEW. (b)Show that ﬁ(u, vi‘)=,6(uv,t). (Show that both functions satisfy thesame differential equation asfunctions ofu,with thesame initial condition.) Byshrinking W,wecanconsequently assume that e=1. (c)Conclude that 35 _35W014‘) -v-w(I,vr). 200 C/zapler 5 (d)Use theintegrability condition onftoshow that %(v,t) and v-f;(vt,;5(v,t)) satisfy thesame diﬁ'erential equation, asfunctions ofv.Use(c)toconclude that thetwofunctions areequal. (e)Deﬁne 0e(t) =,6(l,t). Noting that01(1)!) =,6(v,t), show that orsatisﬁes the desired equation. 8.This problem isforthose who know something about complex analysis. Let f;(C><(C—>(Cbecomplex analytic. Ifwedenote thecoordinate functions in (C><(Cby21,22 =x|,y1,x2,y2, then f=u+ivsatisfies theCauchy-Riemann equations 3u_ 3v ax‘ am I=1,2.3u__ 3v 3y: Br; UseTheorem ltoprove thatwecansolve theequations 30:‘_ _ ,_ 2 __3012 *5?—Mt-\.J.@1(>~,y).v1(x,y))— ay a°'2— oi">01" ))-—-i-ii ax—v(x,y. (M}’- (My —ay inaneighborhood ofOG(C(orofany point :0GC),and conclude that the differential equation ¢'(Z) =f(Z.¢(Z)) (inwhich 'denotes thecomplex derivative) hasasolution inaneighborhood of:0,with anygiven initial condition ¢(z0) =w0. CHAPTER 7 DIFFERENTIAL FORMS We turn ourattention once more totensor ﬁelds, butwewillbeconcerned with aspecial kind oftensor ﬁeld, thediscussion ofwhich requires some more algebraic preliminaries. LetVbeann-dimensional vector space over R.Anelement TG‘Till/) is called alternating if T(v1,...,v,-,...,v_,-,,,.,vk) =0 ifv,-=v_,-(i#1). IfTisalternating, then forany vt,...,vk,wehave O:T(vt,...,v,-+v_,-,...,v,-+v_,-,...,v,t) =T(v|,...,v;,...,v,-,...,v,t)+T(v|,...,v;,...,v;,...,vk) +T(v1,...,v_,-,...,v,-,...,v,t)+T(v1,...,v_;,...,v,.-,...,vk) =O+T(v,,...,v,-,...,v)-,...,v.t)+T(v|,...,v;,...,v;,...,vt)+0. Therefore, Tisskew-symmetric: T(v|,...,v,-,...,v_,-,...,v;,) =—T(v1,...,v;,...,v;,...,v;,). Ofcourse, ifTisskew-symmetric, then Tisalsoalternating. [This isnottrue inthespecial ease ofavector space over afield where I+I=O;inthiscase, skew-symmetry isthesame assymmetry, and thecondition ofbeing alternating isthestronger one.] V\lewilldenote byS2k(V) thesetofallalternating TGTk(V). Itisclear that Q"‘(V) C'Ti"(V) isasubspace of‘Tk(V). Moreover, ifftV—>Wisa linear transformation, then f*: ‘Tk(W) —>T!‘(V)preserves these subspaces- f*;s2'<(w) ->szltv). Notice thatnltv) =r'(v) =1/*,sos2'(v) has dimension I7.Itisalsoconvenient tosetS2°(V) =‘Toll/) =R.Atthemoment itisnotclear what thedimension ofS2k(V) equals fork>1,butonecase is well-known. The most familiar example ofanalternating Tisthedeterminant function detG‘J'"(R“), considered asafunction ofthenrows ofamatrix- wcshall soon seethat thisfunction is,inacertain sense, themost general alternating function. Most discussions ofthedeterminant begin byshowing that ofany two alternating H-linear functions onR",one isamultiple ofthe 201 202 Chapter 7 other; inother words, dimS2"(R") 51.Then oneproves dimQ”(R") =I byactually constructing thenon-zero function det(itfollows, ofcourse, that dimQ"(V) =IifVisanyn-dimensional vector space). The construction of detisusually byamessy, explicit formula, which isaspecial case ofthedeﬁnition tofollow. LetS),denote thesetofallpermutations of{1,...,k}; anelement 0GS),is afunction ii—>o(i). If(v1,...,vk)isak-tuple (ofanyobjects) weset U.(vl:' °'avk) Z (UO'(])$--- lUU(k))' This deﬁnition hasabuilt-in confusion. Ontheright side, theﬁrst element, forexample, isthe0(1)“ ofthev’sontheleftside; ifthese v’shave indices running insome order other than 1,...,k,then theﬁrstelement ontheright isnot necessarily thatvwhose index is0(1). The simplest waytoﬁgure outsomething likeo-(v3,vg. v|,...)is torename things: v3=w|,v2 zw2,v| =w3,... .Thus warned, wecompute 0"(P-(vi,---,vt<))=U'('Jp(1),--..vp(t<)) bysetting vptl) =wh---=vptlr) =wk, sothat 0-(p-(vt,...,v,t))=o-(w1,...,w,t) =(wa(1),»--,wa(.l<)) =(v.0(v(l))> ---i”p(v(k))) Since we=U001)- Thus { 0"(p'(v]s---sVl<))=(pU)'(vls"':vl()- ] Now foranyTG‘J""(V) wedeﬁnc the“alternation ofT” I AltT= FZ: sgno-Too, i055;,- 0 i.e., IAltT(v|,...,v;,) =FZsgno -T(vc,(|),...,v,(;,)), '(IE5), where sgno is+1if0isaneven permutation and -1if0isodd. Dg'ﬁezenlz'al Fortes 203 1.PROPOSITION. )IfTezrttv), thenA1t(T) GS'2"(V). ()Ifcueszktv), thenAltw=cu. ()Ifrerttv), thenA1t(A1t(T)) =mar). L>Ol\D’: 0:0 PROOF. Lefttothereader [orseepp.78-79 ofCalculus onManzjlilis). \'\lenow deﬁne, forcuGQk(V) and nGQi(V), anelement wAr] GQ'i‘+"(V), thewedge product ofwand17,by k+l)!wAn=(—,(T~Alt(w®n). The funny coellicient isnotessential, butitmakes some things work outmore nicely, asweshall soon see. Itisclear that (1)Aisbilinear: (wt+w2)Ar;=cut An+w2An w/\(Y71 -l-'72) =01/"71 +01/\'72 awAr7=wAan=a(wA27) (2)f*(w/\Yt)=f*w/\f*n- Moreover, itiseasy toseethat (3)Ais“anti-commutative”: cuAn=(—1)mi7 Acu. Inparticular, ifkisodd then wAw=Q Finally, associativity ofAisproved inthefollowing way 2.THEOREM. (1)Ifserttv) andTerltv) andA1t(S) =0,then A1t(S®T)2A1t(Tos)=0. UOFO()Alt(Alt(w ®T1)®e)=Alt(w®0®e)=A1t(w®A1t(nts6)). ()1rcuG§z'<(v), 0GS2"(V), eGsmv), then (lc+l+m)l (wA1))At9=cuA(nA6)= Mt(w®n®3). 204 Chapter 7 PROOF. (l)Wlehave (lf-I-l)lAlt(-S ®T)(U1, ...,Uk_|_,t) =Z:sgno-(S®T)-(0-(v1,...,vk_|.;)) GE-5'),-+1 = E 58"“ U°S(va(I)= ---1Ucr(k)) 'T(Ucr(k+1)= ---aUa(k+l'))- GE-5'),-+[ Now letGCSki.) consist ofall0which leave k+1,...,lt+lﬁxed. Then Z58110’ 'Sivan), ---Iva(t<)) 'T(v0(k+I)> ---,va(k+t)) 0&6’ =ZS3116’ -S(v0'(1)i---=U0’(k)) ~7'(v1<+1..--.vt<+t) 0’iES;,- Suppose now that00¢G.Let00G ={a0o’: 0’GG}.Then Z: sgno -(S®T)(o -(v1,.. .,vk+;)) 0'EO't|G =sgIwo- ZSgI10" -(5®T)(0" -(Go-(vi,---.v1<+t))) ll)’(*l- cr"GG Wehave justshown thatthisisO(since 00-(v1,...,v;,._,_,~) isjustsome other (k+l)-tuple ofvectors). Notice that Gﬁo0G =1?),forif0GGH00G, then 0:000’ forsome 0'GG,so00=o(o')“" GG,acontradiction. \lVecan then continue inthisway, breaking S_t+; upinto disjoint subsets, thesum over each being 0.The relation Alt(T ®S)=0isproved similarly. (2)Clearly A11(A1l(n ®6)—n®9)=Ahtn®6)-Ahte®9)=0. so(l)implies that 0=Alrtw®[A1102sf?)—n®6]) =Alt(w ®Alt(r7 ®9))—Alt(w ®I]®3); theother equality isproved similarly. (3)Vilehave tr+l+m)! _(k+1+m)! (k+1)! Gtr41):”?! '/<11!"The other equality isproved similarly. ~2¢Alt(w ®17®3). 1)t'/]t':reiitt'ttl 1‘lJ?'?.?1.$' 205 Notice that (2)just states that Aisassociative even ifwehad omitted the factor (k+l)!/k!l! inthedeﬁnition. Ontheother hand, thefactor l/kl inthe deﬁnition ofAltisessential-—~without it,wewould nothave Alt(Alt T)=AltT, andthefirstequation intheproof of(2)would fail. [Ifwe haddeﬁned Htjust likeAlt, butwithout thefactor I/kl, then Acould bedeﬁned by 1__ This makes sense, even overaﬁeld ofﬁnite cl1aractenlrtz'c, because each term inthe sum Alt(w ®n)(v1, ...,v_t+;) occurs k!l! times (since cuand I]arealternating), and l/kill canbeinterpreted asmeaning that these kill terms arereplaced by justone.] The factor (k+1)!/kill hasbeen inserted intothedeﬁnition ofA forthefollowing reason. Ifv1,...,v,,isabasis ofV,and¢|,...,<,t>,,isthedual basis, then <t>.A---A¢t=5~J§,ii-%IlA1t<¢-s---st») one0 n =ZSgnv-(¢1®---®¢a)°¢ UESJ; Inparticular, (erA---A<t>n)(v|,---.vs)=1- (Soifvi,...,1),,isthestandard basis forR",then ¢|A---A(6,,=det Abasis forQk(V) cannow bedescribed. 3.THEOREM. The setofall <,l>,-,A---/\¢>,-,\_ l51'|<---<z'k5t1 isabasis forQk(V), which therefore hasdimension (I?) _ ti! kTk!(n —k)!' (Inparticular, S2k(V) ={0}fork>tr.) PROOF. If(UeS2l‘(V) crktv), wecanwrite w= 2 at'1...1i;_- ¢'t'] ®''°®¢i;,- - i1,...,t';_- 206 Clzapter 7 So iv=A1l(¢v)= 2¢1t,...t,_. A1t(¢n ®'--®¢t,.)- t'|,...,i,t,- Each Alt(¢,~, ® ®¢,-,_.) iseither 0or=:1:(1/l<!)¢,-, A A<1),-,,forsome j,< <jk,sotheelements ¢,;,A---A<,t>,,-,_. forjt< <jkspan S2"(V). If 0: E at'1...t'k ¢i|A /\¢I't,-i i|<---<0. then applying both sides to(vi,,...,v,-,,,) gives ct,-,___,-,,_ =0.'1' 4-.COROLLARY. Ifwt,...,w;< GQ](V), then cu],...,w_t arelinearly inde- pendent ifandonly if w|A---Awk7éO. PROOF. Ifa)1,...,a)k arelinearly independent, there isabasis vt,...,v;,,...,v,, ofVsuch that thedual basis vectors ¢1,...,¢,t,...,¢,, satisfy ¢>,-=cu;for I51'5k.Then wtA Awkisabasis element ofS2i‘(V), soitisnot0. Ontheother hand, if wt=a2w2+---+akwk, then w|Aw2A---Aw), =(u2wg+---+ukwk)Aw2A---Aw),=0. *1‘ Toabbreviate formulas, itisconvenient tolet1denote atypical “multi-index” (1],. ..,z'k), andletgt);denote gt»,-IA---Aqt),-,__. Then every element ofS2"‘(V) is uniquely expressible as Zat¢t- I Notice thatTheorem 3iniplies thatevery toGQ*(R") isalinear combination ofthefunctions vi (v1,. ..,v;,) i—>determinant ofak><kminor of(f Wt One more simple theorem isinorder, before weproceed toapply oureon- struction tomanifolds. Dg'ﬂerentz'al 1*'0t'm.s' 207 5.THEOREM. Letvi,...,v,,beabasis forV,letcuGQ"(V), and let H w,-=5 oz,-,-vj i=l,...,)t. ]'=] Then CU(lU|, ...,w,,) =1ClClI(Q',:_,') 'CU(U1,. ..,U0). PROOF. Deﬁne nG‘T"(R“) by l](((£||,-..,(ln]),..., (Cln1,...,(I;m))=(tJ(2(I_;1U_,;',-..,2(l_,m1J_,;'). .1I tI Tlieii clearly )7GQ"(R“), sor)=c-detforsome cGR,and 6=ﬁller.---tea) =w(vi,---.va)- '3' 6.COROLLARY. IfVisti-dimensional and0¢cuGS2"(V), then there isa unique orientation itforVsuch that [vt,...,v,,] =/.1ifand only ifw(v|,...,v,,) >0. With ournew algebraic construction athand, weareready toapply itto vector bundles. IfE=rt:E—>Bisavector bundle, weobtain anew bundle Qk(E) byreplacing each ﬁbre rt“'1(p) with Qk(n“'(p)). Asection cuofQk(§) isafunction with cu(p) G§2k(rt""(p)) foreach pGB.Ifnisasection ofQi(E), then wecandefine asection toA27ofQk""'(E) by(cuAr7)(p) =w(p) A17(1)) G Q"+’tn"' tn).Inparticular, sections ofQk(TM), which arejust alternating covariant tensor ﬁelds oforder k,arecalled k-forms onM.Al-form isjust acovariant vector ﬁeld. Since Qk(TM) canobviously bemade into aC°°vector bundle, wecan speak ofC°°forms; allforms willbeunderstood tobeC°°forms unless the contrary isexplicitly stated. Remember that covariant tensors actually map contravariantly: Iff:M—>NisC°°, and cuisak-form onN,then f*w isa k-form onM.Wecanalsodeﬁne cu,+cu;andcuAT].The following properties ofIt-forms areobvious from thecorresponding properties for52*(V); (w1+w2)/\0=w1/\t;+w2A17 w/\(m+nz)=w/\m+w/\nz fw/\n=w/\fn=f(wAn) wAn=(—I)k"r)AoJ f"‘(wAn)=f*w/\f”n- 208 Clzapter 7 If(x,U)isacoordinate system, then thedx"(p) areabasis forM,,"‘, sothe dx"'(p) A Adx'l~' (1))(it< <ik)areabasis forQ"‘(p). Thus every k-form cucanbewritten uniquely as w=Z: w,-,___,-,‘_ dxil A---Adxik fl.q...<,ik or,ifwe denote dxi‘ A Ad),-fr bydxl forthemulti-index I=(1],. ..,z'k), to=Zw]dxf. I The problem ofﬁnding therelationship between thecu;andthefunctions cu’; when cu=20); dxi=Em’; dyi 1 1 islefttothereader (Problem 16),butwewilldoonespecial case here. 7.THEOREM. Iff:M—>NisaC°°function between n-manifolds, (x,U) isacoordinate system around pGM,and (y,V)acoordinate system around q=f(1>)eN,then8E filgtlyl A---Ady") =(£°f)-det( ) dx‘A---Adx". PROOF. ItSUFECCS tosliow that f*(dy' A---Ady”)=det(L};xD,f)) dx'A---A dx". Now, byProblem 4--l, 1|! fl a a f(tl_V'/\---/\(ly)([)) ~¢, ..-3i.,, ,, ,, a a=dy‘(q)A~--Ad)’ (r1)(J’*-6; ..--.f..5;’P .... . _ ,n H3(}”'°f) 3~d)‘(e)A-- Ar!)(e)(giax, (10% (y"<>f) 3TX” (TOW rm»Q.) 3"'=det( (Pl) , byTheorem 5.*3‘ Dtﬂerenttal Forms 209 8.COROLLARY. If(x,U)and (y,V)aretwocoordinate systems onMand gdy‘ A---Ady" =l1dx' A---Adx", 3yih=g-Clﬁt PROOF. Apply thetheorem with f=identity map. *2»then This corolla shows that ti-forms arethe eometric ob'ects corres ondin to1')’ S J P 3 the“even scalar densities” deﬁned inProblem 4--l0.] Iflf =rt:E—>BisanI1-plane bundle, then anowltere zerosection cuofQ“(§) hasaspecial signiﬁcance: Foreach [JGB,thenon-zero w(p) GQ"(rt"'(p)) determines anorientation ,u.,,ofrt"l(p) byCorollary 6.Itiseasy toseethat thecollection oforientations {,u.,,} satisfy the“compatability condition” setforth inChapter 3,sothat p.={pp} isanorientation of§.Inparticular, ifthere is anowhere zero it-form cuonanit-manifold M,then Misorientable (i.e., the bundle TM isorientable). The converse also holds: 9.THEOREM. IfaC°° manifold Misorientable, then there isann-form to onMwhich isnowhere 0. PROOF. ByTheorem 2-l3 and2-15, wecanchoose acover (9ofMbyacol- lection ofcoordinate systems {(x,U)},andapartition ofunity {qty} subordinate to(9.Let p.beanorientation ofM. Foreach (x,U)choose ann-form tug onUsuch that forv|,...,v,, GMp, pGUwehave wt;(v|,...,v,,)>O ifandonlyif {v1,...,v,,]=,u.,,. Nowlet cu=X:¢UwU. UGO Then toisaC°° 12-form. Moreover, forevery p,ifv|,...,v,, GMpsatisfy 3nn03U11] 1 Mp, (¢vwu)(P)(v1..--.vs) 20, andstrict inequality holds foratleast oneU.Thus w(p) ¢0.*2» 210 Clzapter 7 Notice thatthebundle Q"(TM) is1-dimensional. \/Vehave shown thatifM isorientable, then Q"(TM )hasanowhere Osection, which implies that itis trivial. Conversely, ofcourse, ifthebundle S2"(TM) istrivial, then itcertainly hasanowhere Osection, soMisorientable. [Generally, ififisak-plane bundle, then Qk(E) istrivial ifand only ifEisorientable, provided that thebase space B is“paracompact” (every open cover hasalocally-ﬁnite reﬁnement).] _]ust asS2°(V)hasbeen introduced asanother name forR,aO-form onM willjust mean afunction fonM(and fAwwilljust mean f-cu), For every O-form fwehave theI-form df(recall that df(X) =X(f)), which ina coordinate system (x,U)isgiven by ef_ax,dx. cv=Z:cu1dxi, I then each dc.-J; isaI-form, andwecandeﬁne a(lt+1)-form dw,thedifferential ofcu,byIfcuisak-form dw=Z:dw;dx" I =;ﬁ%dx°' Aer’. tI=] Itturns outthatthisdeﬁnition does notdepend onthecoordinate system. This canbeproved inseveral ways. The first way istouseabrute-force computation, comparing thecoellicients cu’;intheexpression cu=2:0)’; dxi I with thecu). The second method isalotsneakier. W'ebegin byﬁnding some properties of dw(still deﬁned with respect tothisparticular coordinate system). 10.PROPOSITION. (l)d(w| +(U2)=dc.-)1+dC!J2. (2)Ifwtisak-form, then d(w| A(U2)=dwt Acu;+(-l)'i‘w| Adrug. (3)d(dw) =0.Briefly; dz=O. PROOF. (l)isclear. Toprove (2)wefirstnote that because of(l)itsullices toDifferential lbrrns consider only OJ]=fdXI w2=gdxJ. Then w,Acu;=fgdxl Adxi and d(w]A(U2)Gd(fg)Aex’Aex’ GgdfAdx'AdxJ+fdgAdx"Adx“' Gdo),Aw,+t~i)*fes-I AdgAex’ =do),A(U2+(-—l)kw, A(R02. (3)Itclearly suﬁices toconsider only k-forms oftheform Then(:J=fdJCI. H 3 dw=Z%dx“Adie" a=l S0rt n 2 dldw) =2(Z: dxﬂ Adx“Adxl). I12] ﬂ=] Inthissum, theterms andazf ﬂ III I dX /\dx /\dX 32f ti 1 dXaA dx /\dX cancel inpairs. *9 Vilenext note that these properties characterize donU. ll.PROPOSITION. Suppose d’takes it-forms onUto(lc+I)-forms onU forallk,andsatisﬁes '-P‘~<.>Ol\D’:"...vvvv( ( (di(w] +(U2)=d’w| +d’wg. d"(w| A(U2)=d"w] A(02+(—I)"‘w| Ad"w2. d"(d"f) =0- d"f=(the old) dfi Then d’=donU. 212 Clzapter 7 PROOF. Itisclearly enough toshow that d'0)=dwwhen cu=fdx!.Now by(2). aqeG)GafAAJ+fAwwM) GefAA¥+fAenaU bym. Soitsuﬁices toshow that d'(dx1) ==O,where dxi =dx"' A Adxi" =dG“A- Adah bymy Wewilluseinduction onk.Assuming itfork—lwehave e%¢H)=e%wx“A.~Adw@) =e%aﬁuAeG@A.~Aewa —-d".\"" Ad"(d'x"' A---Ad’x'l~') by(2) =O—-O, by(3)and theinductive hypothesis. ¢I~ l2.COROLLARY. There isaunique operator dfrom thek-forms onMto the(k+1)-forms onM,forallk,satisfying d(w1+ (1)2)=dw, +dc); d(cu| A(U2)=dwt A(U2+(—-l)kw| Adrug e2=a andagreeing with theolddonfunctions. PROOF. Foreach coordinate system (x,U) wehave aunique dgdeﬁned. Given theform cu,and [JGM,pick any Uwith 1)GUand deﬁne dwtﬂ) =du(wlU)(P)- '5' The third way ofproving that thedeﬁnition ofddoes notdepend onthe coordinate system istogive aninvariant deﬁnition. Dt'ﬂerentz'al F0t'ms' 213 l3.THEOREM. Ifcuisak-form onM,then there isaunique (k+I)-form dwonMsuch that forevery setofvector lields X1,...,X_t_,_, wehave Pl‘) dall-Y] ,---aXl:.|-I) k+1 =Zr-1)"+'x,-(GtX,, ...,X,-,...,X,.+,))[ml +Zt-1)*'+1'<»ttX..X,-1.X-..-_.7r?.....1'?}.....Xt+i)l5f<.t'5l<+l (=E|+E2,say) __.-,,,_ where over X,-indicates that itisomitted. This (lt+1)-form agrees with dtu asdeﬁned previously. PROOF. The operator which takes (X1,...,X_t_,.]) toE1+E2isclearly linear over R.Moreover, itisactually linear overtheC°°functions 37.Infact, ifX,1,is replaced byfX,-,,,then E,becomes fr.+Zr-1)"+'(X.-f)<»tX......1?}.....X).+.)."#l't| andusing theformulas lfX=Yl=flX.Y]-Yf-X lX.fY]==f[X.Y]+Xf-Y, itiseasily seen that E2becomes +E(—])i+:-i|(~1,lf)w(Xf¢]a X]: ---sjili-s ---9:?-frills '''9AIR-+1) 1'-€t'|| _2(_])iii+j(Xjf)w(~m't|aX]a- --:-j;ITF(s- --a:i?s- '':XR+])i l'[)<_)i abriefinspection then shows that E]+E2becomes fE| +fﬁg. Theorem 4-2shows thatthere isaunique covariant tensor ﬁeld dwsatisfy- ing(>l=).Itiseasy tocheck that dwisalternating, sothat itisa(k+I)-f0rI‘r1. Tocompute dtvinacoordinate system (x,U)itclearly suflices tocompute cl(fdxl).Moreover, byrenumbering, wemight aswell assume w=fdx'A---Adxl‘. 214 Chapter 7 Fordw, asforany form, wehave dtu-= Z dw(3/3x°", ...,3/3x""+‘) dx°" A---A dx“'~'+'. ,;,,.=...<0,,_,_, Itisclear from (>l=)that dw(3/3x“', ...,3/3x°'l<+') =O unless some (0t1,...,0'?}-, ...,0tk_,.|) isapermutation of(1,...,l<). Since the0t’sareincreasing, thishappens only if (<1'1,...,0t.t+1)=ll...-,/<,J') J'>l<. inwhich case tlw(3/3x“‘,...,3/3x“*', 3/Elxl) G(-_1)"%. SO dw=Z:(—-I)k;%dx' A---Adxk/\dJCj j>l< x =Z:Lr.dx="/\dx' A---Adxkt pk3x H 3 .=Z:i.dx-l A.dx' A---Adxk,I213):-l which isjusttheolddeﬁnition. *Z* This isourﬁrstrealexample ofaninvariant deﬁnition ofanimportant tensor, andourﬁrstuseofTheorem 4-2. Wedonotﬁnd dtu(p)(v,, .__,vk_,_,) directly, butﬁrst ﬁnd dw(X1,...,X,t_,_1), where X,-arevector ﬁelds extending v,-,and then evaluate thisfunction atp.Bysome sortofmagic, thisturns outtobe independent oftheextensions X1,..,,X,t+t. This may notseem tobemuch of animprovement over using acoordinate system andchecking thatthedeﬁnition isindependent ofthecoordinate system. Butwecanhardly hope foranything better. After all,although dcu(X1,...,Xk_,_1)(p) does notdepend onthevalues ofX,except atp,itdoesdepend onthevalues Ofcuatpoints other than p— thismust enter into ourformula somehow. One other feature ofourdeﬁnition iscommon tomost invariant deﬁnitions oftensors—the presence ofaterm involving brackets ofvarious vector ﬁelds. This term iswhat makes theoperator D_tﬁereizlz'al Forms 215 linear over theC°°functions, butitdisappears incomputations inacoordinate system. Intheparticular casewhere cuisa1—form, Theorem l3gives thefollowing formula. l<1<»<X.r) =X(w(Y)) -Y<<»tX)> -wtl/Y,Y1)l This enables ustostate asecond version ofTheorem 6-5(The Frobenius Inte- grability Theorem) interms ofdiliereiitial forms. Deﬁne thering S2(M) tobe thedirect sumoftherings ofI-forms onM,forall1.IfAisak-dimensional distribution onM,then J(A) CS2(M) willdenote theSubring generated by thesetofallforms towith theproperty that(iftohasdegree l) o)(X1,. ..,X;)=O whenever X1,...,X;belong toA. Itisclear that wt+tu2 G.l(A) if(:)],(U2 G.l(A), and that T)AtoG..l(A) if toG.l(A) [thus, .l(A) isanideal inthering Q(M)]. Locally, theideal .l(A) isgenerated byn—kindependent 1-forms w"+', ...,cu". Infact,around any point pGMwecanchoose acoordinate system (x,U)sothat 3 3 A -—-— ,...,— san _31:1,, 3).l‘,, P P Then dX1(])) A---Adxk(p) isnon-zero onAP. Bycontinuity, thesame istrue forqsulﬁciently close top,which byCorol- lary4implies that dxl(q),...,dxk(q) arelinearly independent inAq. There- fore, tliere areC°°functions f3“such that It dx°'(q)= Z)‘F(q)dA'ﬂ(q) restrlctedto Aq 0t=k-|-1,...,n. ﬂ=l Wecantherefore let TM».\.CU‘: ="-dxa _ ﬁadxﬂ. l4.PROPOSITION (THE FROBENIUS INTEGRABILITY THEOREM; SECOND VERSION). Adistribution AonMisintegrable ifand only if d(.£(A)) ceta). 216 Chapter 7 PROOF. Locally wecanchoose 1—forms wl,___,cu"which span Mq* foreach q such thatwk“, ...,w"generate .1(A), LetX1,...,X,,bethevector ﬁelds with . w*'(X,-) =5}, Then X1,...,Xkspan A.SoAisintegrable ifand only ifthere arefunctions C5 withk [X,-,X,-] =Zcgxﬂ 1,;=1,___,1<. ﬁ=1 Now dw°‘(X,-,X;) =:X,-(w°'(XJ-)) —X,-(w°'(X;)) —w°'([X;,X;]). ForI5i,j5kand oz>k,theﬁrst two terms ontheright vanish. So dw°‘(X,-,Xj) =Oifand only ifw"'([X,-,X;]) =O.Buteach w°'([X,-,X,-]) =O ifand only ifcach [X,-,X;] belongs toA(i.e., ifAisintegrable), while each dw°'(X,-, =Oifand only ifdw“ G.£(A). '1‘ Notice thatsince thew"Aw? (i<j)span Q2(Mq) foreach q,wecanalways write dcu“ =Z03-wi /\cuj i<j =—. /\wj forcertain forms 6}‘. 1' Ifor>k,andi@,j0 5karedistinct, wehave 0=dw“(X;‘,,X,-0) =Zwf Aa)J)(Xl-n, X,-,,) 1' 26JF:;(Xill)> sowecanwrite thecondition d(J(A)) C.£(A) as dw“ =26; /\wﬁ. ﬁr:-k Once wehave introduced acoordinate system (x,U)such thattheslices {fleU:A-"+‘<q> =W‘,...,x"<q> =~"} areintegral submanifolds ofA,theforms dxk"'1, ...,dx" areabasis for.£(A), sow*""], ...,w"must helinear combinations ofthem. Wetherefore have the following. D;'}§lé’r'eniz'al Forms 217 l5.COROLLARY. Ifct-/‘"H,...,w” arelinearly independent I—forms ina neighborhood ofpeM,then there areI-forms 6;‘(oz,,6>k)with dw“ -=Z 3;Awﬁ 13 ifandonlyifthere arefunctions fﬂ“,gﬂ(OI,,5>k)with (Ha Z’ 8 Although Theorem l3warms theheart ofmany aninvariant lover, thecases k>Iwillhardly everbeused [avery signiﬁcant exception occurs inthelast chapter ofVolume V).Problem l8gives another invariant deﬁnition ofdw, using induction onthedegree ofcu,which ismuch simpler. The reader may reﬂect onthedifiiculties which would beinvolved inusing thedeﬁnition of Theorem l3toprove thefollowing important property ofdz 16.PROPOSITION. Iff:M—>NisC°°andcuisak-form onN,then f*(dw) =d(f*<'J)- PROOF. ForpeM,let(x,U)beacoordinate system around f(p). Wecan EISSUTDC cuzgdxl‘ /\---Adxik. Wewilluseinduction onk.Fork=Owehave, tracing through some deﬁni- tions, f*(ds)(X) =ds(fiX) =[f-=X](s) =X(s<>f) =dts<>f)(X) (and, ofcourse, f*gistobeinterpreted asg0f).Assuming theformula for k—I,wehave dtfm») =d((f*g ax“A---Adx‘i"""‘)Af*dx"~") =d(f*(g ax“A--.Aas-*'~~1)) Af*dxi"' +0 sincedf*dx‘l\' =dd(x"’*' Qf)a0 =f*(d(g dxi‘A---A dx"’~'-"1)) Af*dx""' bytheinductive hyposthesis =f*(dg Adx“A---Adx"'*'"-1) Af*dx""' =f*(dg Adxi‘ A---Adxl*\'"-1 Adxik) =f*(dw)- *2‘ 213 C/rapier 7 One property ofdqualiﬁes, bythecriterion oftheprevious chapter, asa basic theorem ofdifferential geometry. The relation d2=0isjust anelegant wayofstating thatmixed partial derivatives areequal. There isanother setof terminology forstating thesame thing. Aform cuiscalled closed ifdw=O andexact ifw=dryforsome form r7.[The terminology “exact” isclassical- difTerential forms used tobecalled simply “differentials”; adiflerential wasthen called “exact” ifitactually wasthediﬁerential ofsomething. Theterm “closed” isbased onananalogy with chains, which willbediscussed inthenext chapter.) Since d2==O,every exact form isclosed. Inother words, dw=Oisanecessary condition forsolving cu=dn.Ifatisal—lorm II cu= 2widxi, ll then thecondition dw=0,i.e., Bw- Btu-#1.; Bxl Bx‘ isnecessary forsolving cu=df,i.e., af__axi._w,. Now weknow from Theorem 6—lthat these conditions arealsosutﬁcient. For 2-forms thesituation ismore complicated, however. Ifcuisa2—form onR3, (U:/id)’/\dZ—Bdx/\dZ—|—CdX/\dy, then cu=d(Pdx+ Qdy+Rdz) ifandonly if BR BQ__i =A By B: BP BR___=B B2 Bx B2_al=C_ Bx By Thenecessary condition, dw=O,is "’i+‘”i+§-0Bx By B:_' Dgférential Forms 219 Ingeneral, wearedealing with arather strange collection ofpartial diiierential equations (carefully selected sothatwecangetintegrability conditions). Itturns outthatthese necessary conditions arealsosufficient: ifcuisclosed, then itis exact. Like ourresults about solutions todifferential equations, thisresult is true only locally. The reasons forrestricting ourselves tolocal results arenow somewhat different, however. Consider thecaseofaclosed 1—form wonR2: 3f asw=fdx+gdy, with Weknow how toﬁndafunction oronallofR2with w=doe,namely @1(>~',y) =fxf(I,yo)dI +fyg(><.I)dI-In yo Ontheother hand, thesituation isvery different ifcuisdeﬁned only onIR2—{0}. Recall thatifLCR2is[0,oo) ><{O},then Bzlliz-L—>llR, L 0 deﬁned inChapter 2,isC°°; infact, (1-,6): R2-L->{F1r>0}><(0,2;¢) istheinverse ofthe map (a,b)|—>(acosb,asinb), whose derivative at(a,b)hasdeterminant equal toa960.Bydeleting adiﬁferent rayL;wecandeﬁne adiﬁferent function B1.Then B1=6intheregion A1and B1=B+2rr intheregion A2.Consequently dBanddB]agree ontheir common 0 630,40x A1 9* A2 8 I40¢ 220 Chapter 7 domain, sothat together they deﬁne al—form cuonR2—{O}. Acomputation (Problem 20)shows that =-»_d +-M-—<1.“Jx2_|_y2 2x2_|_y2-l’ The l—form cuisusually denoted bydB,butthisisanabuse ofnotation, since cu=d6only onR2—L. Infact, cuisnotdfforary C]function f;R2—{O} —>R. Indeed, if0)=df,then df=d6 onR2—L, sod(f —9)=OonR2—L,which implies that Bf/Bx =B9/Bx and Bf/By = Flt]/By andhence f=6+constant onR2—L,which isimpossible. Nevertheless, dw=0[thetworelations d(d6)=0onR2-1. d(d61) =0onR2-1., clearly imply thatthisisso].Sotoisclosed, butnotexact. (Itisstillexact ina neighborhood ofanypoint ofR2—{0}.) Clearly wisalso notexact inanysmall region containing O.This example shows that itistheshape oftheregion, rather than itssize, that determines whether ornotaclosed form isnecessarily exact. Amanifold Miscallcd (smoothly) contractible toapoint [)9eMifthere is aC°° function H:M><[O,1]—>M such that H(p’1)=p forpeM. H(P,O) =P0 Forexample, R"issmoothly contractible to0eR";wecandeﬁne H:R"><[O,1]—> R" by H(p, I)=tp. More generally, UCR"iscontractible to[)0eUifUhastheproperty that 1)g'ﬂereniial Iibrms 221 peUimplies [)9+t(p —pg)eUforO5I5I(such aregion Uiscalled star-shaped with respect topg). \ xv‘>- 33 Ofcourse, many other regions arealso contractible toapoint. Ifwethink W of[0,1]asrepresenting time, t_ien foreach time Iwehave amap pt—>H(p,t) ofMinto itself; attime 1thisisjust theidentity map, and attime Oitisthe constant map. Wewillshow that ifMissmoothly contractible toapoint, then every closed form onMisexact, (Bytheway,thisresult andourinvestigation oftheform d6 prove theintuitively obvious factthat R2——{O}isnotcontractible toapoint; the same result holds forR"--{O},butwewillnotbeinaposition toprove this until thenext chapter.) The trick inproving ourresult istoanalyze M><[0,1] [foranymanifold M),andpayhardly anyattention atalltoH. Fort G[0,1]wedeﬁne i1iM%MX[O,1] by 2-{(12) =(pat)- Weclaim thatifcuisaform onM><[0,I]with dw=O,then z'|*<u —z'0*w isexact; 222 Chapter 7 wewillseelater (and you may trytoconvince yourself right now) that the theorem follows trivially from this. Consider ﬁrsta1-form wonMx[0,I].Wewillbegin byworking inacoordi- nate system onMx[0,I],There isanobvious function! onM><[0,I](namely, theprojection rtonthesecond coordinate), andif(x,U)isacoordinate system onM,while JIMistheprojection onM,then (xi0HM, ...,x"0rtM,t) isacoordinate system onUx[0,1]. Wewilldenote xioHMbyii,forconve- nience. Itiseasy tocheck (orshould be)that H H l'a*(E(U;' (iii —|—fdI) :=2:0-n(-,oe)dx", 11 11A . where w,-(-,0!) denotes thefunction pt—>w,(p,01). Now forcu=EL, cu;di‘+fdt wehave PI PI . . B-. B _.dw=[terms notinvolving dt]—2%d2?‘Adt+Z dx‘Adt. i=1 i=1 X Sodw=0implies that Btu;__Bf BrTart' Consequently, I8(1)‘: w.-(7,1)-wt-(12.0)=fTimid:0 I l =£ ﬁ(P,f)df, SO (1)Zwi(P>1ld$'i—Z:wi(P=OldXi-‘= %tp,r>dr)dA-*10at a -M: i=1 i= = Ifwe deﬁne g:M—>Rby 1 gm=faftimdr. Dgfrrenlial Forms 223 then 1 (2) §tp>=f§—f,-(mar. X 0 X Equations (1)and(2)show that i1*w —i@*cu ==dg. Now although weseem tobeusing acoordinate system, thefunction f,and hence galso, isreally independent ofthecoordinate system. Notice that for thetangent space ofMx[0,1] wehave (=i=) (Mx[0,I])(,,,,) =kerm, EBkerrtM,,.. kei-2tM,,, Mx[0,1] lo’I] 4% kerm, lHM Ifavector space Visadirect sum V=V169V3oftwosubspaces, then any cuES2l(V) canbewritten_........ M w:=wl+cu; where w1(v1 +U2)=w(v1) w2(vi +v2l =("(112)- Applying thistothedecomposition [=t=),wewrite theI-form cuonMx[0,I]as cu]+w2; there isthen aunique fwith cu;=fdi‘. Ingeneral, forak-form cu,itiseasy tosee(Problem 22)that wecanwrite cu uniquely as cu=cul+(dtA7]) where w1(v1,. ..,vk) =0ifsome v,-eker2rM,,,, and27isa(k-1)-form with the 224 C%dpwr7 analogous property. Deﬁne a(k—I)-form IwonMasfollows: 1 Iw(p)(vl 2---avk'—]) Zf 2)(il:|=v] 2'~->il*vk—])d2- 0 Weclaim thatdw=0implies thati'1*w —i'@*w =d(Iw). Actually, itiseasier toﬁnd aformula fori'1*w —i0*w that holds even when dw¢O. 17.THEOREM. Foranyk-form cuonMx[0,1]wehave z']*w —i'@*w 1:d(1w_) +I(dw). (Consequently, i1"‘w —i'0*w =d(Iw) ifdw=O.) PROOF. Since Iwisalready invariantly deﬁned, wecanjust aswell work in acoordinate system (id,...,i'",t). The operator Iisclearly linear, sowejust have toconsider twocases. (1)w5f¢1>t:=1A--.A.<1x-=~ ==fax-1. Then Bdw=L. +%dr Adi’, itiseasy toseethat 1a1<~'w>= (f;,"§tp.r>dr) <1x’O =ftp.1>—ftp,o>1<1>-‘(pi -=z']*w(p) —z'@*w(p).I? Since Iw=O,thisproves theresult inthiscase. (2)cu—-=fdt Ad:?2' A Ad.i"l<~' =fdt Ad5F2. Then z'1*w =t'@*w -—=O. Now I(dw)(p) =1(-Z 4:A.45:-'1Ad2’)(p) am] 1 a I=- =(L ﬁ(p,r)dt) dxAda. d(Iw) =d(fl f(p,t)dI) dxi 0 H a 1 ‘ I _-=2:-Mg; j(p,i')dt) dx“Ad.\‘. 0t1=l a Clearly I(dw) +d(Icu) _-=O,¢:¢Ql*1=and Dgfierenlial Forms 225 18.COROLLARY. IfMissmoothly contractible toapoint pgeM,then every closed form cuonMisexact. PROOF. Wearegiven H:Mx[0,1] —>Mwith H(11»I)=1>fall eM. H(P>0)=P9 or P Thus Hoi1IM ->Mistheidentity Hoig: M—>M istheconstant map pg. So cu:(Hoi1)*(cu) ==:'1*(H*cu) 0=—'(H°1'0)*(w) =io*(H*<v)- But d(H*cu) =H*(dw) ==O, S0 cu—0-=i1*(H*w) —ig*(H*w) :d(I(H*w)) bytheTheorem. ¢§¢ Corollary l8iscalled thePoincare Lemma bymost geometers, while d2=O iscalled thePoincare Lemma bysome (Idon’t even know whether Poincare had anything todowith it.)Inthecaseofastar—shaped open subset UofR”,where wehave anexplicit formula forH,wecanﬁnd (Problem 23)anexplicit formula forI(H*w), forevery form cuonU.Since thenew form isgiven byanintegral, wecansolve thesystem ofpartial diﬁereiitial equations cu=dnexplicitly in terms ofintegrals. There areclassical theorems about vector ﬁelds inR3which canbederived from thePoincare Lemma and itsconverse (Problem 27),and originally dwasintroduced inorder toobtain auniform generalization ofall these results. Even though thePoincare Lemma anditsconverse ﬁtvery nicely intoourpattern forbasic theorems about differential geometry, ithasalways been something ofa mystery tomejust why dturns outtobesoimportant. Ananswer tothisquestion isprovided byatheorem ofPalais, Nalmal Operations onDiifléz-eizlz'al Harms, Trans. Amer. Math. Soc. 92(1959), 125--141. Suppose we have anyoperator Dfrom k-forms tol-forms, such thatthefollowing diagram 226 Chapter 7 commutes forevery C°°map f:M—>N[itactually suﬁices toassume that thediagram commutes only fordiffeomorphisms f]. * k-forms onM<—J-f-— k-forms onN ,.( (Dl-forms onML l-forms onM Palais’ tlieorem says that, with fewexceptions, D=O.Roughly, these excep- tional cases arethefollowing. Ifk_-=l,then Dcanbeamultiple oftheidentity map, butnothing else. Ifl=k+I,then Dcanonly besome multiple ofd. (Asacorollary, d2=O,since d2makes theabove diagram commute!) There is only oneother case where anon-zero Dexists—when kisthedimension ofM andl=O.Inthiscase, Dcanbeamultiple of“integration”, which wediscuss inthenext chapter. Dﬁrenlial Forms 227 PROBLEMS 1.Show thatifwedeﬁne U.(vl a'''1vk) Z(vO""I(l)! --':vO""'I(k)): then 0°ﬂ°(v1,---Mt)=<TP°(v1,---=vk)- 2.LetEbeAltwithout thefactor I/kl, anddeﬁne w7Yiy =Em ®27).Show thatAisnotassociative. (Trycu,27eS2](V)andBeS22(V).) 3.LetS’CSk+; bethesubgroup ofall 0which leave both sets{I,...,k}and {k+ I,...,k+1}invariant. Acrosssection ofS’isasubset KCSk_(_1 containing exactly oneelement from each leftcoset ofS’. (a)Show that foranycross section Kwehave (‘JAWilli» ---=vk+i) =Z$8710 ‘w®Tl(vcr(l)= --->v0(/<+t‘))- oteK This deﬁnition may beused even inaﬁeld ofﬁnite characteristic. (b)Show from thisdefinition thatwAr; isalternating, andwAT]=(--i)k’T]/\(U. (Proving associativity isquite messy) (c)Apermutation 0GSk_|_; iscalled aslzujiepermuialion if0(1) <0(2) < < o(k) ando(k+ I)<o(k+2) < <o(l<+1). Show thatthesetofall shuﬁie permutations isacross section ofS’. 4.ForvGVand cuEQk(V), wedeﬁne thecontraction v_JcuGQ2“ (V)by (U-l°")(vi,---,v.1<-1) =°J(v,vi,---,vk-»1)- This issometime alsocalled theinnerproduct andthenotation igwisalsoused. (a)Show that v_l(w_l cu)=-—w_l(v_J cu). (b)Show thatifvi,...,v,,isabasis ofVwith dual basis ¢>1,...,¢>,,,then 0 j¢anyi'., "’"’(""‘A"'A""*')“ l(-1)""*¢,-,A---A'<i§;"A...A¢,, ifj=i.,. (c)Show that forcu;GQ'l‘(V) and cu;E§2’(V) wehave v_l(w1 A(1)2): (v_J(U1)/\w3 +(-l)"w1 A(v_|(U2). (Use (b)andlinearity ofeverything.) 228 Chapter 7 (d)Formula (c)canbeused togive adeﬁnition ofcu]Aw; byinduction onk+1 (which works forvector spaces over anyﬁeld): IfAisdeﬁned forforms ofdegree adding upto<k+1,wedeﬁne all Aw2(vls-- -av/C-I-1') =“vi --Iml) Au-22:l(v2w ~'avk-|-I) +(-l)k[w1/\(v1-I <v2)](v2,---,vl<+t)- Show thatwith thisdeﬁnition (U1Acu;isskew-symmetric (itisonlynecessary to check thatinterchanging v,and‘U3changes thesign oftheright side). (e)Prove byinduction that Aisbihnear andthatcu]Acu;==(—-l)"’w; Awl. (f)IfXisavector ﬁeld onMandcuak-form onMwedeﬁne a(lc--l)—form X_Jcuby (X-Iw)(1>)=XU1)-I win)- Show that ifcu;isak-form, then X_|(cu1 A(U2)=:(X_l w1)Acu;+(—-l)kw1A(X_| (U2). 5.Show that nfunctions j],.. .,fig:M—>Rform acoordinate system ina neighborhood ofpeMifandonly ifdﬁA---Adf,,(p) ¢O. 6.Anelement cu6Q2‘(V) iscalled decomposable ifcu=¢]A---/\(}’),Q forsome ¢iEV*=Q’(l/)- 2(a)IfdimV53,then every cuGQ(V)isdecomposable. (b)If¢;,i’=I,...,4 areindependent, then w=(<1),A(pg)+(¢3AQ54)isnot decomposable. Hint: Look atcuAw. 7.ForanycueS22‘(V), wedeﬁne theannihilator ofcutobe /lmi(cu)={¢€ V*I¢/\uJ=0}. (a)Show that dimAn7z(w) 5R, andthatequality holds ifandonlyifcuisdecomposable. (b)Every subspace ofV*is/lrm(0.J) forsome decomposable cu,which isunique uptoamultiplicative constant. (c)Ifto]andco;aredecomposable, then Am:(w1) CAmz(w2) ifandonly if cu;:w]AT)forsome q. (d)lfw,-aredecomposable, then Amz(w]) ft/lnn(0Jg) ={O}ifand only if cu,Acu;¢0.Inthiscase, Aﬂnfwl) —|—/l?272(CU2) =Amlfﬁdl /\(U2). (e)lfVhasdimension n,then anycueQ""l (V)isdecomposable. (f)Since v,-eVcanberegarded aselements ofV**,wecanconsider v1A---A vi,eS2"‘(V*). Reformulate parts (a)-—(d) interms ofthisAproduct. Dgﬂerenlial Forms 229 8.(a)LetweQ2(V). Show that there isabasis <,i>1,...,¢,,ofV*such that ('3=(¢l/\¢2) 'l""'l' (¢2r—l "\¢2r)- Hint: If w=Zja.-,~//.- A10;,I'<j choose ¢,involving 1,01,1,03,...,1,0,,and¢3involving 1,03,...,1,0,,sothat "J=¢i/\¢2+w’, where w’does notinvolve 1,01or1,03. (b)Show thatther-fold wedge product wA---Awisnon-zero anddecompos- able, and that the(r+I)-fold wedge product isO.Thus riswell—determined; itiscalled therank ofw. (c)Ifw=ZR, a,-_,-1,0; A1,0,-,show that therank ofwistherank ofthema- lI'iX(ail). 9.Ifv,,...,v,, isabasis forVand w,-=Z;-’=, oz,-,-v,-, show that (lCi(()l5_;)lU*1/\~--/\ w*,,=11*,A Av*,,. 10.Let./l =(a,-;)be annxn matrix. LetI5p5nbeﬁxed, andletq =n—p. ForH=li, < </1,, andK=k, <--- <kq,let "rat, ¢li,h,.- ¢1p+1,1<, ﬂ,g+i,1<., B”-.=det : : , CK=det : I0 I O 1 “pill: ---“.vJIp “ink: ---an,-its (a)Ifv1,...,v,, isabasis ofVand I1‘ w,-= ga,-iv,-, ;=1_ showthat w,A---Aw,,=Z:BHvH H Kw,,_|.,A---Aw,,=Z:C UK. K 230 Chapter 7 (b)LetH’={l,...,n}—-H(arranged inincreasing order). Show that 0 K;éH" UHAUXZ €H'Hi'U1/\-/\'U K*H’ , .. H _ , where eH,H.- isthesign ofthepermutation (1.........71) /21!/221'--shpsklsw-skq . (c)Prove “Laplace’s expansion” detA=285,51 BHCH’. H 11.(Cartan’s Lemma) Let¢>1,...,¢k 6V*beindependent andsuppose that 1,0|,...,1,0,t eV*satisfy (¢’i/\¢i)+"'+(¢k/\ll/k) =0- Then k 1,0;=ZCl_;'i¢J', where a,-,-=a,-,-. J'=| 12.Inaddition toforms, wecanconsider sections ofbundles constructed from TM using Qandother operations. Forexample, ifE==rt:E—>Bisavector bundle, wecanconsider Q"‘(§*), thebundle whose ﬁbre atpisS2"([rt'"' (p)]*). Since wecanregard BF asanelement of (M,,)"’*, anysection ofS2"(T*M) canbewritten locally as B B/1%/\---/\ﬁ. (a)Show thatif 8A A3——/1 3A A3gBy' "aw‘ OX‘ thcn I Byi T. Di-ferential Forms 231 This shows thatsections ofS2"(T*M) arethegeometric objects corresponding tothe(even) relative scalars ofweight -1inProblem 4-l0. (b)Let‘J}k[”’](V) denote thevector space ofallmultilinear functions Vx---xVxV*x---x V*—>Q’"(V).L J Q ..._J V '"'7’W ktimes Itimes Show thatsections of‘J]"["l(TM) correspond to(even) relative tensors oftype and weight I.(Notice that ifv,,...,v,, isabasis forV,then elements of Q"(V)canberepresented byrealnumbers [times theelement 11*,A---Av*,,].) (c)If‘J}‘§m](V) isdeﬁned similarly, except that S2”’(V) isreplaced byQ"'(V*), show thatsections of3”}ﬁ,](TM) correspond to(even) relative tensors oftype andweight —-1. (d)Show that thecovariant relative tensor oftype andweight Ideﬁned in Problem 4-l0, with components 2“"2",corresponds tothemap V*x-~x V*—>Q”(V) ntimes given by((151,...,<,i>,,) 1—>(,0,A---A(,()_q,Interpret therelative tensor with com- ponents £;,___,-,, similarly. (e)Suppose Q"“”(V) denotes allfunctions I7:Vx xV—>Rwhich areof theform T)('U1,...,'lJ,,)=[w(U1,.. .,v,,)]w waninteger forsome weS2"(V). Let‘J}"[”"”]( V)bedeﬁned like‘J]"l”], except thatS2"(V) is replaced byS2"‘“’(V). Show thatsections of‘J}"[”“”](TM) correspond to(even) relative tensors oftype and weight w.Similarly forBjﬁzw]. (f)Forthose who know about tensor products V®Wandexterior algebras A"(V), these results canallberestated. Wecanidentify 7,2(V)with k I ®I/*®® V=Y*2;~~®Yi®K2~~®K-ktimes (times Since §2’"(V) %A"'(V*) %[A"'(V)]*, wecanidentify k I 'r;"l'"l(v) with cg)v*®® V®A"’(V) k I ‘i;(‘,,,(v) with (X)1/*®® V®A”’(V*). 232 C/rapier 7 Consider, more generally, . k 1'T.-'k[m.wl(V) :__®V*®®V®®w /(mw) k I =it...1<v>-® we1/@®““-v*>»Noting thatA”(V)®- --®A"(V) isalways 1-dimensional, show thatsections of ‘J',',"[”“”l(TM) and?]’[‘n,w,(TM) correspond to(even) relative tensors oftype andweight wand-—w, respectively. 13.(a)IfVhasdimension nandA:V—;Visalinear transformation, then themap A*; S2"(V) —>S2"(V)must bemultiplication bysome constant c. Show that c=detA.(This may beused asadeﬁnition ofdetA.) (b)Conclude thatdetAB=(detA)(det B). 14.Recall thatthecharacteristic polynomial ofA:V—>Vis X(A)=det(AI -.4) =A"-(ti-a¢@.4)i""‘ +---+(-1)"detA =at"-@,i""1+ c2A”"2 +--.+(-l)”c,,. (a)Show that ck=trace ofA*: Qk(V) —>Q"(V). (b)Conclude thatc,r<(A B)=c;,(BA). (c)Let beasdefined inProblem 4—5(xiii). IfA:V—>Vhasama- trix(a,-2)(with respect tosome basis), show that 1 1'1ii iiii---iickl/1): D allafz‘Hair; 511---is-' l_1,...,I,i,- Jl$"'!Jk Thus, if5isasdeﬁned onpage 130, and Aisatensor oftype then the function p1—>ck(A(p)) canbedeﬁned asa(2k)-fold contraction of A13;---®A®6.\-_i._.,,....._._.-I ktimes 15.LetP(X,-;) beapolynomial inn2variables. Forevery nxn matrix A=(a,-J-) wethen have anumber P(a;j-). Call Pinvariant ifP(A) =P(BAB"l) for allAandallinvertible B.This problem outlines aproof thatanyinvariant P isapolynomial inthepolynomials c1,...,endeﬁned inProblem l4.Wewill Dgfereiilial Forms 233 need thealgebraic result thatanysymmetric polynomial Q(y,, ...,yn)intheii variables y1,.. .,y,,canbewritten asapolynomial in01,...,o,,, where or;is theill‘elementary symmetric polynomial ofy,,...,y,,. Recall thatthe0,can bedeﬁned bytheequation R ]_[(y-yr)=y"—viy”'"‘ +---+(—1)"-2»1"-=1 Thus, they arethecoefficients, uptosign, ofthepolynomial with roots y1,..., yn.Since theeigenvalues A1,...,knofamatrix Aare,bydeﬁnition, theroots ofthepolynomial X(k), itfollows that Cl'(-A) =Ui(A-li"-iA-n)- Wewillﬁrstconsider matrices Aover thecomplex numbers (C(thecoefficients ofPmay alsobecomplex). (a)Deﬁne Q(y|, .,.,y,,) tobeP(A) where Aisthediagonal matrix (italThen there isapolynomial Rsuch that Qlyli--'=}’H) -:'R(UI(y|:--':yﬂ):|---iUﬂ(y|:--->yﬂ))- Thepolynomial Rhasrealcoefficients ifPdoes. (b)P(A) =R(c1(A),. _.,c,,(A)) foralldiagonalizable A. (c)The discriminant D(A) isdeﬁned as]_],-#1-(A; —A,-)2, where A;arethe eigenvalues ofA.Show that D(A) canbewritten asapolynomial intheentries ofA. (d)Show that P(A) =R(c1(A), ...,c,,(A)) whenever D(A) ¢0.Conclude, by continuity, thattheequation holds forallmatrices Aover (C.(This lastconclu- sion follows even if(Cisreplaced bysome other ﬁeld, since thesetwhere D¢O isZariski-dense; thisis“the principal ofirrelevance ofalgebraic inequalities”, compare pg.V375.) Now suppose thatthecoefficients ofParerealandthat P(A) =P(BAB"l) forallreal Aand real invertible B. (e)The same equation holds forcomplex Aandcomplex invertible B.(Regard theequation asn2polynomial equations inthea,-,-andbi,-.) 234 C/rapier 7 16.(a)Letv1,...,v,, beabasis forV,andletw1,...,w;, GVbegiven by H w,»== goi,-,-v,-. I-"=1 ForweQ"(V) Showthat wfwls---=wR)= E aIw(vi1>---avi;,-)-i 1=i‘1<---<i‘;,- where 0!]isthedeterminant ofthe kxksubmatrix of(&';'j') obtained byselecting rows i1,...,ii<. (b)Generalize Theorem 7andCorollary 8tok-forms. (c)Check directly from (b)that thedeﬁnition ofddoes notdepend onthe coordinate system. 17.Show that d(Z:,-{J 01,-;dxlAdxj) =0ifand only if Boi-- Ba-/< B0:-k __ax: -ax’,-A +3;, =0 foralli <;<l<. 18.InProblem 5-l4 wedeﬁned LXA foranytensor ﬁeld A. (Show thatifwisak-form, then soisLXw. (b)Show thatQ3V Lxlwi Awz) ==Lxwi Awz+wt/\Lxwz. (c)Using 5-l4(e), show that X(w(X|,--->Xi-)-) =LX(w(X1=--->Xk)) =LX0‘J(X1i---=Xk) k +2('_])i+]a2(l:/Y: Xi]: X1: ---iEs '''s/Y/C)» i=1 (d)Deduce thefollowing twoexpressions: d°J(X1,---,Xl<+1) k+1 =2('—l)l+’LX,.(:J(A’],-..,Xi,...,Xk+]) 1'21 +Z1-1)"+*'*'<»tiX.~,X,-1, Xi,_.-._...55.....rat.)i<j Dﬂereiitial Fbriiis 235 dw(X1,- --,-1’t<+i) Ik-I-1 . =52f_2)’+liXz'(w(X1i---iXi>---iXk+1))i=1 "l"LXfw(X1:"-:Xf:---:XR+1)} (e)Show that X_|dw=LXw—d(X_lw), i.e., d<"(-Y1»---,Xk+1) =(1-X1w)(X2, -A-=Xk+1) -dl-Y1-l w)(X2>---,Xk+l)- (This may beused togive aninductive deﬁnition ofd.) (f)Using (e),show that d(LX w)=L,-((dw). 19.LetCljjben2functions onR”with a,-,-=aj,-.Show thatinorder forthere tobefulictions 1.11,. ..,1.1,,inaneighborhood ofanypoint inR"with ,,.-1%+%“T2 Bx!’ Bx‘ itisnecessary andsufficient that 3261,"; 320;‘); B2a,~,- B2a;;, — . I :— .. fll'' .Bx"‘Bx" Bx!Bx’ Bx"‘Bx' BxJBx’ ora1’bk’! Hint: First setuppartial differential equations forthefunctions )3-,1,=Buy/Bxk— Bug/Bx], anduseTheorem 6-l. 20. Compute that -d—d “d6” Z A -J; x. I+J’ (Atmost places B=arctan y/x [+aconstant] .) 21.(a)IfwisaI-form fdx on[0,1]with f(O) =f(I), show that there isa unique number Asuch that w—k dx=dgforsome function gwith g(O) =g(l). Hint: Integrate theequation w—Adx=dgon[0,1]toﬁnd A. (b)Let1':S‘—>R2—{O} betheinclusion, andlet0’=z'*(dB). Ifc: [0,1] —>S] is c(x)=(cos2rtx,sin Zrtx), show that c'*(o’) 2Zndx. (c)Ifwisaclosed 1-form onSlshow thatthere isaunique number Asuch that w-Ito’isexact. 236 Chapter 7 22.(a)Show that every wGQk(V1 EBV3)canbewritten asasum offorms cu;/\0);where cu;hasdegree ozand (U2hasdegree )3=k—ozand w1(v1,...,v,,,) =Oifsome vieV; w3(v;,...,vg) =Oifsome 11,-GV1. (b)Ifdim V3=I,and O¢kEI/3*, then cucan bewritten uniquely ascu;+ (0)2/\A),where cu;isak-form andcu;isa(k—I)-form such that w1(v1,...,v;<) =0ifsome v,-eV; w;;(v1,...,v;<_.;) =0ifsome v,-GV3. 23.LetUCR"beanopen setstar-shaped with respect toO,anddeﬁne H:U>< [0,1] —>UbyH(p,t) =Ip.If w=Z w,-,___,-R dxi‘A---/\dxik i1<~-<51; onU,show that I(H*w) 1* 1 =Z: Z:(—1)°"'l(f tk'"‘w,-,_,_;k(Ix) dt)x""dx" /\-../\dx"<*/\---/\dx"‘. 1']<---<1}; G51 0 24.(a)LetUCR2beabounded open setsuch thatR2—U isconnected. Show thatUisdiﬁeomorphic toR2,andhence smoothly contractible toapoint. (The converse isproved inProblem 8-9.) Hint: Obtain Uasanincreasing union of sets, thekthsetbeing aﬁnite union ofsquares containing thesetofpoints inU whose distance from boundary Uis5I/k. IIU TlifRRR if lllTVila 7 F ¢.J‘":*.. **1" ', fl, ,r_a ,t_n tYRi l _LP4?“ i1*’T-'Tii*4‘;:*;“""i“iI411,o_J__4a..toR;Lj___4L.nIIa _\A:J3Z,tfie’iigikfr"""‘!‘L‘h|..__.4it‘J;;_ iii} -_ 1| a ;IEEE1he"2(b)Find abounded open setUCR3such thatR3—Uisconnected, butUis notcontractible toapoint.1.Jo Dzfe?'en£z'al Forms 237 25.Let UCR"beanopen setstar-shaped with respect toO.IsUhomeo- morphic toR"? (Itwould certainly appear so,butthe“obvious” proof does notwork, since thelength ofraysfrom 0totheboundary ofthesetcould vary discontinuously.) 26.Let(,)betheusual inner product onR", H ((1,1))=ids". iml (a)Ifv1,...,v,,_.1 eR",show thatthere isaunique vector vi><---><v,,_.1 eR” with w v (v1><---xvn-1,w)=det( :1) forallweR”. vn~i (b)Show that x ><eS2""l (R"), andexpress itinterms ofthe2*,-,using the expansion ofamatrix byminors. (c)ForR3show that v><w=(v2w3 —v3w2, 113w‘ —vlw3, vlwz —vzwl). (First ﬁnd alle;><ej 27.(a)Iff1R"—>R,deﬁne avector ﬁeld grad f,thegradient off,onR" by ”afa" a g"adf=Za—x="W=ZD"f'a—x="fa} fml Introducing theformal symbolism '1 8 238 Chapter 7 wecanwrite grad f=Vf. If(grad f)(p)=wp,show that =(U9 w): where DUf(p)denotes thedirectional derivative inthedirection vatp(or simply v,,(f), ifweregard upER",,). Conclude thatVf(p) isthedirection in which fischanging fastest atp. (I3)IfX=Z‘;'$1a'8/ﬁx’ isavector ﬁeld onR",wedeﬁne thedivergence ofXas nBa;d'X= ——.. IV ax; (Symbolically, wecanwrite divX=(V,X).)Wealsodeﬁne, forn=3, curlX(=V><X) 3a3 Baz 8 Ba‘ 8&3 8 3:12 Ba‘ 8 Bx?"aw8.142+3x3"81"’3x2+F"FF‘ Deﬁne forms tux=aldx+a2dy+a3dz 27,1,»==a'dy/\dz+a2dz/\dx+a3dx/\dy. Show that (if==tug,-adf d(wX) :7icurlX d(27,y) =(divX)dx/\dy/\dz. (c)Conclude that Curlgrad f=0 divcurl X=0. (d)IfXisavector ﬁeld onastar-shaped open setUCR“andcurlX=0, then X=grad fforsome function f:U—>R.Similarly, ifdivX=0,then X—_=curlYforsome vector ﬁeld YonU. CHAPTER 8 INTEGRATION The basic concept ofthischapter generalizes lineandsurface integrals, which ﬁrstarose ﬁ'om very physical considerations. Suppose, forexample, that c:[0,1]—>R2isacurve andw=fdx +gdy isa1-form onR2(where f,g:R2—>R,and xand ydenote thecoordinate functions onR2). Ifwe choose apartition 0=to<---<1,,=1of[0,1],then wecandivide thecurve c intonpieces, thei‘hpiece going from c(r,-_;) toc(:,-). When thedifferences I;-[I-QQIIaresmall, each such piece isapproximately astraight segment, with c(I) ¢’(!:) C(€|') <_c2(!,-) —c2(r,-..,) 6(0) ¢'(!|'»1)____ \c'an-c‘<1.--1) horizontal projection cl(I,-)-c'(!,-_|) andvertical projection c2(!,-) —c2(t,-._.|). Wecanchoose points c(§,-) oneach piece bychoosing points E;6[1,-_1,t,-]. For each partition Pand each such choice E-=(E1,...,§,,), consider thesum s<P.s>=Z1"<c<s.-))ta-"<1.-)-c‘<:.~_1>1+gee.-))tczw)-c2<:.-~1)1-I'=l Ifthese sums approach alimit asthe“mesh” ||P|] ofPapproaches 0,that is, asthemaximum of!,--2‘,-..,approaches 0,then thelimit isdenoted by ‘/fdx-l-gdy. (This isacomplicated limit. Tobeprecise, if|]P]| =max(!,- -I,-_|}, then the equation I lim.S'(P,§)=/fdx+gdyHP"-+0 ¢ 239 240 Chapter 8 means: forall.1:>O,there isa5>0such that forallpartitions Pwith ||P]] <5. wehave ‘S(P,§)—-ffdx+gdy <5 forallchoices EforP.) The limit which wehave justdeﬁned iscalled a“line integral”; ithasanatural physical interpretation. Ifweconsider a“force ﬁeld” onR2,described bythe ////4/"” f /‘ then S(P,§)isthe“work” involved inmoving aunitmass along thecurve cin thecase where cisactually astraight linebetween I,-_|andI;andfandgare constant along these straight linesegments; thelimit isthenatural deﬁnition ofthework done inthegeneral case. (Inclassical terminology, thedifferential fdx+gdywould bedescribed asthework done bytheforce ﬁeld onan“in- ﬁnitely small” displacement with components dx,dy;theintegral isthe“sum” ofthese inﬁnitely small displacements.) Before worrying about how tocompute thislimit, consider thespecial case where J/0 6(1)=(bro)- l II]ll1iS C2156, (‘](!;) —(‘](!;'...]) =I;"—!;__,|, Whilﬁ ('2(!;') "—(‘2(!;'...|) =0,S0 II s<P.s>=Zf(E;,yo)(!: -1,-...)-iml lrztegratiorz 241 These sums approach 1 ffdx+gdy =ff(x,yo)dx-c 0 Ontheother hand, if yg O——-it cm=(lb+<1~:)@.y@). __ _ _; 1 a b 111611 ¢"(!r) "1f"(!:~1)=(b "a)(1r "-HH1), $0 $(P,§) =(5"-61')'Zf(§:b +(1"-€i)¢1,yo)(h' -is--1)~ ;1 These sums approach I b (b-a)Lf<xb+<1—x>@.y@)dx=[ f<x.y0>dx- Ingeneral, foranycurve c,wehave, bythemean value theorem, ¢‘](!r) '~C'(h'-1) =¢‘“(¢Yi)(!r "-It-1) 0!:Elb‘-Mil @2(r,-)—~@2(n-*1) =c2'(ﬁ:)(n- -—mi) ﬁrE[1,-._|,r.-]. C/JO "trs<P.s)={f<c<s.-)>c"<a.-) +g<c<s.->)c2’w.-)} <:.--1.--1). Asomewhat messy argument (Problem I)shows thatthese sums approach what itlooks likethey should approach, namely I /0[f<c<mc"<:)+g<c<:))c2’<:)1di. Physicists’ notation (orabuse thereof) makes iteasy toremember thisresult. The components c',c2 ofcaredenoted simply byxandy[i.e., xdenotes 242 Chapter 8 xocand ydenotes yoc;thisisindicated classically bysaying “letx=x(!), y=y(!)”]. The above integral isthen written ' d dffl.-1x+gdJ’=L [f(1',J")E? +s(x,y)?ﬂ dr- lnpreference tothisphysical interpretation of“line integrals”, wecanin- troduce amore geometrical interpretation. Recall that dc/d!(E,-) denotes the CUE) ¢'(¢it) fiﬂ _dl($1) £'(*':‘-1) tangent vector ofcattime E,-.Then thesums (*) Zw<<.~<s-)1 (gen) -<1;»-1.-..,)In-I =Ztree.->)c"<s.-> +g<c<s-))c2’<s.->1 -<:.--1.-H.)IR] clearly alsoapproach 1 f[feenc"<o+g<c<mc2’<:)1d:.0 Consider thespecial case where cgoes with constant velocity oneach (5-1, 1;) 1iitegi'a£i0tt 243 Ifwe choose anyE;E(I,-.,|,!,-), then length of%!€(E,-) =theconstant speed on(I,-..|, 1,-) length ofthe segment from c(!,-..,) toc(t,-) It'~1:“: ' so d[length of?:(§,-)] -(!,--XIDHI) =length ofsegment from c(!,-..|) toc(!,-). Inthiscase, fl Z|:lC}"lgil1 Of -(I;—l';...]) lim] isthelength ofc,andthelimit ofsuch sums, forageneral c,canbeused asa deﬁnition ofthelength ofc.The lineintegral fw=limit ofthesums (*) C canbethought ofasthe“length” ofc,when ourruler ischanging contin- uously inaway speciﬁed byw:Notice that therestriction ofw(c(!)) tothe 1-dimensional subspace ofR269) spanned bydc/dz‘ isaconstant times “signed length”. The natural way tospecify acontinuously changing length along c istospecify alength onitstangent vectors; thisisthemodern counterpart of theclassical conception, whereby thecurve cisdivided into inﬁnitely small parts, theinﬁnitely small piece atc(I), with components dx,dy,having length f(¢‘(!)) dx+g(@(!)) 4)’- Before pushing thisgeometrical interpretation toofar,weshould note that there isnol—form cuonR2such that fw=length ofc forallcurves c. C Itistruethatforagiven one-one curve cwecanproduce aform 0)which works forc;wechoose w(c(!)) ES2'(R2c(,)) sothat R R I: w(c(!))(%) =1, it '\‘ llkernel w(c(r)) -ls (choosing thekernel ofcuarbitrarily), andthen extend wtoR2.Butifcis 244 C/zapter 8 notone-one thismaybeimpossible; forexample, inthesituation shown below, there isnoelement ofS2'(R2,(,.)) which hasthevalue 1onallthree vectors. Ingeneral, given anywonR2which iseverywhere non-zero, thesubspaces Ap==kerw(p) form aI-dimensional distribution onR2;anycurve contained inanintegral submanifold ofAwillhave “length” O.Later wewillseeaway ofcircumventing thisdiiiiculty, ifweareinterested inobtaining theordinary length ofacurve. Forthepresent, wenote thatthesums (*),used todeﬁne this generalized “length”, make sense even ifcisacurve inamanifold M(where there isnonotion of“length”), andwisal—form onM,sowecandeﬁne fccu asthelimit ofthese sums. One property oflineintegrals should bementioned now, because itisob- vious with ouroriginal deﬁnition and merely true forournew deﬁnition. If p:[0,I]—>[0,1]isaone-one increasing function from [0,1]onto[0,1],then the curve copi[0,1]—>Miscalled areparameterization ofc—it hasexactly the same image asc,buttransverses itatadiiierent rate. Every sumS(P,E) forc isclearly equal toasum .S'(P',§") forcop,andconversely, soitisclear from ourﬁrstdeﬁnition thatforacurve c:[0,1]—>R2wehave /0):] ..,r: cop (“the integral ofwover cisindependent oftheparameterization”). This isno longer soclear when weconsider thesums (=1<)foracurve c:[0,1]—>M,noris itclear even foracurve c:[0,1]—>R2,butinthiscase wecanproceed right to theintegral these sums approach, namely 1 f[re-<:1)c"<:)+ g<c<mc2*<:)1d:.0 Integration 245 The result then follows from acalculation: thesubstitution J’=p(u) gives I L1f<c<:11¢-"<11 +g<c<:>1c2’<:11dr p"'(l) =f (0)[f(@(P(H)))@"(P(H))+3(¢‘(P(H)))@2'(P(H))lP'(u)dup—l =Lllﬂv QP(H))(¢‘ °P)"(1-')+ g(¢‘OP(H))(@ °P)2'(H)l du- Foracurve inR",and al—form w=EL] w,~dxl, there isasimilar calcula- tion; forageneral manifold M,wecanintroduce acoordinate system forour calculations ifc([0, 1])liesinonecoordinate system, orbreak cupintoseveral pieces otherwise. Wearebeing abitsloppy about allthisbecause weareabout tointroduce yetathird deﬁnition, which willeventually become ourformal choice. Consider once again thecase ofa1-form onR2,where fw=f'1f<c<i11c“<i1+g<c<i11c2*<:11dr-c 0 Notice that ifIisthestandard coordinate system onR,then forthemap c:[0,1] —>R2wehave c*(fdx+gdy)=(f0@)c*(dr) +(gQ@)¢'*(dy) =(f°C)d(X'=>¢')+(g°@)d(J’°@) =(foc)c"dt +(goc)c2'd!, sothat formally wejustintegrate c*(fdx+gdy); tobeprecise, wewrite c'*(f dx+gdy) =Itcl:(intheunique possible way), and take theintegral ofiton[0,1]. Everything wehave saidforcurves c:[0,1] —>R“could begeneralized to functions ct[0,1]2 —>R".Ifxandyarethecoordinate functions onR2,let as 3c 8c M n="* 5 @_ ( Z’ 3J*—c*/"'_""\ cuCD\.___,./ I":Q:‘-=: as5-: Forapairofpartitions so<---<s,,,andto<---<Inof[0,I],ifwechoose 246 C/zapter <9 E,-yE[5,-_|,s,-] ><[tj-1,t;] andwisa2-form onR”,then IIIII CII /I-5‘1'~1 st we-(s-,-1) (§—;<s.-,-1.§—;<r.-,-1) <s.---5'1‘-1)(!j-1,--119 isa“generalized area” oftheparallelogram spanned by 8c 8c 5(E:j), a—y(€i;)- Thelimit ofsums ofthese terms canbethought ofasa“generalized area” ofc. Tomake along story short, wenow proceed with theformal deﬁnitions. AC°°function c:[O,1]" —>Misealled asingular /<-cube inM(theword “singular” indicates thatcisnotnecessarily one»one). Wewilllet[0,1]"=R0= 0eR,sothatasingular 0-cube cisdetermined bytheonepoint c(0) EM. Theinclusion mapof[0,11*inn’<Willbedenoted by1*;[0,11*_>nk;itis called thestandard /<-cube. Ifwisak-form on[0,1],‘,andxl,...,x" arethecoordinate functions, then cu canbewritten uniquely as w=fdx'/\---Adxl‘. Wedeﬁne =ff(x',...,x")dx' dxk [0=ll"" f(1)tobe ff inclassical notation, which modern _ notation attempts tomimic asfar[0-11* [0-1]" . .aslogic permits Ifwisak-form onM,andcisasingular k-cube inM,wedeﬁne /.~r=110.11‘ where theright hand sidehasjustbeen deﬁned. Fork=0,wehave aspecial deﬁnition: aO-form isafunction f,andforasingular 0—cube cwedeﬁne /Cf=f<¢~<0>>. lntqgmtiorz 247 1.PROPOSITION. Lete:[0,1]”—>R"beaone-one singular n-cube with detc’30on[0,1]".Letwbethen-form I w=fdx'/\---Adx”. Then C /~J=/ft~—> Cc<t0.11"> * T PROOF Bydeﬁnition, g/cu :/Ic*(w) C [Q1111 ==/(foc)(det c’)dxl/\---/\dx” byTheorem 7-7 t0,]]r1 =f(fo0)]detc']dxl/\---/\dx" byassumption [g,|]r| = f f bythechange ofvariable formula. ‘I0 c([0,l]") 2.COROLLARY. Letp:[0,1]"—>[0,1]"beone-one onto with detp’3;0, letcbeasingular k-cube inMandletwbeak-form onM.Then c cop PROOF. Wehave w=(C0pm=fp*<c*w>Cop [°.|l"" [9-1]“ =Ic*(w) bytheProposition, since pisonto [01]“ 0:0 C 248 Chapter e The map cop: [0,1],‘—>Miscalled areparameterization ofcifpi[0,1]"—> [0,1],‘isaC°°one-one onto map with detp’560everywhere (sothat p" is alsoC°°);itiscalled orientation preserving ororientation reversing depending onwhether detp’>0ordetp’<0everywhere. The corollary thus shows independence ofparameterization, provided itisorientation preserving; anori- entation reversing reparameterization clearly changes thesign oftheintegral. Notice thatthere would benosuchresult ifwetried todeﬁne theintegral overc ofaC°°function f:M—>Rbytheformula f foC. t0.11*' Forexample, ifct[0,1] —>Mthen 1 1 ff(c(!))dt isgenerally eéf f(c(p(!))) cit. 0 0 From aformal point ofview, differential forms arethethings weintegrate be- cause they transform correctly (i.e., inaccordance with Theorem 7-7, sothat thechange ofvariable formula willpop up); functions onamanifold cannot be integrated (wecanintegrate afunction fonthemanifold R2‘only because it gives usaform fdxl /\ /\dxk). Our deﬁnition oftheintegral ofak-form wover asingular k-cube ccan immediately begeneralized. AIt-ehain issimply aformal (ﬁnite) sum ofsingular k-cubes multiplied byintegers, e.g.. lC|-262 +363. The It-chain let=l-clwillalsobedenoted simply bycl.WeaddIt-chains, andmultiply them byintegers, purely formally, e.g., 3(f1+31'-0+ (-2l(¢'1 +63+6'2)="262 -263+51'4- Morcovcr, wedeﬁne theintegral oftooverak—chain c=2,.a,-c,»intheobvious way: Q): a‘f OJ. fxafc; Ic,- The reason forintroducing It-chains isthat toevery /<-chain c(which may be justasingular /c-cube) wewish toassociate a(k—-l)-chain Be,which iscalled theboundary ofc,andwhich issupposed tobethesum ofthevarious singular lrztegratiorz 249 (k—1)-cubes around theboundary ofeach singular k-cube inc.Inpractice, it isconvenient tomodify thisidea. Theboundary ofI2,forexample willnotb , e thesum ofthefour singular 1-cubes indicated below ontheleft,butthesum, ~1 -1 +1 with theindicated coefﬁcients, ofthefour singular 1-cubes shown ontheright. (Notice that this ill ' wnotchange theintegral ofa1-form over 812.) Foreach 1' with 151'5nweﬁrstdeﬁne twosingular (n—l)—cubes 13-0)and I1)(the (2,0)-face and(1',l)-face ofI")asfollows: IfxE[0,l]“", then I?-'0)(x) =]“(x',. ..,x"_1,0,x",...,x"_1) =-.(x',...,x'_',0,x',...,x”_'), 1{j._,,(x) =]”(x1,.. .,x"-1, 1,1-",.. .,x"-1) =(xl,...,x"',l,x',...,x“'1). Ii2.1> I I 151.01 lim) I(1.01 2<1.11 --, . ...,- . 1(2,u) 250 Chapter e The (i,or)-face ofasingular n-cube cisdeﬁned C(;_a) =C0(I3-ﬂy). c cm 0'0) ¢'(21) c 90.0) 0(0) C“-‘P Now wedeﬁneH 6c=Z Z (_i)l+aC(f,a). 5:] 1‘I=0,l Finally, theboundary ofann-chain Z,a,-c,-isdeﬁned by a,-er) =23:1,-8(c,-). These deﬁnitions allmake sense only forn2LForthecase ofa0-cube c:[0,1]”—>M,which wewillusually simply identify with thepoint P=c(0), wedeﬁne Betobethenumber 1ER,andforaO—chain Z,arc;wedeﬁne e,-e,-)=Zila,-8(c,-) =Ea,-. Notice thatforal-cube ct[0,1] —>Mwehave 36'=111.1)-¢'t1.0)= so 8(8c)=1— 1=0. Wealsohave, forasingular 2-cube ct[0,l]2—>M, S as=60.1)-(‘(2.1)-60.0)+60.0), C2”) 3(3¢')=(R-‘Ql-(R—-5') P —($— P)+(Q— P)_0 ¢‘(2.0)__ R Q 90.1)¢’{2.1) Integration 25I From apicture itcanbechecked that thisalso happens forasingular 3-cube, agood exercise because thisinvolves ﬁguring outjustwhat theboundary ofa 3-cube looks like.Ingeneral, wehave: 3.PROPOSITION. Ifcisany n-chain inM,then 8(8c) =0.Brieﬂy, 32=0. PROOF. Let1'5j5n—l,and consider (](’:.,a))U.,ﬁ). ForxE[0,1]"_2, we have, from thedeﬁnition (]iie)l<1.ﬁ)(>t) =]ft.e)(]i}1i)(x)) =1,';.,,,,(s-',. ..,xf-',,e,x-1', ...,x"_2) =]”(x',...,x’_',t.r,x',...,x"_1,ﬂ,x*',...,x"_2). Similarly, (1ii'+1.ﬂ))t=".a) =]iit+1.r)(]<'i.ii)(")) =If‘)-+,'ﬁ)(a",...,x'i"l,a,x‘,...,x”'2) =I"(x',...,xi_',a,xi,...,xj_',,3,xj,...,x"_2). Thus (]ﬁ.‘a)){j.,B) =(]6.+,‘B))(,,a) for2'5j5n—1.Itfollows easily forany singular n-cube cthat (t-(,-,,,,))U-H5) =(cu-_,_,,m)(,-_,,,) for2'5j5n—1.Now I1 3(3c)=a(Z Z(-1)"+"q,-_,,,) Ila 0] sr.;M311= "l)i+a+j+ﬂ(C(1',cc))(j,B)- Inthissum, (c(,-,,,,))(y-J3) and(c(y+1,,5))(,-My occur with opposite signs. Therefore allterms cancel inpairs, and 3(3c) =0.Since thetheorem istrue forsingular n-cubes, itisclearly alsotrue forsingular n-chains. '3' Notice thatforsome n-chains cwehave notonly 8(8c) =0,buteven 8c=0. Forexample, thisisthecaseifc=cl—C2,where c1andC2aretwol-cubes 252 Chapter <9 with t-1(0) =c;(0) and c;(l) =c;(l). Ifcisjust asingular 1-cube itself, then C2 C1 8c=0precisely when c(0) =c(l), i.e.,when cisa“closed” curve. Ingeneral, C anyIt-chain ciscalled elosed if30=0. Recall thatadi;f"Terential form wwith dw=0isalsocalled “closed”; this terminology hasbeen purposely chosen toparallel theterminology forchains (ontheother hand, achain oftlieform 3cisnotdescribed, reciprocally, by theclassical term of“exact”, butissimply called “aboundary”). This parallel terminology wasnotchosen merely because oftheformal similarities between cl and 3,expressed bytherelations (12=0and82=0.The connection between fornis andchains goes much deeper than that. Forexample, wehave seen that onR2—{0}there isal—form “d6” which isclosed butnotexact. There isalsoa l-chain cwhich isclosed butnotaboundary, namely, aclosed curve encircling (' lntegratzon 253 thepoint 0once. Although itisintuitively clear that eisnottheboundary ofa 2-chain inR2—{O},thesimplest proof uses thetheorem which establishes the connection between forms, chains, d,and3. 4.THEOREM (STOKES’ THEOREM). Iftoisa(/<-1)-form onMand cis ak-chain inM,then fdw =‘/I w. c 3c PROOF. Most oftheproof involves thespecial casewhere cuisa(k-1)-form onR2andc=I".Inthiscase, wisasum of(k—l)-forms ofthetype 1 """= 1.-fdx /\---/\dx'/\---/xdx, anditsuﬁices toprove thetheorem foreach ofthese. Wenow compute. First, alittle notation translation shows that IiAI1@_,,*(fex'A---Aéiiiiw---week)0,11‘- 0 ifj#z' =ff(-\°],...,oz,...,.xk)dx'...dxk ifj=z'.[0,]]1'\ The1'eforc ffdx' /\---Agiil/\---/\dx" 3!!‘ k “Z Z:(—l)j+“£0]] 112 *(fdx'/\... /\dx"/\---Adxk) 1_ .(.-=1) ;'=1oz=0, -"' J =(-1)"+' If(x‘,...,1,...,x")dx'...dx" I0-ll" +(~1)"/fo.-',...,o,...,)<")¢1>.-'...d).-‘R t0.11'* 254 Chapter 8 Ontheother hand, £kd(fdx' /\---/\ iii?’/\---/\(l.rk) __--.., =ID,-f dxi/\dX]/\---/\ dx‘/\---/\d.\-" t0.11*' =(-1)‘-' ID,-f. l0-11"‘ ByFubini’s theorem and thefundamental theorem ofcalculus wehave fd(fdx'/\---/\3iil'/\---/\dx")[Ii 1 1 ,,___ =(—l)i_'f D,-f(.)t",...,.rk)clxi) dx'...da‘l...(1.r" 0 0 =(-1)‘-'L]...L] (f(x',...,1,...,)t-") I k I j _k—f(x ,...,0,...,x )1dx...dx ...d:1 =(—l)"_' ff(.*c',...,l,...,x")dx'...dx" t0.11’~' +(—l)" ff(x',...,0,...,.rk)cla-1...dx". [0-11* fdo)=I w. Z” art Foranarbitrary singular k-cube, chasing through thedeﬁnitions shows that [(1):] c*w. Be 31* /clw=f c*(dw)=/ d(c*a))=/ c*w=f w. c Z2 1* 3!“ 3c Thetheorem clearly follows fork-chains also. +1»Thus Therefore Iiztegration 255 Notice that Stokes’ Theorem notonly uses thefundamental theorem ofcal- culus, butactually becomes thattheorem when c=I1andw=f. Asanapplication ofStokes’ Theorem, weshow thatthecurve c:[0,1]—> R2—{O}deﬁned by c c(t)=(cos 2rrt,sin 2m), although closed, isnot3c2forany2-chain c2.Ifwedidhave c=302,then we would have fee=/ee=f d(dt9)=/ o=o.c Bcz (:2 c2 Butastraightforward computation (which willbegood forthesoul) shows that _-J) A ‘£((6=‘lC. dX+ dy=2H. [There isalso anon-computational argument, using thefactthat “dd” really isd9forI9:R2—([0,oo) ><{O})—>R:Wehave f d6l=l9(l —s)—6l(s), c|{s,l—t=:] and6l(l-5) —9(5) —>211'ass—>0.] Although weused thiscalculation toshow that cisnotaboundary, wecould justaswell have used ittoshow that to=“:16” isnotexact. For, ifwehad to=dfforsome C°°function ftR2—{O}—>R,then wewould have Wewere previously able togive asimpler argument toshow that “d6” isnot exact, butStokes’ Theorem isthetoolwhich willenable ustodealwith forms onR"—{O}.Forexample, wewilleventually obtain a2-form cuonR2’—{O}, ifxdy Adz —ydx /\dz+zdx Ady “J (x2_|__,,2 +32):-1./2 256 Chapter <9 which isclosed butnotexact. Forthemoment wearekeeping theorigin ofwa secret, butastraightforward calculation shows that do)=0.Toprove that wis notexact wewillwant tointegrate itover a2-chain which “ﬁlls up”the2-sphere S2CR2’-{O}.There arelotsofways ofdoing this, butthey allturn outtogive thesame result. Infact, weﬁrstwant todescribe away ofintegrating n-forms over n-manifolds. This ispossible only when Misorientable; thereason will beclear from thenext result, which isbasic forourdeﬁnition. 5.THEOREM. LetMbea11n-manifold withanorientation pt,andletC1,C2 : [0,1]“—>Mbetwosingular n-cubes which canbeextended tobediffeomor- phisms i11a11eigl1borl1ood of[0,1]".Assume that clandC2areboth orientation preserving (with respect totheorientation ptonM,and theusual orientation onR").Ifwisann—form onMsuch that support wCcl([0,1]")tic2([0, 1]”), PROOF. Wewant touseCorollary 2,andwrite cg c2o(c;- 10C1) c] Theonlyproblem isthatcf‘0e,isnotdeﬁned onallof[0,1]"(itdoes satisfy det(c2‘"' ocl)"30,since cla11de2areboth orientation preserving). However, a glance attheproof ofCorollary 2willshow thattheresult stillfollows, because oftl1efactthatsupport cuCc|([0, 1]”)Oc2([0, 1]”). '3'then The common number fa), forsingular n-cubes c:[0,1]”->Mwith sup- C port toCc([0, 1]")a11dcorientation preserving, willbedenoted by /Ma). lftoisa11arbitrary n-form onM,then there isacover (9ofMbyopen setsU. each co11tai11ed insome e-([0, 1]"), where cisasingular n-cube ofthissort; if(D isapartition ofunity subordinate tothiscover, then or Integration 257 isdeﬁned foreach oiE(D.Wewish todeﬁne /../~=g/...¢-‘~-Wewilladopt thisdeﬁnition only when whascompact support, inwhich case thesum isactually ﬁnite, since support tocanintersect only ﬁnitely many ofthe sets{p:<;5(p) ;=é0},which form alocally ﬁnite collection. Ifwehave another partition ofunity ll!(subordinate toacover £9’),then Z]¢-<»=Zjf Zr»-¢-<»=Z 2/l1"¢'w;¢>e M <t>e M1,tre\It ¢>e1,tre\I1 M these sums areallﬁnite, and thelastsum canclearly also bewritten as Z Q‘)-30-w= 1/1-w, 1,tre\I1 <t>e M 1,tre\I1 M sothatourdeﬁnition does notdepend onthepartition. (Wereally should denote this sum by (M.110 fortheorientation -itofMweclearly have /I to=—f cu. (M.—t1) (M41) However, weusually omit explicit mention ofpt.) With minor modiﬁcations wecandeﬁne IMweven ifMisann—manifold— with—boundary. IfMCR”isann—dimensional manifold-with-boundary and f:M—>Rhascompact support, then fMfdx'/\---/\dx"=]!j‘. where theright hand sidedenotes theordinaiy integral. This isasimple conse- quence ofProposition l.Likewise, iff:M“—>N“isadiﬁeomorphism onto, andtoisann-form with compact support onN,then jiv M . . . . . —fco iff1sorientation reversing. Nto iffisorientation preserving 258 Chapter 8 Although n-forms canbeintegrated only over orientable manifolds, there is awayofdiscussing integration onnon-orientable manifolds. Suppose that wis afunction onMsuch that foreach pEMwehave w(p) =lnpl forsome upEQ"(Mp), i.e.,foranynvectors v1,...,12,,EMpwehave w(p)(vl: ~'-iv?!) =i7ip(v1,-- ':vﬂ)i 20' Such afunction cuiscalled avolume element—on each vector space itdeter- mines away ofmeasuring n—dime11sional volume (notsigned volume). If(x,U) isacoordinate system, then onUwecanwrite w=f|dx'/\---/\dx”] forf3;0; wecallwaC°°volume element iffisC°°.One way ofobtaining avolume element istobegin with ann-form ryandthen deﬁne w(p)=lry(p)[. However, notevery volume element arises inthisway—the form 17,,may notvary con- tinuously with p.Forexample, consider theMobius strip M,imbedded inR3. Si11ce Mpcanbeconsidered asasubspace oflR3p, wecandeﬁne w(p)(vp, wp) =area ofparallelogram spanned byvand w. ltisnothard toseethatwisavolume element; locally, wisoftheform cu=I17] forann-form ry.Butthiscannot betrue onallofM,since there isnon-form ry onMwhich iseverywhere non-zero. Theorem 7-7hasanobvious ITl0diﬁCE1ti0I‘l forvolume elements: 7-7'. THEOREM. Iff:M—>NisaC°° function between n-manifolds, (x,U)isacoordinate system around pEM,and (y,V)acoordinate system around q=f(p)EN,then fornon-negative gtV—>IRwehave f*(g|d};] /\.../\dy”]) :(gof) .det .|d_)(] /\... Ada-"|_ PROOF. Gothrough theproof ofTheorem 7-7,putting inabsolute value signs intheright place. *2* lntegmzzim 259 7-8’. COROLLARY. If(x,U)and (y,V)aretwocoordinate systems onM and g]dy'/\~~-/\dy"]=12|dx' /\---/\dA'”l g,/120 then _ By‘h=g- det(F)]. [This corollary shows that volume elements arethegeometric objects corre- sponding tothe“odd scalar densities” deﬁned inProblem 4--10.] Itisnow aneasy matter tointegrate avolume element cuover anymanifold. First wedeﬁne f cu: ffforw=f]dx'/\---/\dx"|, f?_0.[g,|]n t0,]]n' Then forann-chain c:[0,1]"—>Mwedeﬁne ‘/cu zf c*w. c [0,1]" Theorem 7-7"shows thatProposition Iholds foravolume element cu=fldxl/\ ---/\dx"| even ifdetc’isnot30.Thus Corollary 2holds forvolume elements even ifdetp’isnot30.From thisweconclude that Theorem 5holds for volume elements cuonanymanifold M,without assuming c1,c2 orientation preserving (oreven thatMisorientable). Consequently wecandeﬁne fMw foranyV0ll.11'l'lC element wwith compact support. Ofcourse, when Misorientable these considerations areunnecessary. For, there isanowhere zero n-form 17onM,andconsequently anyvolume element w canbewritten w=fW,f20 lfwechoose anorientation p.forMsuch thatw(v|, ...,v,,)>Oforv1,...,v,, positively oriented, then wecandeﬁne Volume clements willbeimportant later, butfortheremainder ofthischapter weareconcerned only with integrating forms over oriented manifolds. Infact, ourmain result about integrals offorms over manifolds, ananalogue ofStokes’ Theorem about theintegral offorms over chains, does notwork forvolume elements. 260 Chapter 8 Recall from Problem 3-I6thatifMisamanifold-with-boundary, andpE HM, then certain vectors vEMpcanbeCiiStiI‘lg"|_1iShCCi bythefactthat forany coordinate system x:U—>ll-ll"around p,thevector x,,.(v) Ell-ll";-(1,) points “outwards”. Wecallsuch vectors vEMp“outward pointing”. IfMhasan 5,__ orientation pt,wedeﬁne theinduced orientation Hp,forBMbythecondition that [v|,...,v,,...|] E(3;,t)p ifandonly if[w,v|,...,v,,..|] Eupforevery outward pointing wEMp.Ifitistheusual orientation of1H1",then forp==(a,0) Elrll” wehave lipI[(9l)p, ...,(@n)pi =(“lim-][(@n)pi(9l)p: ---1(en--llpi =(~1)”[(~@~)p,(@1),,,-.-,(@s~1)p]- Since (-e,,),, isanoutward pointing vector, thisshows that theinduced orien~ tation onlR"'“' x{0}=an" is(-I)” times theusual one. Thereason forthis choice isthefollowing. Letcbeanorientation preserving singular n~cube in (M,;,t) such that BMOc([0, 1]") =c(,;,_0)([0, ]]"'“1). Then c(,,,0): [0,]]“'“1 —> ('(n.0) (3M,8;,t) isorientation preserving foreven n,and orientation reversing for oddn.Ifwisan(n-1)~form onMwhose support iscontained intheinterior lnfegratioti 25I oftheimage ofc(this interior contains points intheimage ofc(,,_0;,), itfollows that ﬂ .;11_-.-.:-_‘._-=. ;l'.\-. support cu Butc(,,_0; appears with coefﬁcient (-1)" in3c.So (*) /0):/i w=(--l)"/ 0):‘/i w. 36 ("-1)"¢‘m,tii Conn) HM Ifitwere notforthischoice ofBuwewould have some unpleasant minus signs inthefollowing theorem. 5.THEOREM (STOKES’ THEOREM). IfMisanoriented n-dimensional manifold-with~bounda1y, and BM isgiven theinduced orientation, and wisan (n—})—form onMwith compact support, then ‘/dw=f w. M BM PROOF. Suppose ﬁrstthatthere isanorientation preserving singular n—cube c inM—BMsuch thatsupport wCinterior ofimage c.Then fdw=/dw=fcu byTheorem 4 M c 3c =0 since support 0)Cinterior ofimage c, I w=0. BM Suppose next that there isanorientation preserving singular n-cube cinM such that3Mﬁc([O, 1]")=c(,,,0;([0, }]”""), andsupport wCinterior ofimage c. Then once againwhile weclearly have 252 Cfzajltesr :5’ Ingeneral, there isanopen cover (9ofMandapartition ofunity <1)sub- ordinate to(9such that foreach Q5E(Dtheform Q5-wisoneofthetwosorts already considered. Wehave 0=d(}) =d(Z¢) =Zdtp, ¢>e¢ ¢e¢ S0 Zd¢/\0)=0. ¢€¢ Since whascompact support, thisisreally aﬁnite sum, and weconclude that Z] d(;5/\w=:O. ¢e¢ M Therefore /M(la): ‘/q‘;-(Irv: fdqﬁ/\w+¢-dw M M =2] d(¢.w)= z~l€i.M¢.w:f8Mw. 0:.¢>e¢ M ¢>e¢ One ofthesimplest applications ofStokes’ Theorem occurs when theoriented n-manifold (M,;,t)iscompact (sothat every form hascompact support) and BM :=U.Inthiscase, ifr;isany (11—i)-form, then I {(7):} M HM Therefore wecanﬁnd ann-form wonMwhich isnotexact (even though it must beclosed, because all(n+i)~forms onMare0),simply byﬁnding anw with /Iw:,=é0. M Such aform cualways exists. Incleccl wehave seen that there isaform wsuch thatforvi,...,v,,6Mpwehave (*) w(v;,...,v,,)>0 if[v1,...,v,,]=;i,,. Ifc":[0,1]"—>(M,ii)isorientation preserving, then theform c*won[0,1]”is clearly _ gdxiA--»/\dx” forsome g>0on[0,l]", 1rttegr'at2'0n 263 sofaw>0.Itfollows that IMw>0.There is,moreover, noneed tochoose aform wwith (*)holding everywhere—we canallow the>sign tobereplaced by3.Thus wecaneven obtain anon-exact n-form onMwhich hassupport contained inacoordinate neighborhood. This seemingly minor result already proves atheorem: acompact oriented manifold isnotsmoothly contractible toapoint. Aswehave already empha- sized, itisthe“shape” ofM,rather than its“size”, which determines whether ornotevery closed form onMisexact. Roughly speaking, wecanobtain more information about theshape ofMbyanalyzing more closely theextent towhich closed forms arenotnecessarily exact. Inparticular, wewould now liketoaskjusthowmany non-exact n-forms there areonacompact oriented n-manifold M.Naturally, ifwisnotexact, then thesame istrueforw+dqfor any(n—l)—form 17,sowereally want toconsider wandw+digasequivalent. There is,ofcourse, astandard way ofdoing this, byconsidering quotient spaces. Wewillapply thisconstruction notonlyton~forms, buttoforms ofanydegree. Foreach k,thecollection Zk(M) ofallclosed k—forms onMisavector space. The space B"(M) ofallexact k-forms isasubspace (since dz=0),so wecanform thequotient vector space H’<<M>=z"<M>/B"<M>; thisvector space Hk(M)iscalled theIt-dimensional dcRham cohomology vector space ofM. [deR/zamia" T/zeorem states that thisvector space isisomorphic to acertain vector space deﬁned purely interms ofthetopology ofM(forany space M),called the“k-dimensional cohomology group ofMwith real coef- ﬁcients”; thenotation Zk,Bl‘ischosen tocorrespond tothenotation used in algebraic topology, where these groups aredeﬁned.] Anelement ofHi‘(M)isanequivalence class [cu]ofaclosed k-form w,two closed k—forms cu;andcu;being equivalent ifandonly iftheir difference isexact. Interms ofthese vector spaces, thePoincare Lemma says that H"UR" )=0(the vector space containing only 0)ifk>0,ormore generally, H"(M )=0ifM iscontractible andk>0. Tocompute H°(M) wenote firstthatB°(M) =0(there arenonon-zero exact 0—forms, since there arenonon-zero (—l)—forms forthem tobethedif- ferential of). SoH°(M)isthesame asthevector space ofallC°° functions f:M—>IRwith df=O.IfMisconnected, thecondition df=0implies thatfisconstant, soH°(M) QR.(Ingeneral, thedimension ofH°(M) isthe number ofcomponents ofM.) Aside from these trivial remarks, wepresently know only oneother factabout H"(M)—if Miscompact andoriented, then H“(M) hasdimension 3l.The further study ofHk(M)requires acareful lookatspheres andEuclidean space. 254 Chapter 5 On .S'""" CR”—{0}there isanatural choice ofan(n—l)-form 6’with fS,,__; 0’>0;for(v;),,,...,(v,,...;),, E.S'"'"1,,,, wedeﬁne P oJ(p)((v1)ps ---:|(v!l--l)P) :det( Li] )' Un-I Clearly this is>Oif(v;),,,, ...,(v,,...;),, isapositively oriented basis. Infact, wedeﬁned theorientation of.S'"'"1inprecisely thisway—this orientation isjust theinduced orientation when .S'”“‘ isconsidered astheboundary oftheunit ball {pEIR”:Ip|5I}with theusual orientation. Using theexpansion ofa determinant byminors along thetoprow weseethat 0’istherestriction to S""1oftheform 0on1R"deﬁned by H tr=Z:(—I)’i""x"dx] /\---Adxi /\ Adx”. i=1 The form tr’on.S'”"" willnow beused toﬁndan(n—I)-form onR"—{0} which isclosed butnotexact (thus showing thatH"‘“‘(R"—{O})=,£0).Consider themap r:R"—{0}—>S”"" deﬁned by P P Hp)=——=—-lpl11(1)) Clearly r(p) =pifpE.S'”"l; otherwise said, ifit.S'”'"l —>R"—{0}isthe inclusion, then roi=identity of.S'”“‘. (Ingeneral, ifACXand r:X—>Asatisﬁes r(a) =aforaEA,then ris called aretraction ofXonto A.) Clearly, r"'o" isclosed: d(r*a’) =r*dcr" =0. However, itisnotexact, forifr*a’=dry,then tr’=i*r*a’ =di*q; hutweknow that0’isnotexact. ltztegratzm 255 Itisaworthwhile exercise tocompute bybrute force that ‘d—d d—-dforn=2, r*o"’=—A yy~7x=x y x=d6 forn:3’ﬁg, :Xdyfs dz—ydig/\ dz+zdx /\dy (x2+P2+:2)” =Li-§[xdy/\dz—ydx/\dz+zdx/\dy]. Since wewill actually need toknow r*o" ingeneral, weevaluate itinanother way: 7.LEMMA. Iftristheform onR"deﬁned by II tr=Z:(—I)""'1x"dx‘ /\---/xdxi /\---Adx”, II andtr’istherestriction 1'*aoftrto.S'""“, then (*) »-WP)= So H I - - --~. r*o’ =FZ(—])’"'x’ dxl/\---/\dX' /\---Adx”. i=1 PROOF Atanypoint peR”—{O},thetangent space lR"pisspanned bypp andthevectors vpinthetangent space ofthesphere .S'”'"'(| pl)ofradius Ip|. Soitsufﬁces tocheck thatboth sides of(=1=)give thesame result when applied ton—Ivectors each ofwhich isoneofthese twosorts. Now ppisthetangent vector ofacurve ylying along thestraight linethrough 0and p;thiscurve is taken tothesingle point r(p) byr,sor...(pp) =0.Ontheother hand, P P cr(p)(pp, (v;)p, ...,(v,,..;)p) -=det ‘ll =0. Uni-2 Soitsuffices toapply both sides of(*)tovectors inthetangent space of .S'"“' (lpl). Thus (Problem l5),itsufﬁees toshow thatforsuch vectors vpwe have "*(vP) =Tg|'vr(p)- 266 Chapter 8 Butthisisalmost obvious, since thevector vpisthetangent vector ofacircle y lying in.S'”"](]p|), and thecurve royliesin.S'""'l and goes I/|p| asfarinthe same time. '§' 8.COROLLARY (INTEGRATION IN“POLAR COORDINATES”). Let frB—>R,where B={PER”=lpls 1}. and deﬁne g:.S'"'"l —>Rby 1 g(P)=[0it"-‘f<~ -mu. Then ff=ffdxl/\---/\dx”=/ go’.B B Sn--I PROOF. Consider .S'”‘"' x[0,I]andthetwoprojections 1 ,l 1 [9,1]H12.S'”"] X[0,1] —>Sn“: H21Sm“! X[0,1] —>[0 . 0% LetususetheabbreviationSn--1 6"Ad1: :rr1*a' /\7r;*dI. If(y,U)isacoordinate system onS""1, with acorresponding coordinate sys- tem (jg!) =(y0J'1'1,.'!1'_'._\) onS""'l X[0,I],and 6"=cedyl /\ /\dy”'"l, then clearly cr"Ad! =Exo:r1d;7‘/\---/\dj"“‘/xdt. From thisitiseasy toseethatifwe deﬁne h:.S'”“' X[0,I]—>1Rby h<p.~)=~"*‘f<~ -P). then f go’=(—])”"1 f ho’Adz.Sn--I Sn--E x[0, 1] Now wecandefine adiffeomorphism Q5:B—{0}—>.S'""l X(0,I]by Mp)=(Hp),11(9))=(P/lp|,lp|)- 1zzzegi'azz'0n 257 Then ¢>*(o’ Adz)=¢*(rr,*a" Arrfdz) =¢*rr;*o" A¢*rr2*dz‘ =(YF1° <z">)*0"' /\(H20Q5741 =r*6" Av*dz‘ I(H D- l T-T " nxi ,- v" I=- Z:(—I)’ ‘x‘dx‘A---/\dx'A---Adx )AZ:~;dx '=l i=1 (_])n I" _ I =W Z(Xl)2d.?Ci /\"- Adi” lim] (_])rI-1 =——--—dx‘ A---Adx".vn--I Hence ¢*(lto" Adz)-=(Ito¢)¢*(o’ Adz) (_])n--I =v"'"lf-——-»—dx]/\---Adx"vn“"] -=(—])"""fdx‘ A---Adx”. So, ffdx'A---Adx"=(—I)""lf ¢*(ha’Adz)B B-{0} =(—])""'/ ho’Adz s"-*=<(0,i] =f go’.Sn-I (This last step requires some justification, which should besupplied bythe reader, since theforms involved donothave compact support onthemani- folds B—{0}a11dS""1X(0,I]where theyaredeﬁned.) '1' Weareabout ready tocoinpute Hi‘(M) inafewmore cases. Wearegoing to reduce ourcalculations tocalculations within coordinate neighborhoods, which aresubmanifolds ofM,butnotcompact. Itistherefore necessary tointroduce another collection ofvector spaces, which areinteresting intheir own right. 258 Chapter a The dcRham cohomology vector spaces with compact supports Hp“(M) are deﬁned as H.f<M>=Zi‘<M>/B§<M>,where Z2‘(M) isthevector space ofclosed k-forms with compact support, and B§(M) isthevector space ofallIt-forms dr]where 17isa(k—I)-form with compact support. Ofcourse, ifMiscompact, then H__,"(M) =Hi‘(M). Notice that Bf(M) isnotthesame asthesetofallexact k-forms with compact support. Forexample, onIR”,iff3Oisafunction with compact support, and f>0 atsome point, then w=fdx‘ A---Adx” isexact (every closed form onR"is)andhascompact support, butwisnottiff] foranyform qwith compact support. Indeed, ifw=drywhere ryhascompact support, then byStokes’ Theorem f0):‘/I dr]=f 1;.-=0.Rn Rn 3R1: This example shows that H,f(lR”) 79O,and asimilar argument shows that ifMisanyorientable manifold, then Hc"(M );=é0.\zVearenow going toshow that foranyconnected orientable manifold Mweactually have H§‘(M) en. This means thatifwechoose aﬁxed wwith _fMwgé0,then foranyn—form cu’ with compact support there isarealnumber asuch thatcu’——aw isexact. The number acanbedescribed easily: if w’—aw =dn, v/w"—]_ aw:-I dr;=O, M’ M M a=fw’/I w; M M theproblem, ofcourse, isshowing that17exists. Notice thattheassertion that Hg‘(M) %Risequivalent totheassertion that isanisomorphism ofH,f(M) with R,i.e., totheassertion that aclosed form w with compact support isthedifferential ofanother form with compact support ifIM0.)=O.then so 122legration 259 9.THEOREM. IfMisaconnected orientable n-manifold, then H§(M )W»IR. PROOF. Vilewillestablish thetheorem inthree steps: (l)The theorem istrue forM=IR. (2)Ifthetheorem istrue for(n—I)-manifolds, inparticular forS““], then itistrue forR”. (3)Ifthetheorem istrue forR",then itistrue foranyconnected oriented n-manifold. Step1.LetwbeaI-form onRwith compact support such thatfpcu=0.There issome function f(not necessarily with compact support) such that cu=df. Since support wiscompact, df=0outside some interval [—N, N],sofisa f —N N constant c;on(-00, —N) andaconstant C2on(N,oo). Moreover, O=‘/w=f df=ff'(z‘)dz‘=c2-—c;. R R R Therefore c1=C2=candwehave w=d(f-C) where f—chascompact support. Step2.Letw=fdxlA---Adx”beann-form with compact support onR" such thatfR,,w=0.Forsimplicity assume thatsupport cuC{pER":|p|<I}. Weknow that there isan(n—II)-form rjonR”such thatcu=dq.Infact, from Problem 7-23, wehave anexplicit formula forrj, H i n(z>)=D-1>""' ’”“'.f(z -P)tr’)X"dx‘A---AdxiA---Adx“.{=1 0 270 Chapter <9 Using thesubstitution tz=|p|z thisbecomes llq(p)=(Lpu”_']f(u-£1-)du)ﬁ; H X2(—I)i“lxi dxlA---AdxiA---Adx" 17:] lpl i P =f z.z”'" f(u -—-) du -r*o’(p) byLemma 7. 0 |P| Deﬁne g:.S'”"'l —>Rby I g(z>)=f0~”"'f(~ -z>)d~- OnthesetA={pER":|p|>I}wehave f=0,soonAwehave I ..,.)=(/0.~-.»(11.,-5;).1.)~=-*<z’(p). 1'?=(gOr)-r*<z'=z'*(g<z’)-or Moreover, byCorollary 8wehave forthe(n—I)-form go’onS""1, f go"=ffdx‘A---Adx".Sn—! B _ =:‘/I w=O.RH Thus, bythehypothesis forSte]:2, go’-=c/A forsome (n—2)-form AonS'”"'1. Hence ry=z'*(dA) =d(r*l). Let/2:R"—>[0,1] beany C°° function with It=IonAand it=0ina neighborhood ofO.Then l1z'*A isaC°° form onR”and w=dry=d(t7—d(1zz'*A)): 1ntegi'a£z'0zz 2'7I theform n—d(/2z'*A) hascompact support, since onAwehave ry—d(lzz'*A) =ry—d(z'*A) ==0. Step3.Choose anzz-form wsuch that IMw;éOandcuhascompact support contained inanopen setUCM,with Udiffeomorphic toR”.Ifw’isany other n-form with compact support, wewant toshow that there isanumber c andaform 17with compact support such that cu’=cw+dr). Using apartition ofunity, wecanwrite w’=<z‘>iw’+---+<z‘>zzw’ where each ¢,~w" hascompact support contained insome open setU;CM with U,-diffeomorphic toR”.Itobviously suffices toﬁndc,-andti;with ¢,-w’ = c,-cu+dry,-, foreach z‘.Inother words, wecanassume cu’hassupport contained insome open VCMwhich isdiffeomorphic toR“. Using theconnectedness ofM,itiseasy toseethatthere isasequence of open sets U=HWqH=V cliﬁfeomorphic toR”,with V;I’)V,-+1 :,._~‘-(5.Choose forms cu;with support wtC support cu; V;Ol/,-+1 and IV,cu;:,é0.Since weareassuming thetheorem forR”wehaveI cu;-c;w =dig; cu;—C20); =dz}; w’—c,w,_.i =dnf: where all:7,have compact support (CV,-). From thisweclearly obtain the desired result. *§* 272 Chapter 6’ The method used inthelaststep canbeused toderive another result. IO.THEOREM. IfMisany connected non-orientable n-manifold, then H_f(M) =0. PROOF. Choose ann-form wwith compact support contained inanopen setU diffeomorphic toR",such thatfuwqé0(thisintegral makes sense, since Uis orientable). Itobviously sufﬁces toshow thatw=dnforsome form 27with compact support. Consider asequence u=mwUm=v ofcoordinate systems (V,-,x,-) where each x,-ox,-_|.;'"' isorientation preserving. Choose theforms cu;inStep3sothat, using theorientation ofV;which makes x,-:V;—>R”orientation preserving, wehave fp,0);>0;then alsoIV,+1w,->0. Consequently, thenumbers C;:=v/i (1)5 (:);'...] GT6 POSIIIVC. Vi Vi cu;=cw+dn where c>0.Itfollows that Now ifMisunorientable, there issuch asequence where V,.-=V;butx,ox;"‘ isorientation ret:ersirz_g. Taking cu’=-0), wehave —w=cw+d17 forc>0 so ' (—c—])w=dr] for—c—];é0.¢Z¢ Wecanalsocompute H"(M) fornon-compact M. ll.THEOREM. IfMisaconnected non-compact n-manifold (orientable or not), then H"(M) =0. PROOF. Consider ﬁrst ann-form wwith support contained inacoordinate neighborhood Uwhich isdiffeomorphic toR".Since Misnotcompact, there isaninﬁnite sequence U=U1,U2,U3,U4,--- lmlegration 2'73 ofsuch coordinate neighborhoods such that U;F)U;-+1 96Q),andsuch that the sequence iseventually inthecomplement ofanycompact set. rt support wt suppo w U= support (02 Now choose n-forms w,-with compact support contained inU;OU;-+1, such that furcu;;=éO.There areconstants c;and forms q;with compact support CU;such that w=Ciwt +dﬂi wt=Cf+lwz'+l +d7lz'+1 i21- Then w=dr]1—|-C1(U; =d171+C1dY72 +Ciﬁzwz =dni+Cidr72 +61624173 +CiC2<1'3w3 Since anypoint pEMiseventually inthecomplement oftheU,-’s, wehave w=dm+cidnz +ClC2d7i3 +C1C2C3d7l4 +---, where theright side makes sense since theU;areeventually outside ofany compact set. Now itcanbeshown (Problem 20)thatthere isactually such asequence U1,U2,U3,...whose union isallofM(repetitions areallowed, and U;may intersect several U_;forj<1',butthesequence isstilleventually outside ofany compact set). The cover (9={U}isthen locally ﬁnite. Let{zpy} beapartition ofunity subordinate to(9.Ifwisann-form onM,then foreach U;wehave seen that ¢U,w =dn; where I];hassupport contained inU;UU,-+1 UU;-+3 U---. Hence2 2 IP12st w:Z:¢Utw=§:d’7I'=d( ')-°:‘ i=1 i=1 = 274 Chapter :5’ SUMMARY OFRESULTS (l)ForR"wehave k Rk=0 H(R”) "Ar0k>0. (2)IfMisaconnected n-manifold, then H°(M) A;n RifMisorientable H.;'<M>~{ .. .0ifM15non-orientable Hé‘ ifMiscompact H"(M) W{ , _0 ifMisnotcompact. Wealsoknow thatH""" (R”—{0}) :,é0,butwehave notlisted thisresult, since wewilleventually improve it.Inorder toproceed further with ourcomputations weneed toexamine thebehavior ofthedeRham cohomology vector spaces under C°°maps f:M—>N.Ifcuisaclosed k-form onN,then f*w isalso Ill closed (df*w =fdw=0),sof*takes Z"(N) toZ"‘(M). Ontheother hand, f*also takes Bk(N) toB"(M), since f*(dn) 2d(f*rp). This shows that f* induces amap z"<N>/B"<N) ->z’<<M)/B"<M). alsodenoted byf*: f*:H"(N) _>H"(M). Forexample, consider thecase/<=0.IfNisconnected, then H°(N)isjust thecollection ofconstant functions c:N->R.Then f*(c)=c0fisalsoa constant function. IfMisconnected, then f*: H°(N) —>H°(M) isjust the identity map under thenatural identiﬁcation ofH°(N) and H°(M) with R. IfMisdisconnected, with components Mp,aEA,then H°(M) isisomorphic tothedirect sum @Ra, where each Ra%R; a€A themap f*takes cERintotheelement ofEBRQ, with ad‘component equal toc.IfNisalsodisconnected, with components Np,B<5B,then f*=69%—>EBA“BGB aeA takes theelement {C5}ofQEBGB R5to{CL}, where cg,=C5when f(M,,) CNp. 122tegration 2'75 Amore interesting case, and theonly onewearepresently inaposition to look at,isthemap f*: H“(N) —>H”(M) when MandNareboth compact connected oriented n-manifolds. There is nonatural waytomake H"(M)isomorphic toR,sowereally want tocompare fMf*w and /Iva) forwanzz-form onN.Choose onewt;with INwo;é0.Then there issome number asuch that /f*w0=a-/I wo. M N Since w|—>_fMwisanisomorphism ofH"(M) andR(and similarly forN)it follows that forevery form wwehave /Mf*w=a-jivw. The number a=degf,which depends only onf,iscalled thedegree off. IfMandNarenotcompact, butfisproper (theinverse image ofanycompact setiscompact), then wehave amap f*1H§(N)—> HUM) andanumber degf,such that [M/*0»=(dssf)[Nw forallforms cuonNwith compact support. Until onesees theproof ofthe next theorem, itisalmost unbelievable that thisnumber isalways aninteger. 12.THEOREM, Let f;M—>Nbeaproper map between two connected oriented It-manifolds (M,;t) and (N,v).Letq6Nbearegular value of Foreach pEf"'l(q), let I ifftp: Mp—>Npisorientation preserving signp f= (using theorientations upforMpandupforN4) —] ifﬂtpisorientation reversing. 275 C/zapter :5’ Then degf= Z signpf (=0iff_l(p)=@). Pcf-*0?) PROOF Notice ﬁrst that regular values exist, bySard’s Theorem. Moreover, f“'(q) isﬁnite, since itiscompact andconsists ofisolated points, sothesum above isaﬁnite sum. _ Letf_'(q) ={p1,...,pk}. Choose coordinate systems (U,-,x;) around p; such that allpoints inU;arereg-ular values off,and theU;aredisjoint. ‘We want tochoose acoordinate system (V,y)around qsuch that f“l(V)=U;U UUk. Todothis, ﬁrst choose acompact neighborhood Wofq,and let _. .. .1,- él‘-<"’+'*'-§?1";;fiJ " ¢-.-.--rt '..._ . ..-,;_§'?r.\» W’CMbethecompact set W’=f"(W)—(U; u---ut/,,). Then f(W’)isaclosed setwhich does notcontain q.Wecantherefore choose VCW—f( W’). This ensures that f"!(V) CU;U---UU;;. Finally, redeﬁne U; tobeU;I’)f“'(V). Now choose cuonNtobew=gdy‘ /\---Ady"where g30hascompact support contained inV.Then support f*0)CU;U---UUk. So z<etA\xE(~..§ sign.2 15* Since fisadiffeomorphism from each U;toVwehave ff*w=f wiffis orientation preserving U, V =—fwiffisorientation reversing. V Since fisorientation preserving [orreversing] precisely when signpf:.-I[or -1]thisproves thetheorem. *1» Integiatiozz 277 Asanimmediate application ofthetheorem, wecompute thedegree ofthe “antipodal map” AIS"—>S"deﬁned byA(p) .-=-—p. 'We have already seen that Aisorientation presenting orreversing atallpoints, depending on whether z:isoddoreven. Since A"‘l(p)consists ofjustonepoint, weconclude that degA=(—I)""I. Wecandraw aninteresting conclusion from thisresult, butweneed tointro- duce another important concept ﬁrst. Two functions f,gtM—>Nbetween twoC°°manifolds arecalled (smoothly) homotopic ifthere isasmooth function HIM X[0,1] —>N with H(P,0) =f(P) foranPEM_ H(P=l)=8(P) ’ themap Hiscalled a(smooth) homotopy between fandg.Notice that Mis smoothly contractible toapoint pgEMifandonly iftheidentity map ofMis homotopic totheconstant map pg.Recall thatforevery k-form toonMX[0,I] wedeﬁned a(k—])—form IwonMsuch that z'1*w -—t0*w =d(]w) +](dw). Weused thisfacttoshow thatallclosed forms onasmoothly contractible man- ifold areexact. Wecannowprove amore general result. I3.THEOREM. Iff,g:M—>Naresmoothly homotopic, then themaps f*;H“(N) ->H"(M) g*:H"(N) ->H"(M) areequal, f*-:=g*. PROOF Byassumption, there isasmooth map H:MX[0,I]—>Nwith f=Hotg g=H01'1. Any element ofH"(N) istheequivalence class [cu]ofsome closed k-form to onN.Then g*w —f*0J =(HoI';)*w --(Hoit-;)*w =z.*<H*w> -a*<H*w)=d(IH*w) +](dH*w) =d(]H*w) +0. Butthismeans that g*([w]) -=f*([w]). '1' 278 C/lapter 8 14.COROLLARY. IfMand Narecompact oriented n-manifolds and the maps f,g: M—>Narehomotopic, then degf =degg. 15.COROLLARY. IfItiseven, then there does notexist anowhere zero vector ﬁeld onS". PROOF Wehave already seen thatthedegree oftheantipodal map A:S"—> S"is(—I)""". Since theidentity map hasdegree I,Aisnothomotopic to theidentity forneven. Butifthere isanowhere zero vector ﬁeld onS",then wecanconstruct ahomotopy between Aandtheidentity map asfollows. For each p,there isaunique great semi-circle ypfrom ptoA(p) =—pwhose tangent vector atpisamultiple ofX(p).Deﬁne 11(1),!) =101(1)» :3‘ ForP?oddwecanexplicitly construct anowhere zero vector ﬁeld onS”.For p=(x;,...,x,,_|.;) ES”wedeﬁne X(P) =(—-Ti. X0.—-Y3. X2.---=—Xa+i= In); thisisperpendicular top=(x;,x2, ...,x,,+;), andtherefore in.S""p. (On S‘ thisgives thestandard picture.) The vector ﬁeld onS"canthen beused togive I 1 7 \ ahomotopy between Aandtheidentity map. Foranother application ofTheorem l3,consider theretraction I-=R"-to->Sr‘ to)=P/|P|- Ift:.S'”'"' —>R"—{0}istheinclusion, then roi:S'”'"l —>.S'”"" istheidentity lofS'“"'l. 1rztegr'a!2'0zz 279 The map z'or:R"—{0}—>R"——-{0} toz'(p)=p/|p| is,ofcourse, nottheidentity, butitishomotopic totheidentity; wecandefine thehomotopy Hby I/.p(I$i] I I . ’3i(z>)n HQ) (i=0)H(P.!)=!P+(1 —1)z"(z1)cR -—{0}- <1=@> Aretraction with thisproperty iscalled adeformation retraction. Whenever z‘ isadeformation retraction, themaps (roi)*and(z'or)* aretheidentity. Thus, forthecaseofS”"] CR”—{O},wehave H"<s"*‘> 11>H"<s"-{O}) Hﬂs"—on-’—>H"<$""> and 7*or=(1'o)°)* =identity ofH"(]R" -to}; 1*o)'*=<,-oI')*.-=identity ofH’<(s"~‘). Sot*andr*areinverses ofeach other. Thus H"(s""‘) wH"(n" -{opforallz<. Inparticular, wehave H”'"l(R” -- "1"!R.Agenerator ofH”'“l(R" — is theclosed form r*o". Wearenowgoing tocompute Hk(R”—-{O})forallk.Weneed onefurther observation. The manifold MX{0}CMXRI isclearly adeformation retraction ofMXR’. SoH"(M) %Hk(M XR!) forall1. 280 C/zapter :5’ 15.THEOREM. For0<k<I?-Iwehave Hk(R" —{O}) =Hk(.S'"_1) .-=0. PROOF. Induction onn.The ﬁrst case where there isanything toprove is n=3.WeclaimH'(]R3 -{op=0. Letwbeaclosed I-form onR3.LetAandBbetheopen sets (0,0,I) A=R’-{(0.0)><(-00.01) B=R3—{(0.0)><I0.00)}. *"" (0=0s_'l) Since Aand Bareboth star-shaped (with respect tothepoints (0,0, I)and (0,0, —I),respectively), there are0-forms f4andfgonAandBwith w=df4 onA w=df5- onB. Now (((_}:q—f3)=0 0nA|')B, and AF)B=[R2—{0}]XR, soclearly f4—fgisaconstant conAI’)B.Thus toisexact, for to-=d(f.; —c) onA to=d(j,'5-) onB andf.;—c=f5- onAI’)B. Iftoisaclosed I-form onR4,there isasimilar argument, using A=n4-{(0,0,0) ><(-00,01} B=n4-{(0,0,0) ><[0,oo)}. Ifcuisaclosed 2-form onR4,then weobtain I-forms 17,4and 173with w=d11,q onA w=d173 onB. 1ntegrat2'0rz 281 Now d(rjA—n3)=O onA|')B and H}(AnB)=H1([]R3 -{on><n)isH'(]R3 ~{op=0. Sor;,.;-rig=dlforsome 0-form AonAI’)B.Unlike theprevious case, we cannot simply consider 17,4—dl,since thisisnotdeﬁned onA.Tocircumvent thisdifficulty, notethatthere isapartition ofunity {¢,;,¢B} forthecover {A,B} ofR3-—-{O}: Q5/t+$3=I d(,(>,4 +dgbg =0 support<,b,4 CA supportqﬁg CB. NOW, if ¢BA OI]AOB Q5371 denotes {O onA_(AnB), andsimilarly for<,(>,4A, then (I231 isaC°° form onA (,(),4A isaC°°form onB. OnAI’)Bwehave WA—¢((<t>B/I) =77A-953 all-"‘($3Al =77A+(¢A —1)dl+d¢A A1 =77A—dl+d(<t>A/U =ms+d(¢Al)- Sowecandeﬁne aC°°form onR"—{0}=AUBbyletting itberig—d(¢;,-A) onA,and 173+d(¢,;A) onB.Clearly, (U=d7],4 =61(1),; — 011A =dﬂe=‘((175 +d(<t>/ill) 01’!3, socuisexact. The general inductive step issimilar. *1‘ 282 Chapter 8 Weendthischapter with onemore calculation, which wewillneed inChap- terll. 17.THEOREM. For05k<nwehave Hc‘."(R”) =0. PROOF The proof that H_§(R") =0islefttothereader. Letcubeak-form onR”with compact support, 0<k<n.Weknow that cu=dnforsome (k—])—form r;onR".LetBbeaclosed ballcontaining support w.Then onA=R"—Bwehave dq=0.Since Aisdiﬂfeomorphic to 0=w=dr) it: R”—{O}and k-I<rt—Iwehave from Theorem 15that 17=dA forsome (k—2)—form AonA. Letf:R"—>[0,I]beaC°°function with f=0inaneighborhood ofBand f=IonR"—2B,where 2Bdenotes theballoftwice theradius ofB.Then d(fA) makes sense onallofR"and w=40=do-am); theform ry—d(fA)clearly hascompact support contained in2B.'2' 1zztegz'atz'0r2 283 PROBLEMS l.TheRiemann zhtegrat versus I.'1zeDarboux tntegrat. Letf:[a,b]—>Rbebounded. Forapartition P={:0< <z,;}of[a,b], letm;=m;-(f) betheinfoff on[z;_.;,z;] and deﬁne M;=M;-(f) similarly. Achoice forPisann-tuple E=(E1,...,E;;) with E;E[z;_.;,z;]. Wedeﬁne the“lower sum”, “upper sum”, and “Riemann sum” forapartition Pand choice Eby Lo’.P)-Zia.-<1") -<1.»-z.-_.t>l'=l rm’.P)=M.-<1’)-0;-z.--1)i=1 so".an-Zf<s-><z.- -!I'-l)-[=1 Clearly L(f, P)5.S'(f, P,E) 5U(f, P).WecallfDarboux integrable ifthe sup ofallL(f, P)equals theinfofallU(f, P); this sup orinfiscalled the Darboux integral offon[a,b]. \'VecallfRiemann integrable if l' S P '' upilplo (f, ,5;') exists, thelimit iscalled theRiemann integral offon[a,b]. (a)Wecandefine S(f,P,§) even iffisnotbounded. Show however, that Hjilpi 0.S'(f, P,§) cannot exist iffisunbounded. (b)Iffiscontinuous on[a,b], then fisRiemann andDarboux integrable on [a,b],andthetwointegrals areequal. (Use uniform continuity offon[a,b].) (c)IffisRiemann integrable on[a,b],then fisDarboux integrable on[a,b] and thetwo integrals areequal. (d)Lettn5f5Mon[a,b]. LetP={so< <Sm}and Q={:0< < z,,}betwopartitions of[a,b]. Foreach t=-1,...,n,let e;=length of[z;_.1, 1;] —sum oflengths ofall[s;,__;, sp]which arecontained in[t;__;, z‘;]. It-i fzllwrl III 1 [s;,_1,s;,]’s contained in[I‘;_.;, 1;] shaded lengths -=adduptoe; 284 Chapter e Show that, ifM;denotes thesupoffon[z;_.1,z;], then H U(f,P)5U(f.Q)+Z014 -M.-)e; i=1 5U(f,Q)+(M —zn)i:e;. i=1 There isasimilar result forlower sums. (e)Show that EL, e;—>Oas||P|| —>0,anddeduce Darboux’s Theorem: j)iﬁ1;1>0U(f, P)=inf{U(f, Q)1Qapartition of[a,b]} lUl)il{l)>0L(f, P)=sup{L(f, Q): Qapartition of[a,b]}.ll (f)IffisDarboux integrable on[a,b],then fisRiemann integrable on[a,b]. (g)(Osgood’s Theorem). Letfand gbeintegrable on[a,b]. Show that foi choices E,E" forP, H z> ,,;i,g0§f<s;)s<s'.><zt-z._;> ==fe- HW? Iflgl SM01’l[¢1'=bl.thﬁfl|f(§"z)8(§’t)—f(§i)8(§'t)| SM|f(§'z)—f(§r)|- (h)Show thatfafdx +gdy, deﬁned asalimit ofsums, equals (blf(c(r))@"(z) +s(c(¢))c2'(!)ld1- 2.Compute fad6==f[0,,]c* dd,where c(z)=(cos2rtz,sin2m) on[0,1]. 3.Foritaninteger, and R>O,letcR,,,: [0,I]—>R2—{O}bedeﬁned by cR,,,(I‘) =(RcosZnrrz‘, Rsin Znm‘). (a)Show thatthere isasingular 2-cube c:[0,I]2—>R2—{0}such thatcR,,,, — CR2," =Be. (b)Ifc:[0,1] —>R2—{O}isany curve with c(0) =c(I), show that there is some nsuch thatc-—c1,,,isaboundary inR2—{O}. (c)Show that nisunique. Itiscalled thewinding number ofcaround 0. lrttegratiart 285 4.Letf:(C—>(Cbeapolynomial, f(z) =z"+a;z”_l-I----+a,,,wheren 3I. Deﬁne cR,f: [0,1] —>(CbycR,f =foCR,1. (a)Show thatifRislarge enough, then cg’; —-cR,,., istheboundary ofachain in(C—{O}.Hint: Note thatcR~,;;(z‘) =[cR,; (t)]”, andwrite C1 C1 f(z)=z"(]+~;_l+..._|_;_.;)_ (b)Show thatf(2)=0forsome zE(C(“Fundamental Theorem ofAlgebra”). Hint: Iff(z) #0forallzwith |z|5R,then cR,f —c0,f isaboundary. 5.Some approaches tointegration usesingular simplexes instead ofsingular cubes. Although Stokes’ Theorem becomes more complicated, there aresome advantages inusing singular simplexes, asindicated inthenext Problem. LetAnCR”bethesetofallxER"such that F1 O5x'5l, 2.135]. t'=I \-,-.\_ A3 at at . _,_. .. ________ 0 I Asingular it-simplex inMisaC°° function c:Ap—>M,and anzz-chain isaformal sum ofsingular H-SlI‘l)plCXCS. Asbefore, letI":An—>R”bethe inclusion map. Deﬁne 8;:A,,._] —>Anby aoo-)=([1-3;,‘x=‘],s', ...,x""‘) 8,-(x)=-(x',...,x"’"',0,xf,...,,\'"“') 0<z'5n, andforsingular zz-simplexes c,deﬁne 8,-c=-co8;.Then wedeﬁne H zBc=Z(—]) 8;-c. i=0 (a)Describe geometrically theimages 3;-(A,,_.;) inAn. (b)Show that 82.-=0. 286 Chapter 8 _,_,--C H (c)Show thatifwsfdxl /\---/\dX2/\---/\dX“ isan(n— 1)-form onR, then fdw=f cu. I" at" (Imitate theproof forcubes.) (d)Deﬁne few foranyk-chain cinMandk-form toonM,andprove that ‘/dw-:,/2 w c 3c forany(k—1)-form cu. 6.Every xEA;;_,_; canbewritten asIx’,forO5t51,and x’E80(A;;). 1! X Morcover, x’isunique except when z‘===0.Foranysingular k-simplex c:A1;—> R”,deﬁne E:A;;_|_, —>R"by L. E(.\') ---z-c(x'). C Wethen deﬁne Eforchains cintheobvious way. (a)Show that dc=0implies that c=85. (b)Letc:[0,1] —>R2beaclosed curve. Show that cisnottheboundary of anysum oofsingular 2-cubes. Hint: If8o-=Z,a;c;, what canbesaid about Zr‘??? (c)Show that wedohave c=Bo+c"where c’isdegetzerate, thatis,c"([O, 1])is apoint. (d)Ifc;(0) =cg(0) and 01(1) =cg(]), show that c1——cgisaboundary, using either simplexes orcubes. Irztegratz'02z 287 7.Letwbea1-form onamanifold M.Suppose thatftw==0forevery closed curve cinM.Show thatwisexact. Hint: Ifwedohave w=df,then forany curve <3wehave fw=f<<:<m-f<<:<<>>>- 8.Amanifold Miscalled simply-connected ifMisconnected and ifevery smooth map f:S1—>Missmoothly contractible toapoint. [ActuaIly, any space M(notnecessarily amanifold) iscalled simply-connected ifitisconnected and anycontinuous f:S1—>Mis(continuously) contractible toapoint. Itis nothard toshow thatforamanifold wemay insert “smooth” atboth places.] IfMissmoothly contractible toapoint, then Missimply-connected. .S'Iisnotsimply-connected. (cS"issimply-connected forn>I.Hint: Show thatasmooth f:S‘—>S" isnotonto. (d)IfMissimply-connected and peM,then anysmooth map f:S‘—>M issmoothly contractible top. (e)IfM=UUVwhere Uand Varesimply-connected open subsets with UOVconnected, then Missimply-connected. (This gives another proof that S"issimply-connected forn>1.)Hint: Given f:SI—>M,partition SIinto aﬁnite number ofintervals each ofwhich istaken into either UorV. (f)IfMissimply-connected, then H‘(M)=0.(See Problem 7.) 9.(a)LetUCR2beabounded open setsuch that R2-—Uisnotcon- nected. Show that Uisnotsmoothly contractible toapoint. (Converse of--/€'5Ql Problem 7-24.) Hint: Ifpisinabounded component ofR2-—U,show that there isacurve inUwhich “surrounds” p. (b)Abounded connected open setUCR2issmoothly contractible toapoint ifandonly ifitissimply-connected. (c)This isfalse foropen subsets ofR3. 288 Chapter 8 l0.Letwbeann-form onanoriented manifold M".Let<15and II!betwo partitions ofunity byfunctions with compact support, andsuppose that Zf4;-lwl<oo. ¢ed> M (a)This implies that Zqbeq, fM¢-wconverges absolutely. (b)Showthat ZfM¢-w=Z Z]1//-¢-w,¢ed> ¢e¢1,tre\l1 M andshow thesame result with wreplaced bylwl.(Note thatforeach qt),there areonly ﬁnitely many 1,»which arenon-zero onsupport ¢.) (c)Show that Zwew fM1/1-lwl<00,andthat /l -w--: I1)-cu. M; MQ3 Q, M Wedeﬁne thiscommon sumtobeIMcu. (d)LetAnC(n,n+1)beclosed sets. Letf:R—>RbeaC°°function with IA"f==(-—])"/n and support fCU"An. Find twopartitions ofunity <1) andIIIsuch that 21¢“, fn¢-fdxand Ewew IR1,9-fdxconverge absolutely todifferent values. ll.Following Problem 7-12, deﬁne geometric objects corresponding toodd relative tensors oftype andweight w(wanyrealnumber). 12.(a)LetMbe{(x,y) ER2:|(x,y)l <I},together with aproper portion ofits boundary, andletw=xdy.Show that /.., even though both sides make sense, using Problem IO. (Nocomputations needed—~note thatequality would hold ifwehadtheentire boundary.) (b)Similarly, ﬁnd acounterexample toStokes’ Theorem when M=(0,1) andwisaO-form whose support isnotcompact. (c)Examine apartition ofunity for(0,1)byfunctions with compact support to seejustwhytheproof ofStokes’ Theorem breaks down inthiscase. Integration 289 13.Suppose Misacompact orientable rt-manifold (with noboundary), and6 isan(n—1)-form onM.Show thatd6is0atsome point. 14.LetM1,M2 CR"becompact n-dimensional manifolds-with-boundary with M2CM1--8M1. Show thatforanyclosed (n--1)-form wonM1, 3M1 /f..,.HM] 3M2 15.Account forthefactor 1/lpl" inLemma 7(wehave r,,.(v,,) =(1/lp|)v,(p), butthisonly accounts forafactor of1/lp|"'"1, since there aren--1vectors v1,...,v,,..1). 16.Usetheformula forr*dx‘ (Problem 4-1)tocompute r*0". (Note that r*o" =r*i*o =(ior)*o; themap ior:R"-—{0}—>R”--{O}isjustr,considered asamap intoR"--{0}.) 17.(a)LetM”andNmbeoriented manifolds, andletwandrybeann-form andanm-form with compact support, onMandN,respectively. Wewill orient M><Nbyagreeing thatv1,...,v,,,w1,...,w,,, ispositively oriented in (M><N)(p,q) HMpQBNqifv1,...,v,, andwt,...,wm arepositively oriented inMpandNq,respectively. Ifrt,-:M><N—>MorNisprojection ontheill‘ factor, show that I .T1'1*(1)/\I1'2*I]-"-=‘l'(U"[7]. MxN M N (b)Ifh: MXN—>Ris C°°, then f hrr1*w/\rrg*r;=[ gw, MxN M where g(P)=fN/MP,-)m MP,-)=q1->/1<P,q)- (c)Every (n1+n)-form onM><Nis/t1r1*w /\1r2*17 forsome wand27. 290 Chapter 8 18.(a)LetpER"--{O}.Letw1,. ..,w,,..g ER"),andletv6R",,be(1119);, for some AER.Show that r*cr'(v, w1,. ..,w,,..g) =0. (b)LetMCR"-{0}beacompact (n—1)-manifold-with-boundary which is theunion ofsegments ofrays through 0.Show that fMr*o" =0. I I r tl ' I I I I ll’ If (c)LetMCR"-—{O} beacompact (rt-—1)-manifold-with-boundary which inter- sects every raythrough 0atmost once, andletC(M) =-{lp:pEM,A I30}. C(M) ‘to~Killl \ \.i \ ~. C(M)ﬁS2 Show that fr*6'=f r*U". M C(M)ﬁ.S‘3 The latter integral isthemeasure ofthesolicl angle subtended byM.Forthis reason weoften denote r*o' bydG),,. 19.Forall(x,y,:) ER3except those with x=0,y=0,zE(--00,0], we deﬁne ¢(x,y,:)tobetheangle between thepositive 2-axis andtherayfrom 0 through {x,y,z). In£egrazz'0n 291 (I,J’,I) ‘¢\ (X,y) (a)¢(x,y, z)=21I‘Cl21I‘l(\/X2 +yz/z) (with appropriate conventions). (b)Ifv(p) ==lpl,and6isconsidered asafunction onR3,6(x,y,z) = arctan y/x, then (v,6, Q5)isacoordinate system onthesetofall points (x,y,z) inR3except those with y=O,xE[0,00)orwith x-=0,y=0,zE(—o0,0]. (c)Ifvisalongitudinal unit tangent vector onthesphere .S'2(r) ofradius r, then d¢(v) =1.Ifwpoints along ameridian through p=(x,y,z) E.S'2(r), ‘Q? then I d6(w )=-Mi.p /X2_,_y2 (d)If6and¢aretaken tomean therestrictions of6andqﬁto[certain portions of]S2,then 6'=h(19/\dqﬁ, where h:S2->Ris lt(x,y, 2)=--vxz +yz (theminus sign comes from theorientation). (e)Conclude that 0"=d(-cosQ5d6). 292 Chapter 8 (f)Letrg:R2—{0}—>S1betheretraction, sothatd6=--?'2*t'*o, fortheform 0 onR2.Show that ?'g*d6 ="-"d6. Ifrt:R3—>R2istheprojection, then theform d6on[part of]R3isjustrr*d6, fortheform d6on[part of]R2.Usethistoshow that r*d6 =-"d6- (g)Also prove thisdirectly byusing theresult inpart (c),and thefact that r...(v,,) =v,(,,)/[pl forvtangent to.S'2(|p|). (h)Conclude that d(-D3 =r*a" =d(—cos(¢ or)d6) =d(—cos¢d6). (i)Similarly, express d®,, onR"—{0}interms ofd®,,._1 onR”'"‘ —{O}. 20.Prove that aconnected manifold istheunion U1UU2UU3U ,where theU;arecoordinate neighborhoods, with U;OU;qé(5,andthesequence is eventually outside ofanycompact set. 21.Letf:M”—>N”beaproper map between oriented n-manifolds such that f,.:Mp—>Nf(,,) isorientation preserving whenever pisaregular point. Show that ifNisconnected, then either fisonto N,orelseallpoints are critical points of 22.(a)Show thatapolynomial map f:(C—>(C,given byf(z) =z"+a;z"""I+ +a,,, isproper (n3I). (b)Letf’(z) =nz”'"I +(n—l)a1S""'2+- -'+an-_1. Show thatwehave f"(z) = lim0[f(:+w)—f(z)]/w,where wvaries over complex numbers.w—> (c)W'rite _/'(x+r'y) -=u(x,y)+iv(x, y)forreal-valued functions uand v.Show that , _ 6 _8v f(X+1y) =a—_i(X,y) +1$(X.y) #61) )£31.: ) “T ° Hint: Choose wtobearealIi,andthen tobefit. (d)Conclude that |f’<x+0812=dct1>f<x.y>. Integration 293 where f"isdeﬁned inpart (b),while Dfisthelinear transformation deﬁned foranydiﬂ"erentiable f:R2—>R2. (e)Using Problem 21,giveanother proof oftheFundamental Theorem ofAl- gebra. (f)There isastillsimpler argument, notusing Problem 2l(which relies onmany theorems ofthischapter). Show directly thatiff:M—>Nisproper, then the number ofpoints inf""1(a)isalocally constant function onthesetofregular values of Show that thissetisconnected forapolynomial fI(C—>(C,and conclude that ftakes onallvalues. 23.LetM""1CR”beacompact oriented manifold. ForpER"—M,choose an(n—I)-sphere Earound psuch thatallpoints inside EareinR"—M.Let rp:R"—{p}—>Zbetheobvious retraction. Deﬁne thewinding number w(p) ofMaround ptobethedegree ofr,,lM. 0 E 0@0I Show that thisdeﬁnition agrees with thatinProblem 3. (b)Show thatthisdeﬁnition does notdepend onthechoice ofZ. (c)Show thatwisconstant inaneighborhood ofp.Conclude thatwiscon- stant oneach component ofR"—M. (d)Suppose Mcontains aportion Aofan(72—I)-plane. Letpand qbepoints":55"‘~t_/ /A@" 0U 294 Chapter 3 close tothisplane, butonopposite sides. Show that w(q) =w(p) :|:l.(Show thatr,,,|M ishomotopic toamap which equals rp[MonM—Aandwhich does nottakeanypoint ofAonto thepoint xintheﬁgure.) (e)Show that, ingeneral, ifMisorientable, then R"—Mhasatleast 2com- ponents. The next fewProblems show how toprove thesame result even ifM isnotorientable. More precise conclusions aredrawn inChapter ll. 24. LetMand Nbecompact n-manifolds, and letf,g: M—>Nbesmoothly homotopic, byasmooth homotopy H:M><[0,1]—>N. (a)LetqENbearegular value ofH.Let#f"I(q) denote the(ﬁnite) number ofpoints inf'“1(q). Show that #f'"I(q) ﬁ#3'“I(q) (mod 2)- HinI: H""1(q)isacompact l-manifold-with-boundary The number ofpoints initsboundary isclearly even. (This isoneplace where weusethestronger form ofSard’s Theorem.) (b)Show, more generally, thatthisresult holds solong asqisaregular value of both fandg. 25.Fortwomaps f,g: M—>Nwewillwrite f2gtoindicate that fis smoothly homotopic tog. aIf 2:g,then there isasmooth homoto H’:M><[0,l —>Nsuch that PY H"(p,I) =f(p) forIinaneighborhood of0, H’(p,I) =g(p) forIinaneighborhood ofl. (b)2isanequivalence relation. 26.Iffissmoothly homotopic togbyasmooth homotopy Hsuch thatpi-> H(p,I) isadiffeoniorphism foreach I,wesaythat fissmoothly isotopic tog. (a)Being smoothly isotopic isanequivalence relation. (b)Let¢:R"—>RbeaC°°function which ispositive ontheinterior ofthe unitball, and0elsewhere. ForpES""1, letH:R><R”—>R"satisfy 2””) -¢<H<r8))-P H(0,x) =x. (Each solution isdeﬁned forallI,byTheorem 5-6.) Show thateach .1‘1->H(I,1') isadiffeomorphism, which issmoothly isotopic totheidentity, andleaves all points outside theunithallﬁxed. Integm {ion 295 (c)Show thatbychoosing suitable pandIwecanmake H(I,0) beanypoint intheinterior oftheunit ball. (d)IfMisconnected andp,qEM,then there isadiffeomorphism f:M—> Msuch that f(p)=qandfissmoothly isotopic totheidentity. (e)Usepart (d)togive analternate proof ofStep3ofTheorem 9. (f)IfMand Narecompact n-manifolds, and f:M—>N,then forregular values (]],(]2 ENwehave #f"l(q1) E#f"I(Q2) (mod 2) (where #f'"'(q) isdeﬁned inProblem 24). This number iscalled themod 2 degree off. (g)Byreplacing “degree” with “mod 2degree” inProblem 23,show thatif MCR"isacompact (n--l)-manifold, then R"—-Mhasatleast 2components. 27.Let{X'} beaC°°family ofC°°vector ﬁelds onacompact manifold M. (Tobemore precise, suppose XisaC°°vector ﬁeld onM><[0,1];then X"(p) willdenote rrM,.X(,,,,,-).) From theaddendum toChapter 5,andtheargument which wasused intheproof ofTheorem 5-6, itfollows thatthere isaC°°family {wt}ofdiffeomorphisms ofM[notnecessarily a1-parameter group], with qbo= identity, which isgenerated by{X'},i.e.,foranyC°°function f:M—>Rwe have <Xrf)(p) Z f(¢r+h(P))h"“ f(¢r(P))_ Forafamily cu,of/<-forms onMwedeﬁne thek-form . . (I) [1""‘(I) 8.-.=hmh—>0 /2 (a)Show thatfor17(I)=¢,*w, wehave Ylr=¢f*(LX'wr +68)- (b)Letwoandcu;benowhere zero 22-forms onacompact oriented n-mani- fold M,anddeﬁne cu,=(I-—I)w0+Iw|. Show that thefamily qﬁ,ofditcfeomorphisnis generated by{X'}satisﬁes ¢,*w,- =wt) forallI ifandonly if LXIOJ; =(U0--0J1. 295 Chapter 3 (c)Using Problem 7-l8, show that thisholds ifand only if d(Xr_lQ)y) 2(U9-(1)1. (d)Suppose that IMwo=IMwt,sothat wo-cu;=allforsome Il.Show that there isadiffeomorphism ﬁzM—>Msuch thatwt)=f1*w;. 28.Letf:M2—>R"andg:Ni—>R"beC°° maps, where Mand Nare compact oriented manifolds, II=k+1+I,andf(M) Og(N) =(5.Deﬁne af,g: M><N—> S”_' CR”-{0} by <1/.g<p.q>= age)—mi)== Wedeﬁne thelinking number offandgtobe Sdeg“,/-,8: where M><Nisoriented asinProblem l8. la)5(f=8)=(~1)"’+‘t3(8=f)-(b)LetH:M><[0,1] —>R"and K:NX[0,1] —>R"besmooth homotopies with H(I>.0) =f(P) K(q.0) =8(8) H(P=1)=f(P) K(q.1)=8(8) such that {H(p,I) IpEM}O{K(q,I) :qEN}=Q)forevery I. Show that wig)---I<f.§>- (c)Forf,g: S1—>R3show that ____ I I Irztegratiarz 297 where "(",v) =I301)" f(")l <f‘>'<~> <f2>*<~> <f3>’<~>/1(~.v) =det (8‘)’(v) (82)’(v) (83)'(v) g‘<v>-/1<8)82(1))-1"’-<8)8%»)-1%) (thefactor I/4:rrcomes from thefactthat I52a’=4zr[Problem 9-14]). (d)Show that €(f,g) =0iffand gboth lieinthesame plane (ﬁrst doit for(x,y)—plane). The next problem shows how todetermine Z(f,g)without calculating. 29.(a)For(a,b,c) ER3deﬁne (x—a)dy/\dz—(y-b)dx/\dz+(z—c)dx/\dy "“’<“-be [<8-8>2+<i‘1e8ri<: -82138 - Foracompact oriented 2—manifold-with-boundary MCR3and (a,b,c) géM, let Q(a=b>C) =/Md@(a.b.¢)- Let(a,b,c) and (a",b’,c") bepoints close topEM,onopposite sides ofM. Suppose (a,b,c) isonthesame sideasavector wpER3p--Mpforwhich the l-UP o(£1,(?, C) triple wp,(vi)p,(U2)pispositively oriented inR3pwhen (v1)p,(v2)pispositively oriented inMp. Show that lim Q(a,b,c)--S2(a',bl,C’)=-—4rr. (=1.b.¢)—>p (ﬁlly.-'I")—>P Hint: First show that ifM=6N,then Q(a,b,c) =-41: for(a,b,c) EN—M andS2(a,b,c) =0for(a,b,c) ffN. 298 Chapter 8 (b)Letf:S‘—>R3beanimbedding such that f(.S'l) =BMforsome com- pact oriented 2—manifold-with-boundary M.(AnMwith thisproperty always exists. SeeFort, Yﬁpology Qf3-Jlﬂinﬂiilds, pg.I38.) Letg:S1—>R3andsuppose The ﬁgure ontheleftshows anon-orientable surface whose boundary isthe“trefoil” knot, /’ -\/2/ . \Q67 .,| butthesurface ontheright--including the hemisphere behind theplane ofthepaper- irorientable. thatwhen g(I) =pEMwehave dg/dI 55Mp. Let21+bethenumber ofinter- sections where dg/dI points inthesame direction asthevector wpofpart (a), andI1“thenumber ofother intersections. Show that __ --lIi.-=II+--n =—f g*(dQ). 471' SI (c)Show that 6Q __ ,,,(yg--gb) dz--(2 dy %("’b"°) "flif|<x.y.:>P itTl 3Q __ ,,,(z-—c)dx--(x--a)dz @—8("’b’C"'(.<=f( l(>~'.y.r)l3 l BS2 ___ *(x--a)dy,—-(y--b)dx T1?(“’2")"A112 l |<x.3».z>P l‘ (d)Show that n=€(f,g). Compute t?(f,g) forthepairs shown below. <\* r§ ‘ QC Irztegratitm 299 30.(a)Letp,q ER"bedistinct. Choose open setsA,B CR"—{p,q} so thatAandBarediffeomorphic toR"—-{O},andA|"|Bisdiﬁeomorphic toR”. Using anargument similar tothat intheproof ofTheorem I6,show that =,;t';?.Q'»f=._-.=.=."§£=.=-"t-'.-3'?»-‘Q_,,._,-.__ »=§e%e\_'.‘.',.p‘>‘ f, ."__'.. .-§'._ }%¢_;.;( ~ "1 -- /‘1 2,8*-.it.-7%.‘-ifi 3\.2:-22;)‘ i >2xiI‘:V-3.-2 :22' . .1.5’._.'_ ' s~W1'-é~‘:'/'2-1\~,~7Pf.'.,_t"-;='~£\‘0 11-. 0P .1-3;'.- q .4.-,_.,~t-,'f_\-.e\,;. .;ri2vg- are-:=~.=~.-.-Te,__.,.._,_. ,,,,_. ,_. ii»2‘"-3E* "i\_ .1 :- Qmwg '%@%m:_wt. f"'_'."' tr; 8.-.16: ‘,3:K.1-: .*'.\;,t.->"v‘'.~'a"~\vZ\.:t3‘-‘.') 2-‘*'-=I'.\‘Y2.'.‘\ -0.'%_' .,_-.-.~>.-2: 1-. sq,">>a",1-.<,~QE-$..-gf. ..-~..l-.-""_:* _".-;.\' .>.‘-C._.-;’...-"_.‘__.)_,,,I:_,/.-'-t.-t‘¢=*._-’t~>.'-1”l2'§'='3“r-J4?~"?2'€”‘ H"(R" ~{p,q}) =0for0<I<<IT-1,andthatH"“‘(R” ~{p,q}) has dimension 2. (b)Find thedeRham cohomology vector spaces ofR"-Fwhere FCR”is aﬁnite set. 31.Wedeﬁne thecupproduct U:H2(M) ><Hl(M) —>H"+'(M) by [w]v[It]=[w/\rt]- Show thatuiswell-deﬁned, i.e.,cuA17isexact ifwisexact and17isclosed. Show thatUisbilinear. If8EH"(M) and8EH"(M), then8U8=<-1)“8 UCY. Iff;M_>N,and8eH"(N), 8EHl(N), then /"""\-<I""'~<I"'“'-<&o€& f*<v~ﬁ) =f”‘<Wf”‘ﬁ- (e)Thecross-product ><;H"(M) ><H'(N) _>H'<+’<M ><N)isdeﬁned by [w]><[11]=[1rM*w /\H~*n]- Show that><iswell-deﬁned, andthat or><6=:rtM*cz u7r,v*ﬁ. 300 Chapter 3 (f)IfA:M—>M><Misthe“diagonal map”, given by5(P) =(P,p),show that (IL/ﬂ:A*(O!Xﬁ). 32,Onthen-dimensional torus T”=.S'l><---XS!\..i_......V_i._/ ntirnes letd6ldenote rt,-*d6, where 1:,-:T"—>S1isprojection onthe1"”factor. (a)Show thatalld62'/\---/\d6"'~' represent different elements ofH"(T"), by ﬁnding submanifolds ofT"over which they have different integrals. Hence dimH"(T") 3 Equality isproved intheProblems forChapter ll. (b)Show thatevery map f:S"—>T”hasdegree 0.Hint: UseProblem 25. CHAPTER 9- RIEMANNIAN METRICS Inprevious chapters wehave exploited nearly every construction associated with vector spaces, and thus with bundles, butthere hasbeen onenotable exception—we have never mentioned inner products. The time hasnow come tomake useofthisneglected tool. Aninner product onavector space Vover aﬁeld Fisabilinear function from V><VtoF,denoted by(v,w)1->(v,w),which issymmetric, (vsw)-'=(wlv), andnon-degenerate: ifv=,£0,then there issome w960such that (w,v):,=é0. Forus,theﬁeld Fwillalways beR. Foreach rwith 05r_-5n,wecandeﬁne aninner product (,),onR"by F‘ H (8,8),=Z898" -Z8%‘, i=1 in:-+1 thisisnon-degenerate because ifa;=é0,then N ((al, ...,a”), (al,...,a',-—ar+1,.. .,-—a"))r -.=Z(al)2 >0. i=1 Inparticular, forr=nweobtain the“usual inner product”, (,)onR", TI (8,8)=Zalbl. [ml Forthisinner product wehave (a,a)>0foranya;=éO.Ingeneral, asymmetric bilinear function (,)iscalled positive deﬁnite if (v,v) >O forallv:,-‘-0. Apositive deﬁnite bilinear function (,)isclearly non-degenerate, andconse- quently aninner product. _ 30] 302 C/tapter .9 Notice that aninner product (,)onVisanelement of'T2(V), soif f:W—>Visalinear transformation, then f*(,)isasymmetric bilinear function onW.This symmetric bilinear function maybedegenerate even iff isone-one, e.g.,if(,)isdeﬁned onR2by (8,8)=8181- 828?, andf:R—> R2is f(Q)=(ale)- However, f*(,)isclearly 11on-degenerate iffisanisomorphism ontoV.Also, if(,)ispositive deﬁnite, then f*(,)ispositive deﬁnite ifand only iffis one-one. Foranybasis 111,...,vpofV,with corresponding dual basis v*1,. ..,v*,,,we canwrite H (=)= Z8=';v*i®v*i-z',j=l Inthisexpression. gt;=(v='=“il= sosymmetry of(,)implies thatthematrix <g;'j) issymmetric, 31'}zgif- Thematrix (g,-,8) hasanother important interpretation. Since aninner product (,)islinear inthesecond argument, wccandeﬁne alinear functional (ppEV*, foreach v6V,by 8(w) =(v=w)- Since (,)islinear intheﬁrstargument, themap vI-->(ppisalinear transfor- mation from V1oV*.Non-degeneracy of(,)implies that85,,;é0ifv950. Thus, ifVisﬁnite dimcnsional, aninner product (,)gives usanisomorphism or:V—>V*,with (v,w) =ar(v)(w). Clearly, thematrix (g,-y) isjust thematrix ofas:V—>V*with respect tothe bases {vi} forVand {v*,-} forV*.Thus, non-degeneracy of(,)iscquivalenl tothecondition that (g,-1-) isnon-singular. det{g,-;) #0. Positive dcﬁniteness of(,)corresponds tothemore complicated condition thatthematrix (g,-;) be“positive deﬁnite”, meaning that H Z:gtjalaj >0 foralla1,...,a,, with atleast oneaj;é0.[:1 Itttmtartiziati 1l48tt'ic.\ 303 Given anypositive algizzite inner product (,)onVwedeﬁne theassociated norm by ||v||=t/(v,v) (thepositive square root istobetaken). InR"wedenote thenorm corresponding to(,)simply by Ial=t/(8.8)=|§j<8"r.i=1 The principal properties of||||arethefollowing. l.THEOREM. Forallv,wGVwehave ||(1v||= lr1|-|lvl|- (2)|(v,w)| 5||v||-]|w||, with equality ifand only ifvand warelinearly dependent (Schwarz inequality). <3)||v+wn5uvn+||w||(Triangle i11¢qu=<1Iiw)-/-'-~.hull\-_/ PROOF (l)istrivial. (2)Ifvandwarelinearly dependent, equality clearly holds. Ifnot,then 051$ }\v—wforall)\eR,so 0<||)\v—w||2=(Av—w,}\v —w) =»\2||v||2—28(1).w)+||wu2. Sotheright side isaquadratic equation inLl.with noreal solution, and its discriminant must benegative. Thus 48.».w)2—4||v||2||w||2 <0. (3) ||v+w||2 =(v-l-w,v+w) =||v|l2+ ||w||2+2(v,w) 5llvllz+llwllz+2||v|| -llwll by(2) =(llvll+||w||)2- *3‘ The function ||||hascertain unpleasant properties--for example, thefunc- tion|IonR"isnotdifferentiable at0ER”—-which donotarise forthefunction ||||2.This latter function isa“quadratic function” onV-—in terms ofa basis {v,-}forVitcanbewritten asa“homogeneous polynomial ofdegree 2”inthe components, H 2 H Ialt)’ =gg,-,.-a'a". I'=I i,j=1 304 C/zapter 9 More succinctly, Pi 2ll :Z gtjl-:'*:‘ ‘V2,?- Is]-=l Aninvariant deﬁnition ofaquadratic function canbeobtained (Problem l)from thefollowing observation. 2.THEOREM (POLARIZATION IDENTITY). IfII||isthenorm associ- ated toaninner product (,)onV,then ll)(U1w): illlv+w|l2—|lv||2—||w||2l <2)<v.w>=~lt||v+wnt—nv—wl|2]- PROOF. Compute. *1‘ Theorem 2shows that twoinner products which induce thesame norm arc themselves equal. Similarly, iffIV—>Visnorm presenting, that is,||f(v)|] = ||v||forallveV,then fisalsoinner product preserving, thatis,(f(v),f(w)) = (o.w)forallv,weV. ‘Newillnow seethat, “uptoisomorphism”, there isonly onepositive deﬁnite inner product. 3.THEOREM. If(,)isapositive deﬁnite inner product onanI1-(lilI'lt3l1- sional vector space V,then there isabasis 1.11,...,upforVsuch that (tr,-,tr,-)= 5,-,;. (Such abasis iscalled orthonormal with respect to(,).)Consequently, there isanisomorphism f:R"—>Vsuch that (¢1.b)= (f(a)» f(b)), albGR"- Inother words, f*(8)=(J)- PROOF Let w1,. ..,wpbeany basis forV.VVcobtain thedesired basis by applying the“Gram-Schmidt orthonormalization process” tothisbasis: Since w]560,wecandeﬁne wtv]=Z, llwlll and clearly ||v|||=l.Suppose that Wehave constructed tr],...,v;, sothat lvi=vil =5:; I5-",15R Rzenzantziarz /l4em'c.s 305 and spanv|,...,v;< _—=span w1,...,w;<. Then wk_|_; islinearly independent ofU],...,v;<. Let wi<+1 =wk+1 —(1-‘IN-’k+1l'~’l — '"(1%?-’k+1l1J/< 5*0- Itiseasytoseethat (w;{_H,v,-)=0 z'=l,...,l<. Sowecandeﬁne I wk+1 1-’1<+1 =WHIP andcontinue inductively. *I~ Apositive deﬁnite inner product (,)onVissometimes called aEuclidean metric onV.This isbecause weobtain ametric ponVbydeﬁning p(l-)2 w) =llv — The“triangle inequality” (Theorem 1(3))shows thatthisisindeed ametric. We alsocall||v||thelength ofv. W’ehave only onemore algebraic trick toplay. Recall thataninner product (,)onVprovides anisomorphism atV—>V*with ¢Y(v)(w) =(v,w)- Using thenatural isomorphism 1':V—>V**, deﬁned by i(v)(?~) =»$v), weobtain anisomorphism (Fl 1' 5;v*—>V-->(1/*)*. Wecannow use19todeﬁne abilinear function (,)*onV"‘by Wu)‘ =ﬂ(l)(#) =1'vf"(»$(#)= u(<>f"(»\))- Now, thesymmetry of(,)canbeexpressed bytheequation <r(v)(w) =<r(w)(v)- 306 Chapter 9 Letting or(v)=R,rr(w)=M, thiscanbewritten 3$w"‘(u)) =utv-'_‘(1)), which shows that (,)*isalsosymmetric, (t¢,»$* =(i\,#)*- Consequently (,)*isaninner product onthedual space V*(infact, theone which produces 15). - Toseewhat thisallmeans, choose abasis {v,-}forV,let{v*,-} bethedual basis forV*,andletH (, )= Z g;jU*;'®U*j. ;',j=I Then (g,-j)isthematrix of cc:V->V* withrespect to{v,-} and {v*,-} so (8i,t)“1 isthcmatrix ofof‘: V*—>Vwith respect to{v*,-} and {vi} SO (g,-j)_l isthematrix of B:V*—>V** with respect to{v*,-} and {v**,-}. Thus, ifweletgijbetheentries oftheinverse matrix, (gill)=(gt;)_1,Sothat H Zgikgkj =5}, /{=1 then P? (,)*= Z g:;U*=|=i®v=|==|=]_ 1',J'=1 rt =Zg”v;®v,-, ifweconsider v,-eV“‘*. ;,]:l One can check directly (Problem 9),without theinvariant deﬁnition, that this equation deﬁnes (,)"‘independently ofthechoice ofbasis. 1€ienzm2:2z'art /l4em'c.~ 307 Notice thatif(,)ispositive deﬁnite, sothat ar(v)(v) >0 forv#0, then, letting o:(v) =A,wehave Ate-1(,t)) =e(;t)(,t) >0forA¢0, so(,)*isalsopositive deﬁnite. This canalso bechecked directly from the deﬁnition interms ofabasis. Inthepositive deﬁnite case, thesimplest way to describe (,)*isasfollows: The basis v*1,...,v*,,ofV*isorthonormal with respect to(,)*ifandonlyif1.11,...,v,, isorthonormal with respect to(,). Similar tricks can beused (Problem 4)toproduce aninner product onall thevector spaces 9"<(v), 5',,(t/) =:"<(v*), ands2’<(v)_ However, weare interested inonly onecase, which wewillnotdescribe inacompletely invariant way. The vector space Q"(V) is1-dimensional, sotoproduce aninner product onit,weneed onlydescribe which twoelements, wand—w,willhave length 1. LetU],...,L1,,and w1,. ..,w,,betwo bases ofVwhich areorthonormal with respect to(,).Ifwewrite Pi’ 11);’=Zap-v,-, 1=I then H H I1 5;;=(wiswj> = ﬁt/er?-’k»Z:0ftjvt> =ZI-Ykt°ltj(vl<»vll /{=1 f=l /f,f=l II‘ =Z0H<t0H<,t- k=1 Sothetranspose matrix A‘ofA=(oz,-j) satisﬁes A-A‘=1,which implies that cletA=:|:l.Itfollows from Theorem ?—5thatforanywEQ”(V) wehave w(v1,...,v,,) =:|:w(w1,...,w,,). Itclearly follows that 11*]A---/\v*,, =:|:w*1/\---/\w*,,. Wehave thus distinguished twoelements ofQ"(V); they areboth oftheform v*i/\---/\v*,, for{vi}anorthonormal basis ofV.Wewillcallthese twoelements 308 Chapter 9 theelements ofnorm 1inS2"(V). Ifwealsohave anorientation ,tt,then wecan further distinguish theonewhich ispositive when applied toany (v1,. ..,v,,) with [v1,...,v,,] =/tt;wewillcallitthepositive element ofnorm 1inS2"(V). Toexpress theelements ofnorm 1interms ofanarbitrary basis wt,...,w,,, Wechoose anorthonormal basis v1,...,1),,andwrite H U);=2(X),-,-Uj. 1'=1 I-I Problem 19implies that det(crz,-,;) w*1A---Aw*,,=11*]A---Av*,,. Ifwewrite iH‘ (a )= Z 8z'jw*i®w*j, 1'.f=1 then rt rt 8;;=(wt,w,i'l = akivki Z3‘-1ft,tv1> k| t1 H =ZIrzetmej, k=1 soifA=(tre,-J,-). then clet(g,~j) =det(A' -A)=(detA)2. Inparticular, det(g,-j) isalwqrs positive. Consequently, theelements ofnorm 1 inQ"(V) art‘ i~/dettge-) w*1A---/\wﬁt gt;=(wt,we)- Wenow apply ournew tooltovector bundles. If§=rt:E—>Bisavector bundle, wedeﬁne aRiemannian metric onEtobeafunction (,)which assigns toeach pGBapositive deﬁnite inner product (,)1,on7r_l(p), andwhich iscontinuous inthesense thatforanytwocontinuous sections s1,S21B—>E, thefunction (S1,S2l =P'—>(S|(P),S2(Pllp isalsocontinuous. IfEisaC°°vector bundle over aC°°manifold wecanalso speak ofC°°Riemannian metrics. Rzkwzmzrzzian 1l4ezrics 309 [Another approach tothedeﬁnition canbegiven. LetEuc(V) bethesetofall positive deﬁnite inner products onV.Ifwe replace each rr“'(p)byEuc(rr_' (p)), andlet Bets)=UBeet-‘tpii.peB then aRiemannian metric onEcanbedeﬁned tobeasection ofEuc(E). The onlyproblem isthatEuc(V) isnotavector space; thenewobject E:ta(E) thatwe obtain isnotavector bundle atall,butaninstance ofamore general structure, aﬁbre bundle] 4.THEOREM. Let5=rt:E—>Mbea[C°°] It-plane bundle over aC°° manifold M.Then there isa[C°°]Riemannian metric onE. PROOF. There isanopen locally ﬁnite cover (9ofMbysetsUforwhich there exists [C°°]trivializations IU:rr_1(U) —>U><litk. OnUxlitk,there isanobvious Riemannian metric, ((P,a),(1>,b))p =(ab)- Forv,werr"(p), deﬁne <v.w>j;’=(1‘u(v)>fu(w))p- Then (,)Uisa[C°°] Riemannian metric forEIU. Let{@511} beapartition of unity subordinate to(9.Wedeﬁne (,)by (v,w)p =Z¢u(P)(v=w)ff v,werr"(P)- U50 Then (,)iscontinuous [C°°] andeach (,),,isasymmetric bilinear function on7T_](p). Toshow that itispositive deﬁnite, note that twp=Z¢eo>)<v.v>,‘,:’;U50 Ueach ¢U(p)(v,v),, 20,andforsome Ustrict inequality holds. *1‘ [The same argument shows that anyvector bundle over aparacompact space hasaRiemannian metric.] Notice that theargument intheﬁnal step would notwork ifwehad merely picked non-degenerate inner products (,)U.Infact(Problem 7),there isno (,)onT82 which gives asymmetric bilinear function oneach S2},which is notpositive deﬁnite ornegative deﬁnite butisstillnon—degenerate. 310 Chapter 9 Asanapplication ofTheorem 4-,wesettle some questions which have tillnow remained unanswered. :1.COROLLARY. IfE=rt:E—>Misak—plane bundle, then E1'E*. PROOF. Let(,)beaRiemannian metric forE.Then foreach pGM,we have anisomorphism %I1r"‘(p)—> tit-'tp>1* deﬁned by Oipwllwl =<1».we v.weIr‘(p)- Continuity of(,)implies thattheunion ofall upisahomeomorphism from E toE’=UpGMl7T_](P)l*- +1» 6.COROLLARY. IfE =rt:E—>Misal—plane bundle, then Eistrivial if andonly ifEisorientable. PROOF. The “only if”partistrivial. IfEhasanorientation itand(,)isa Riemannian metric onMthen there isaunique s(1>)err'(P) with (-5113'):-5'(Pllp =1; l5(Pll =lip- Clearly sisasection; wethen deﬁne anequivalence ftE—>M><IRby f(5\S(P)) =(PA)- ALTERNATIVE PROOF Weknow (seethediscussion after Theorem 7-9)that ifEisorientable, then there isanowhere 0section of Q‘ts)=s*. sothat E*istrivial. ButE2E*.*I* Allthese considerations take onspecial signiﬁcance when our bundle isthe tangent bundle TM ofaC°° manifold M. Inthiscase, aC°° Riemannian metric (,)forTM, which gives apositive deﬁnite inner product (,),,on Itlientrtmzznrz 1l4emr.\ 31I each MP,iscalled aRiemannian metric onM.If(x,U)isacoordinate system onM,then onUwecanwrite ourRiemannian metric (,)as H (,)= Zg,-,-dx"®dx1', I'.f=1 where theC°°functions gr;satisfy g,-j=gj,-,since (,)issymmetric, and det(g;j) >0since (,)ispositive deﬁnite. ARiemannian metric (,)onM is,ofcourse, acovariant tensor oforder 2.Soforevery C°°map f:N—>M there isacovariant tensor f*(,)onN,which isclearly symmetric; itisa Riemannian metric onNifandonly iffisanimmersion (f,.,, isone-one for allpGN The Riemannian metric (,)*,which (,)induces onthedual bundle T*M, isacontravariant tensor oforder 2,andwecanwrite itas H ..3 3*___ lj_i i_ <.>—UZ=:]e weaxj. Our discussion ofinner products induced onV‘shows thatforeach p,the matrix (g‘~’l(p)) istheinverse ofthe matrix (gr;(p)); thus H Zst-re“ =5,?- /{=1 Similarly, foreach pGMtheRiemannian metric (,)onMdetermines twoelements ofS2"(M,,), theelements ofnorm I.Wehave seen thattheycan bewritten i~/dettet,-(p)) dX'(1>) /\AdX"(1>)- IfMhasanorientation ,U.,then ,tt,,allows ustopick outthepositive element of norm 1,andweobtain anit-form onM;ifx:U—>IR"isorientatiﬂrt preserving, then onUthisform canbewritten ~'det(g,-_;) dxlA---Adx”. Even ifMisnotorientable, weobtain a“volume element” onM,asdeﬁned inChapter 8;inacoordinate system (Jr,U)itcanbewritten as vdet(g,;,-) ldxlA---AdA'”|. This volume element isdenoted bydV,even though itisusually notdof anything (even when Misorientable anditcanbeconsidered tobeann—form), 312 C/zaflier 9 andiscalled thevolume element determined by(,).Wecanthen deﬁne the volume ofMas fav.M This certainly makes sense ifMiscompact, andinthenon-compact case (see Problem 8—lO) iteither converges toadeﬁnite number, orbecomes arbitrarily large over compact subsets ofM,inwhich case wesaythatMhas“inﬁnite volume”. IfMisann-dimensional manifold (-with-boundary) inIR",with the“usual Riemannian metric”H <»=ZMew.i=1 then g,-j=5,-j,so dV=Idx‘A---Adx"|, and“volume” becomes ordinary volume. There isaneven more important construction associated with aRiemannian metric onM,which willoccupy usfortherestofthechapter. Forevery C°° curve y:[a,b] —>M,wehave tangent vectors dy }"(=’) =EEMytt), andcantherefore use(,)todeﬁne their length ||dl = Q,dl = Q,dl ,tobeprecise .dt dtdt dtdzW) Wccanthen deﬁne thelength ofyfrom atob, b b worif giw <=/'hvmw) Ifyismerely j1zk'ceu12'se smoot/1, meaning thatthere isapartition a=to<---< tn=bof[a,b] such that yissmooth oneach [t,-_|,t,-] (with possibly different Riemetmzian 1l/Iemes 313 left—andright—hand derivatives at:1,...,r,,_,), wecandeﬁne thelength ofyby Lfitr) ~=2I-f§_,(VI[rt-1,til)-i=1 Whenever there isnopossibility ofmisunderstanding wewilldenote Lgsimply byL.Alittle argument shows (Problem l5)thatforpiecewise smooth curves in IR",with theusual Riemannian metric tdxl®dxl, t'=l thisdeﬁnition agrees with thedeﬁnition oflength astheleast upper bound of thelengths ofinscribed polygonal curves. Wecanalsodeﬁne afunction s:[a,b] —>IR,the“arclength function ofy” by"dI=LL If d. s() (Y) 0dz I Naturally, (ii) s’(:)=||‘j]—:. Consequently dy/dt hasconstant length 1precisely when s(t)=t+constant, thusprecisely when s(t)=I—a.Then b—a=s(b) =Lg(y). Wecanreparameterize ytobeacurve on[0,b-—a]bydeﬁning PU)=}’(F—61)- Forthenew curve ywehave news(t)1-:L§(}7) =Lf,+"(y) =olds(t+a)-olds(a) =r. If}/satisﬁes s(t‘) -=Iwesaythat yisparameterized byarclength (and then often usesinstead ofItodenote theargument inthedomain ofy). 314 Chapter 9 Classically, thenorm ||llonMwasdenoted byds.(This makes some sort ofsense even inmodern notation; equation (=t=)says thatforeach curve yand corresponding s:[a,b] —>IRwehave Ids!=J/“(ll ll) on[a,19].)Consequently, inclassical books oneusually seestheequation H d-5'2 =Z ggj dx'dx’. l,}:] Nowadays, thisissometimes interpreted asbeing theequivalent ofthemodern equation (,)= J,-=1gt;dx’®dxt’, butwhat italways actually meant was N ||3|-2=Zg,-,-dxidxl. IsJ'=1 The symbol dxldx-l appearing here isnotaclassical substitute fordxl®dxj— thevalue (dxida-j)(p) ofdxidxj atpshould notbeinterpreted asabilinear function atall,butasthequadratic function v|—>dx"(p)(v) -dXj(p)(U) vGMp, and wewould usethesame symbol today. The classical way ofindicating dxl®dxf wasvery strange: onewrote H‘ Zgt;dxltix-" where dxandtixareindependent inﬁnitesimals. l',j:1 (Classically, theRiemannian metric wasnotafunction ontangent vectors, but theinner product oftwo“inﬁnitely small displacements” dxandcix.) Consider now aRiemannian metric (,)onaconnected manifold M. If p,qGMareanytwopoints, then there isatleast onepiecewise smooth curve y:[a,b] —>Mfrom ptoq(there iseven asmooth curve from ptoq).Deﬁnc d(p, q)=inf{L(y): yapiecewise smooth curve from ptoq}. Itisclear thatd(p,q) 30andd(p, p)=0.Moreover, ifrGMisathird point, then forany.9>0,wecanchoose piecewise smooth curves yl:[a,b] —>Mfrom ptoqwith L(}/1) —d(p,q) <a }/2:[b,c-] —>Mfrom qtorwith L(}/2) -~d(q,r) <a. Rz'emanniatz 1l4em'c.i 315 Ifwedeﬁne y:[a,c] —>Mtobeylon[a,b] and }/2on[b,c], then yisa piecewise smooth curve from ptorand I-(1/)= I-(1/0+ Lo/2)<dtptq)+dtq,1')+Ze- Since thisistrueforalle>0,itfollows that d(Pn') Sd(P,Q)+d(q,t')- [Ifwedidnotallow piecewise smooth curves, there would bediﬂiculties inﬁtting together y,and}/2,butdwould stillturnouttobethesame (Problem l7).]The function d1M><M—>IRhasallproperties forametric, except thatitisnotso clear that d(p,q) >0forpséq.This ismade clear inthefollowing. 7.THEOREM. The function dtM><M—>Risametric onM,and if p:MxM—>IRistheoriginal metric onM(which makes Mamanifold), then (M,d)ishomeomorphic to(M,p). PROOF. Both parts ofthetheorem areobviously consequences ofthefollowing 7’.LEMMA. LetUbeanopen neighborhood oftheclosed ballB={pGR": [pl51},let(,)8bethe“Euclidean” orusual Riemannian metric onU, n (,>,=Ede‘ ®dxi, t=1 andlet(,)beanyother Riemannian metric. LetII=llll,and llIIbethe corresponding norms. Then there arenumbers m,M>0such that m~ll5lll|5M'll ORB, andconsequently foranycurve y:[a,b] —>Bwehave ml-e(r) SI-(1')5MLeo’)- PROOF. Deﬁne GtB><.S‘"_l —>Rby G(P=(1) =llapllp- Then Giscontinuous andpositive. Since Bx8"“! iscompact there arenum- bers m,M>0such that m<G<M onB><S"_'. Now ifpGBand0aébpGlR",,, letaGS"" bea=b/lb). Then mlbl <lbIG(p,a) <Mlbl; since lblG(1M1) =lbl-llapllp =ll(l/Jlalpllp =llbllp, thisgives thedesired inequality (which clearly alsoholds forb=0).'1' 316 C/zaflter 9 Notice thatthedistance d(p,q) deﬁned byourmetric need notbeL(y) for anypiecewise smooth curve from ptoq.Forexample, themanifold Mmight be1R2-{O},andqmight be—p. Ofcourse, ifd(p,q) =L(y) forsome 1/, Ml”—P then yisclearly ashortest piecewise smooth curve from pto:1(there might be more than oneshortest curve, e.g., thetwosemi-circles between thepoints p and-11onSl). lnorder toinvestigate thequestion ofshortest curves more thoroughly, we have toemploy techniques ﬁ'om the“calculus ofvariations”. Asanintroduction tosuch techniques. weconsider ﬁrstasimple problem ofthissort. Suppose we aregiven a(suitably differentiable) function F:lR><IR><IR—>IR. Weseek, among allfunctions ft[a,b] —>IRwith f(a) =a’andf(b) ~=b’one 1,»; I ct’ _ |_1' | W ‘ ct b which willmaximize (orminimize) thequantity fbF(r,f(r)=f'(I))dt- Forexample, if F(r,1<,.v)= ~/1+92, Rziemannzian /l4e£rics 317 then wearelooking forafunction fon[a,b] which makes thecurve t|—> (I,f(t)) between (a,a') and(b,b') ofshortest length b f,/1+[f’(t)]-2 dr. Asasecond example, if F(t,x,'y) =2rrx\/1+y2, then wearetrying tominimize thearea ofthesurface obtained byrevolving thegraph offaround thex-axis, which isgiven (Problem l2)by 21:fbf(I),/1 +[f'(t)]2 dt. Toapproach thissortofproblem werecall ﬁrstthemethods used forsolv- ingthemuch simpler problem ofdetermining themaximum orminimum of afunction f:IR—>IR.Tosolve thisproblem, weexamine thecritical points off,i.e.,those points xforwhich f’(x)=0.Acritical point isnotnecessarily amaximum orminimum, oreven alocal maximum orminimum, butcritical points arctheonlycandidates formaxima orminima iffiseverywhere differ- entiable. Similarly, forafunction f:IRZ—>lRweconsider points (x,y)GIRE forwhich (*) Dif(¢\',J’) =D2f(XJ) =0- '(»\'=J'l / A This isthesame assaying that thecurves I'—>f(»\'+r=y) f*—>f(x,y+t) 318 chapter 9 have derivative 0at0.\'Vemight trytogetmore information byconsidering thecondition 0=(foc)'(0) forevery curve c:(—£,s) —>IREwith c(0) =(x,y), butitturns outthat these conditions follow from (rt),because ofthechain rule. Toﬁndmaxima andminima for b Jtf)=fFt:.ft¢),f’tt))dr wewish toproceed inananalogous way, byconsidering curves inthesetofall functions f1[a,b] —>lR.This can bedone byconsidering a“variation” off, that is,afunction or:(—s,s) ><[a,b] —>IR such that 0z(O,!) = The functions t|—>ot(u,r) arethen afamily offunctions on(—e,a) which pass through fforu=0.Wewilldenote thisfunction bycir(u). Thus 6:isa function from (—s,s) tothesetoffunctions f:[a,b] —>IR.Ifeachcir(u) satisﬁes cir(u)(a) =a’,5z(u)(b) r:b’,inother words if _ b’- ot(u,a) =a’ ur_. ar(u,b) =bl'\\ - te~ |~ lg a b foralluG(-—-s,£), then wecalloravariation offkeeping endpoints ﬁxed. Foravariation ctwenow compute dJ(6t(u)) ___dfl’ 80:du 0-»E;H0GF(t,oe(u,t),E(u,t)) dz 11': Z Rz'en2annz'an 1l4elric.s" 319 b -"=1; “=0F(t,a(u,t),i?3—c:(u,t))] dt bBot 3F , zﬁ 82o: 3F , +mto.r>5;tr./(0.1 on]eh. Since 82a/Built =820:/Btdu, wecanapply integration byparts tothesecond term intheintegrand, thus obtaining tr)“J(if"” 0:fabgjt0.r)[%§tt.ftt),f'<¢)) N: d3F , —E(5';(5rf(5lrf (5)))] dl aa or ,”"l"%(01'i)5(r>.f(f)>f(f)) Forvariations ctkeeping endpoints ﬁxed, thesecond term is0,andweobtain - b i ‘”“‘“‘” ef5130.1)Bithftti. rm)0 0 XH: ilu d3F , —E('53:(taf({)1f (l')l)] di- Inclassical treatments ofthecalculus ofvariations, thevariations orwere taken tobeofthespecial form ¢Y(H,f) -‘=f(=’)"I"110(1), forsome 27:[a,b] —>IRwith 17(a)~.-.=r,\(b)=-0.Then weobtain dJ(5r(u)) 1’ or dBFu:0= 7l(r)[$(f1f(t)>f’([))_E($([:f(£)1f!(t))):| dr- Theﬁnalresult is,ofcourse, essentially thesame. Thederivative % 0J(ti:(11)) iscalled the“ﬁrst variation” ofJandisdenoted classically by "- banoar51:.-...-‘L 320 Chapter 9 Asisusual inclassical notation, thearguments offunctions areeither putin indiscriminately orleftoutindiscriminately--in thiscase, notonly arethear- guments tand (I,f(I),f’(1))omitted (resulting inthedisappearance ofthe function fforwhich wearesolving), butthedependence of5Jonccisnot indicated (which canmake things pretty confusing). Iffistomaximize orminimize J,then 5J(ce) must be0forevery variation or offkeeping endpoints ﬁxed. AsinthecaseofI-dimensional calculus, there is noreason toexpect that thecondition 5J(ur) =0forallaswillimply that fis even alocal maximum orminimum forJ,andweemphasize thisbyintroducing adeﬁnition. VVecallfacritical point ofJ(oranextremal forJ)if5J(cz) =0 forallvariations 0:offkeeping endpoints ﬁxed. Theparticular form (=|==|=)into which wehave putSJnow allows ustodeduce animportant condition. 8.THEOREM (EULER’S EQUATION). The C2function fisacritical point ofJifandonly iffsatisﬁes %§<:,_/1:), rm)-%(%<:.f<:),r’u)>) =0. PROOF. Clearly fmust make theintegral in(**)vanish for6061]’ 3 r1(f)=gm») which vanishes ataandb.Sothetheorem isaconsequence ofthefollowing simple 8’.LEMMA. Ifacontinuous function g:[a,b] —>IRsatisﬁes b fr}(r)g(I) df=O forevery C°°function 2;on[a,b] with 11(0) =r;(b) =0,then g=0. PROOF Choose Y]tobeqﬁgwhere Q5ispositive on(a,b)and¢(a) =¢(b) =0.*Z* Asanexample, Consider thecase where F(r,.\-, y)=VI+yz.The Euler cquation is 0:g( rm ) ‘”~/1+[f'm1’ ’ Rz'en2am2z'cm 11/Ietnrs 321 S0 X/i,.@_.=,_,_ f”1+f ff‘/M hence 0:(1_+_f:2)f.v _fr]-:1 :(1_ fr_+_ f!2)f!f, which implies that f""=0,sofislinear. Notice thatwewould have obtained thesame result ifwehadconsidered the case F(t,x, y)=1+yz,forthen theEuler equation issimply d .»0=E(2f (5))- This isanalogous tothesituation in1-dimensional calculus, where thecritical points of\/farethesame asthose off,since r_ ff (~/7)—$- Forthecase ofthesurface ofrevolution, where F(£,x, y)=xv1+yz,the Euler equation is 0:f1+[f,(,)]2 jf( f(r)f»(r) ); \/1+[f’(I)]2 1+ j-:2 _ffu =0, which wewillalsowrite intheclassical form dy2 dzyl"l" —_}W = Tosolve this,weuseoneoftheR0standard tricks (leaving justiﬁcation ofthe details tothereader). Weletthisleads totheequation dy_ ‘Z io PMy dx Then dz)’ dﬁIdpdy_Q dig_dx'dydx"P41-’ 322 Chapter 9 soourequation becomes dp1+p2—yp;)_=0, p 1 id =-ed’, 1+’); P yJ 15log(1 +pg)=logy +constant y=-constant-\/1 +112 d: pz6Ti:VCJJ2—1 L}, —dx Vcyz—1 andthus (seeProblem 20forthedeﬁnition and properties ofthe“hyperbolic cosine” function cosh anditsinverse) 59L;‘2=,,+,,_ Replacing Cby1/C, wewrite thisas a) y=cm%(i%£). The graph of _ex—l-2""__-T isshown below; itissymmetric about they-axis, decreasing forx50,and increasing forx30.cosh x cosh Rzemannian 1l4e£rz'cs 323 Sooursurface must look liketheonedrawn below. Itis,bytheway, not trivial todecide whether there areconstants kandcwhich willmake thegraph of(=i=)pass through (a,a’) and(b,b’). Problem 2linvestigates thespecial case where a’=b’. \ iiU"- /'—'\F’Itiseasy togeneralize these considerations tothec andasewhere f:[a,b] —>IR” b J(f)=f F(r,f(r),f"(:))dr forF:lR><lR"><IR"—>IR. G Inthiscaseweconsider oz:(—a,s) ><[a,b] —>IR"with 51(0) =f,andcompute that dJ(5!(u)) 317' ‘***>T ' — =0- = I)l5F(r,f(r),f’(r)) -%(§§<:,f<:),f*<::»))] dr 317 b + _M=sii.(0,r)a—y,(r, ftr),f’(¢)) Thus ,anycritical point fofJmust satisfy then 3F , d 3F , 5x—,(r,f(r).f (ID—5(5?-(r,f(r),f (r))=0- Wearenow going toapply these results totheproblem ofﬁnding shortest paths inamanifold M.Ify:[a,b] —>Misapiecewise smooth curve, with }/(a) =pand }/(b) ~=q,wedeﬁne avariation ofytobeafunction as:(—s,£) x[a,b] —>Mequations 324 Chapter 9 forsome s>0,such that K)rr(0,I) =}’(l), (2)there isapartition a=10 <1, < <IN=.-bof[a,b] sothat ozisC°° oneach strip (...s,s) ><[1,-_1,r,-].v--a-I V\lecallatavariation ofykeeping endpoints ﬁxed if to s@~~)=1’ fl -—- . a(u,b)__=q oralue( e,s) ,....>\\>\ll/4,—-*___€_______,-- _,./ /' \ “” “1:::::s\ \l§§§Eg? _ ‘~._\ -a_____,. /itAsbefore, weletci?(u) bethepath r|—>ce(u,I). Wewould liketoﬁnd which paths ysatisfy dl-(5¢(M)) T“ u=0 forallvariations ozkeeping endpoints ﬁxed. However, wewilltake ahint from ourﬁrstexaniple andﬁrstfind thecritical points forthe“energy” 1bdy2 1bdydyE=- -d=- --0’) 2L dz I2Lldfdrldi’ which hasamuch nicer integrand; afterwards wewillconsider therelation between thetwointegrals. V\lecanassume that each y|[r,-_|,t,-] liesinsome coordinate system (x,U) (otherwise wejust reﬁne thepartition). If(L1,!) isthestandard coordinate system in(—£,£) X[(1,1)] wewrite a£(u r)-—-at 831.1 ’ iiBu(“,0 Ba 8——(u,I) =C6,, — . 8: (8:(,,,,)) Rz'crnanmmz /l4e£n'c.s 325 Then Elur/3t‘(u,I) isthetangent vector attime Itothecurve cir(u). IfWeadopt theabbreviations 1-Y"(H,r) =»Y’i(¢Y(H,f)), mi)-'=Xi(}’(¢’)) =¢Yi(0,=')= then Soa "aw a dy "dy‘a;(HJ)= ilhll-F , - I l=l A duh!) I I-=l I x Y0) 1lldy dy E(}’| [ii-1,Iil) "= (E= dl 17-1 1‘IH dyldyj::- -- ‘id .,_]wig]gut}/(1)) d,d,1 Ifweusethecoordinate system xtoidentify Uwith IR",andconsider theg,-j asfunctions onR",then weareconsidering ft;F(i'(f),i/"(I))d¢ where 1" .. F(»r.y) =5Zgt,-(Jr)-J=’y’-i,j=l Then __ aF( dy 1"ag,-, d)/‘d1/J 3.1’ 211'} SO-»¢-i/(1).?) =5Z,—A_,<i»ui>W7,i,j=I d OF dy " dy’,—y,(yo).7,7)=ggrrti/(I))T, dBF dy " 41;» "ag” dyfdyt»,;;(;,~;;; (1/<:).;))--ggi.<i/(ti) dz,+2]We/vi) d,ch. 326 Chapter 9 Inorder toobtain asymmetrical looking 1‘esult, wenote that alittle index juggling gives ijEEK- i:%~<’_Y""L".. @¢’_>"'<’L’”_=|as-Iatdz,,j_=|Eixfatat"Uzi 3x"atat 50 i:‘lgidLjﬂ=li:%d_1’i"_1’j+li:@"_1’idLj3x3drdz 2__ Bxfdrdr 2,_|8x‘drdr'F-J=| l,J'=l *»J'= From wenow obtain dam): 9.-_..it-1)duu=0 I," "ad! " d2]/r -—- jg!“ gwﬂlill [g;8rr(}’(fllf,2 H138,-: 3.9;: ast; dy‘di/1''l'i’J_Z=l 5(w(l/(1')) "l"ax;(HO) —ﬁll/(Flow? df Z +-F1$3’d (0.1)Zgm/(ii) 7’;r=lr1:‘ ft‘ Remember thatyisonlypiecewise C°°.Let 1 _ 1'0’:-1) Q _ %(r,-+) =right hand tangent vector ofyatI; drin) d—y(:-‘) .-=lefthand tanentvector of att- %(ti+)(II i g y i’ y(H+:l Notice that theﬁlial suni intheabove formula issimply 3 d 3 d fl‘)? -F fl'—|)97:,(ff—l+)>- Toabbreviate theintegral somewhat weintroduce thesymbols 3 3 .- 1 i '3s‘ l[””'] Z5(ail'1"ail: aitill RZ'6ﬂ?(I???ZZiQ?Z /l4e£rz'cs 327 These depend onthecoordinate system, buttheintegral "'H33:’ H dz}/’ n dyldyj - — .-— “.1 —— -<1. /{Hg 3,,(0.1)gs.»(1/(1))6,,+fJZ:j][=i 1041))d,d,I which appears inourresult, clearly cannot. Consequently, wewillusetheexact same expression foreach [1,-_|,I,-],even though different coordinate systems may actually beinvolved (and hence different g,-;andyl). Now wejusthave toaddupthese results. Let d d d _ Af|'?]l',/=?]g,’(l.I_+)_?Jg,’(!|:_) !=19'-':N—1 <1?9'?AW L. dr dr_Ant-I =—E(7N Then weobtain thefollowing formula (where there isaconvention being used intheintegral). 9.THEOREM (FIRST VARIATION FORMULA). Forany variation ct, wehave dE(5z(u)) duu=0 b H H H ‘. -—g$(0.r) gg.».<i»ui>——~—d,, +UZ=§|[v.11<i/<r>>7,,—7 -4: N -Z(ai<@»~>~>#1l~l_0 3u dt (Inthecase ofavariation atleaving endpoints ﬁxed, thesumcanbewritten from 1toN—1.) This result isnotverypretty, butthere itis.Itshould benoted that[I'j,3]are notthecomponents ofatensor. Nevertheless, later onwewillhave aninvariant interpretation oftheﬁrst variation forniula. Forthetime being wepresent, with apologies, thiscoordinate dependent approach. From theﬁrstvariation formula itis,ofcourse, simple toobtain conditions forcritical points ofE. 328 Chapter 9 l0.COROLLARY. Ifyt[a,b] —>MisaC°°path, then yisacritical point ofE2ifandonly ifforevery coordinate system (x,U)wehave d2 r H dId 1 Zgi,<y<r>>,—§ +Zti./.11<m>>7’;7’, =0fornoev.J"=] |",j=I PROOF. Suppose yisacritical point. Given 1with y(I) eU,choose apartition of[a,b] with IE(I,-_;,I,-) forsome i,andsuch that yl[t,-_1,I;] isinU.Ifor isavariation ofykeeping endpoints ﬁxed, then intheﬁrst variation formula wecanassume that thepart oftheintegral from I,--1toI;iswritten interms of(x,U).The ﬁnal term intheformula vanishes since yisC°°. Now apply themethod ofproof inLemma 8',choosing all30//3u(0, t)tobe0,except one, which is0outside of(I,-_|,I5),butapositive function times theterm inbrackets on(f:'—I>fl)- .:‘ Inorder toputtheequations ofCorollary l0inastandard form weintroduce another setofsymbols k H1,.. Hkl 38'! 38'! 38" "=v'=Zg'l””1=Zg'5(ﬁ""ﬁ-ﬁl'l 1 ' [=1 I: Our equations cannow bewritten d2]/k ll dyi dyj 7;?"t‘"5°"‘”W7 =°- Weknow from thestandard theorem about systems ofsecond order differential equations (Problem 5-4), that foreach peMand each vGMP, there isa unique yr(—s,£) —>M,forsome 8>0,such that 1/satisﬁes H0)=P dy=U d2}/< " dyi dyj __s_ rt.__= .dig+UZ=:] Ulyllll dz.4.»0 Rz'emannz'an 1l4etrics 329 Moreover, thisyisC°°on(—a,s). This lastfactshows thatify,:[0,s) —>M and1/21(—£,0]—>MareC°°functions satisfying thisequation, andifmoreover 71(0) =I’:(0) dyl +___dl’2 ._ then y,and 1/2together give aC°° function on(—£,£). Naturally, wecould replace 0byanyother I.Wenow have themore precise result, ll.COROLLARY. Apiecewise C°°path y:[a,b] —>Misacritical point forE3ifandonly ifyisactually C°°on[a,b] andforevery coordinate system (x,U)satisﬁes ____ d2yk " dyidyj /< ._-'-a';"2—-l~ E POI" i,j=l PROOF. Letybeacritical point. Choosing thesame as’asbefore (allcxlare0 outside of(1,-_,,r,-)), weseethat y][I,-..;,I,-] satisﬁes theequation, because the ﬁnal term intheﬁrstvariation formula stillvanishes. Now choose orsothat 30: dy _%(0,ff)=A;I-E, I=l,...,N—l. Wealready know thattheintegral intheﬁrstvariation formula vanishes. So weobtain IP13.--""‘-""--PQ. 3xDQ. 5--<--..._______...--0Z— i Pi which implies thatallAh.%are0.Byourprevious remarks, thismeans that y .01isactually C°° onallof[a,b]. +¢ Asthesimplest possible case, consider theEuclidean metric onIR", H (,)=Zdxl®dxi. I'=l Here g,-_;=5,-j,soall3g,-J-/3x" =0,andPf,=0.The critical points yforthe energy function satisfy d2yk Tﬂ=0. 330 caqpei9 Thus yliesalong astraight line,soyisacritical point forthelength function aswell. The situation isnow quite different from theﬁrst variational problem weconsidered, when weconsidered only curves oftheform I|—>(t,f(1')). Any reparameterization ofyisalso acritical point forlength, since length is independent ofparameterization (Problem 16).This shows thatthere arecritical points forlength which deﬁnitely aren’t critical points forenergy, since wehave justseen thatforytobeacritical point forenergy, thecomponent functions ofymust belinear, and hence ymust beparameterized proportionally toarc- length. This situation always prevails. 12.THEOREM. Ify:[a,b] —>Misacritical point forE,then yisparam- eterized proportionally toarclength. PROOF. Observe ﬁrst, from thedeﬁnitions, that 52“ ...."E"?-1'E'lir£' = Now wehave ddy2d" dytdyfZ;d, —E(ijZlglj(y(I))T!T) _ n nagij dy; dy,-dyj H dz}/,. dyj -— Z"5_"_r(l’(*')l dfd,"E,""""l" 8.-*.»'(}’(fl)T,2 7;" 1,,/=1 [=1 r,;=1 " dyi d2yr + gir(Y(f))WTf2 - :,r=l Replacing 3g,-_;/3x] bythevalue given above, thiscanbewritten as ddr2__Hdrj H div’ ".. dyldr’d,d,-2;dt(2;8oU40}dﬁ'+w?€Dh}hYUD?ﬁ"E7)_ r: J‘= ndy; n dz]/r rt __ dyj dy; +ZW(Z 21-rt}/(f))T2 +Z[i1.11<y(1>>»;;--5;).l'=l I‘-1] J',f:] Since yisacritical point forE,both terms inparentheses are0(Corollary 10). Thus thelength lldy/dr 1]isconstant. *1‘ Riernannian Metrics 331 The formula 38:; .. ..(*l 5";=l1/<>Jl+lJ/Ml occurring inthisproof willbeused onseveral occasions later on.Itwillalsobe useful toknow aformula for3g’-i/3x". Toderive one,weﬁrstdifferentiate H Zsimgm‘ =5;’m=I toobtain H H 32”” dgim,,,-Zs1m—i=-Zﬁg ’-m=i m'=l Thus wehave 38” if 38”” Hmj33-‘may,-22 glmayk-gigs2ay, =_Zg”g”’~’l([!k,m] +[mk,I]) by(=|=) l',m =—Zs”1"fi —Zs"“'1".§.i.,I H1‘ OT 38” H:1J 1;z(*=l<) W =—Z(g l"”,+g FM). Il Wecanﬁndtheequations forcritical points ofthelength function Linex- actly thesame way aswetreated theenergy function. Forthemoment we consider only paths y:[a,b] —>Mwith dy/dr ;,é0everywhere. Forthepor- tion yI[t,-_1,r,-] ofycontained inacoordinate system (x,U),wehave r,- " dyrdyj I-(1/l[ri~.si.r=-1):] Zst-.1-(1/(r))— ——-—dr-,,.,,, “=1 drdr Considering ourcoordinate system asR",wearenow dealing with thecase F(x,y) =’Zsi".r(>¢)y"J""- \i..r=1 332 Chapter 9 Weintroduce thearclength function 8(1)=Lit?)- ds dy __ dy 5-"; -r(i»o.;).Then Sowehave "dgi; drldi/I Z"c;r‘*"’”?i? atE U)dy :li"’l=' 3x*'1'’d: 2 Q dz H d r 2gilt)/(I))"—y-~ar,dy__r=l_,J???W l’(),dt — - dz After alittle more calculation weﬁnally obtain theequations foracritical point ofL: H , dzs dz?" "1. di/‘di/" did‘F “E1i'5“+=-,2: PU-(y(l)) drid] —Sdiiids-0. dz Itisclear from thisthat critical points ofEarealsocritical points ofL(since they satisfy dzs/dtz =0).Conversely, given acritical point yforLwith dy/dr =,é0everywhere, thefunction s:[(1.1)]—>t9.1-in/>1 isadiffeomorphism, and wecan consider thereparameterized curve 1/08"‘:t9.L5u/>1» M- This reparameterized curve isautomatically alsoacritical point forL,soit must satisfy thesame differential equation. Since itisnow parameterized by arclength, thethird term vanishes, soyos_1isacritical point forE. There isonly one detail which remains unsettled. Conceivably acritical point forLmight have akink, butbeC°°because ithasazero tangent vector Rz'emam2z'an 1l4e£i"z'c5 333 there, asintheﬁgure below. Inthiscase itwould notbepossible toreparame- terize ybyarclength. Problem 37shows that thissituation cannot arise. Henceforth wewillcallacritical point ofEageodesic onM(fortheRie- mannian metric (,)).This name comes from thescience ofgeodesy, which isconcerned with themeasurement oftheearth’s surface, including surveying andthemeasurement ofdegrees oflatitude andlongitude. Ageodesic onthe earth’s surface isasegment ofagreat circle, which istheshortest path between twopoints. Before wecansaywhether thisistrueforgeodesics ingeneral, which aresofarmerely known tobecritical points forlength, wemust initiate alocal study ofgeodesics. The most elementary properties ofgeodesics depend only onfacts about differential equations. Observe thattheequations forageodesic, dilyk " kdyr dyj “diam +i,iZ=:i P’-"i?1r_'F.7 =0’ have animportant homogeneity property: ifyisageodesic, then I|—>1/(ct) is also clearly ageodesic. This feature oftheequation allows ustoimprove the result given bythebasic existence and uniqueness theorems. 13.THEOREM. LetpGM.Then there isaneighborhood Uofpand a number e>0such thatforevery qGUandevery tangent vector vGMqwith ||v||<8there isaunique geodesic }/1,:(—2,2) —>M satisfying d 141(0)=Q.%<9>=v. PROOF. The fundamental existence anduniqueness theorem saysthatthere isaneighborhood Uofpand 81,6; >0sothat forqGUand vGMqwith Ilvll<stthere isaunique geodesic Yvi(-2!-?2,2~‘-I2) —>M 334 Chapter 9 with therequired initial conditions. Choose .9<81822 Then ifIvl<eand |I|<2wehave Ilv/e2]| <e1and IE-Igfl <222. Sowecandeﬁne y,,(I) tobey,,/,,(s;t). +$* IfveMqisavector forwhich there isageodesic yr[0,1] —>M satisfying d1/1/(0)=9.5(0) =U5 then wedeﬁne theexponential ofvtobe exp(v) =CXpq(l.J) =y(1). (The reason forthisterminology willbeexplained inthenext chapter.) The geodesic ycanthusbedescribed as 1/(I)=@XPq(='v)- Since Mqisann—dimcnsional vector space, there isanatural waytogiveit aC°°structure. If(9CMqisthesetofallvectors vGMqforwhich CXpq(lJ) isdeﬁned, then themap expq: (9—>M isC°°,since thesolutions oftheclifferential equations forgeodesics have aC°° flow. Identifying thetangent space (Mq),_, atvGMqwith Mqitself, wehave an induced map (@XPq)v*I Mq _*M€XPq(U)‘ Iiiparticular, weclaim that themap (expq)o,,: Mq—>Mq istheidentity. Infact, toobtain acuive cinthemanifold Mqwith dc/dI(0) =vGMq= (M,,)0, wecanletC(t‘)=rv.Then expq 9c(z)=expq(rv), thegeodesic with tangent vector vattime 0,so d (@XPq)0*(v) =E0@><Pq(¢‘(F)) =P-I: R2'emam22'az2 1l4e£i'ics" 335 Before proving thenext result, werecall some facts about themanifold TM. If(x,U)isacoordinate system onM,then forqGUwecan express every vector vEMquniquely as H ,3U=.ZCl :=i Wewilldenote aibyi'f(v), sothat _- 3 U=Zx’(v)F,i=1 Irtv) where rt:TM—>Mistheprojection. Then (x1orr,...,x" 07I,)lfl,...,.>l'n)=(Jf'1,...,.7f",.>t1,...,.>lfn) isacoordinate system onrr""l(U). ForveMq,qeUwetherefore have tangent vectors TM 3 3 5;“, G(TM)v> U ll? QM thevectors 3/3J't"|v areallinthetangent space ofthesubmanifold MqCTM, while thevectors 3/39?(Uspan acomplimentary subspace. I4.THEOREM. Forevery pGMthere isaneighborhood Wand anumber e>0such that (l)Any twopoints ofWarejoined byaunique geodesic inMoflength <s. (2)Letv(q,q") denote theunique vector veMqoflength <esuch that expq(v) =q’.Then (q,q") |—>v(q,q") isaC°°function from WxW—> TM. (3)Foreach qeW,themap expq maps theopen e—ball inMqdiffeomor- phically onto anopen setUq3W. 336 Chapter 9 PROOF. Theorem l3saysthatthevector 0eMphasaneighborhood Vinthe manifold TMsuch thatexpisdeﬁned onV.Deﬁne theC°°function F:V—> MxMby F(v)=(Yf(v)»@><P(v))- Let(x,U) beacoordinate system around p.Wewillusethecoordinate system (.t‘,...,.t",.»21,.. .,x"), -1 -thdescribed above, forrt(U). Ifrt,-:M><M—>Misprojection onthe1 factor, then (x1o1r;,...,x" om,x‘ orr;,...,x" orrg) =(x1‘,...,x1”,x21,...,x;”) isacoordinate system onUxU.Now, using thefactthat (expp)o,,.I Mp—>Mp istheidentity, itisnothard toseethatat0eMpwehave (8) 8 8Fir “T 5“_"",T "l"i-3X‘ 0 3X1 (pip) OX2! 3 ts>--it9ax0 axl (pm) Consequently, F,isone-one at0GMp,soFmaps some neighborhood V’ of0diffeomorphically onto some neighborhood of(p,p)GMXM.Wemay assume that V’consists ofallvectors vGMpwith qinsome neighborhood U" ofpand [lull<s.Choose Wtobeasmaller neighborhood ofpforwhich F(V") 3W><W.'2'(rap) Given aWasinthetheorem, andqeW,consider thegeodesics through q oftheform r|—>expq (Iv)for[lull<.9.These ﬁlloutUp.The close analysis of geodesics depends onthefollowing. / {¢><p,.(v) Illvll=9} R2'emannz'an /ldezrics 337 15.LEMMA (GAUSS’ LEMMA). InUp,thegeodesics through qareperpen- dicular tothehypersurfaces {expp(v) :[lull=constant <e}. FIRSTPROOF LetU2IR—>Mpbeasmooth curve with ||v(I)ll =aconstant k<.9forallI,anddeﬁne a(u,;) .-=@xpp(u-u(I)) -1<u<1. Weareclaiming thatforevery such atwehave 3<g—:(u,I), £(u,I)> =0 forall(u,I). Acalculation precisely likethatintheproof ofTheorem 12proves thefollowing equation, inwhich thearguments (u,I)andar(u, I)areomitted, forconvenience: 33a:3a: H3a:-l H 320:’ H__3al3c.r' ‘llnln=El"ga(§g"—ai: +,§l"=1la7nl H30:‘ H 32¢’ H 3021'30:"_ -— '1,‘—-- ."lg311(gg”auai +j,_,Z=:,[’ ’]all31‘) The ﬁrstterm ontheright is0since each curve u|—>ar(u,I) isageodesic. Similarly, weobtain 330:30: H30:5 " 32a’ "__3321'33:" <2)aln=nl"2§n(;g"Tai"*",§,l1’*’lns?)> which isjusttwice thesecond term ontheright of(1).But301/3u(u, I)isjustthe tangent vector attime utothegeodesic u|—>expp(u -v(I)), where ||v(I)|| =-k; so||3cx/3u|| =k.Thus thesecond term ontheright of(2)isalso0.So 3a:3a:<5, isindependent ofu. Buta(0,I) =expp(0) =q,so3a!/3I(O,I) =0.Itfollows that aa=0 forall(Ll,f). 338 Chapter 9 SECOND PROOF. Letv:R—>Mpbeanysmooth curve with ||v(I) ll-=aconstant k<.9forallI,anddeﬁne ,B(u,I) =expp (I-v(u)) (note carefully theroles played byrandu). Then ,6isavariation ofthegeodesic y(I) =expp (I-11(0)), deﬁned on[0,1].By ,1I /'1 ,//K / Z’/’ /// /11» z/9 theﬁrst variation formula, wehave dE(ﬁ(M)) __35 gig; 36 5’); ifdbl “=0” lat/0’1)’ d.=ml_l5(0’0)’ atloll _35 dr“-1)» 'aT(1)>i theintegral vanishing since yisageodesic. Buteach curve ﬂ(u) hasenergy - 2 E(5(u))=fl i§.l.‘.fli"_). d;=fl/(id,-=k1,0 df 0 SO __dE(B(~)), __915 d_i/0‘ dlling): <au(°’1)’di l/'_'\._.\—/0:0 l6.COROLLARY. Letc:[a,b] —>Up—{q}beapiecewise smooth curve, ¢‘(=')=@><Pq(H(I) -11(1)). Rz'emaiIm'an /l4eIrI'cs' 339 for0<u(I) <.9and ||v(I)|| =l.Then Lie2Mb)—~(d)l. with equality ifand only ifIIismonotonic and visconstant, sothat Cisaradial geodesic joining twoconcentric spherical shells around q. PROOF Ifc.r(u, I)=expp(u -v(I)), then c(I)=or(u(I),I) and dc__3a,I 3a dI-3uu()+3I' Since 3a:3o: 30: 5’5l=°’ ln=‘= wehave dcz ,2 33:2 ,2 |p;—|d(=')l—i-'5 ?.l~(=*)l. with equality ifandonly if3a/3I =0,andhence v"(I) =0.Thus bdc b /i —- dI2/P |u'(I)|dI 2|u(b) —u(a)l, 0dl 0 with equality ifand only ifIIismonotonic and visconstant. +I+ l7.COROLLARY. LetWand.9beasinTheorem l5,lety:[0,1]—>Mbe thegeodesic oflength <9joining q,q’ EW,andletc:[0,1]—>Mbeany piecewise C°°path from qtoq’.Then 1-(1/)5I-(¢'). with equality holding ifandonly ifcisareparameterization ofy. PROOF. Wecanassume thatq’-=expp(rv) eUp—{q}(otherwise break cup intosmaller pieces). For5>0,thepath cmust contain asegment which joins thespherical shell ofradius 5tothespherical shell ofradius r,andliesbetween them. ByCorollary 16,thelength ofthissegment haslength 3r—3.Sothe length ofcis3r,andclearly cmust beareparameterization ofyforequality tohold. +I+ 340 Chapter 9 Wethus seethatstﬂicienthi small pieces ofgeodesics areminimal paths forarc- length. WecanuseCorollary 1?todetermine thegeodesics onafewsimple surfaces, without any computations, ifweﬁrst introduce anotion which will playacrucial rolelater. If(M,(,))and(M’,(,)’)areC°°manifolds with Riemannian metrics, then aone-one C°°function f:M—>M’iscalled an isometry ofMinto M’iff"‘( ,)’=(,).Forexample, reﬂection through a plane E2CR"+' isanisometry I:S"—>S".Itisclear thatifc:[0,1]—>M isaC°°curve, then thelength ofcwith respect to(,)isthelength offoc withrespect to(,)’;andifcisageodesic, then f<>cislikewise ageodesic. Fortheisometry I:S"—>S"mentioned above, theﬁxed point setisthegreat circle C=S"('1E2. Letp,q GCbetwopoints with aunique geodesic C’ ofminimal length between them. Then I(C’)isageodesic ofthesame length asC’between 1(p) =pand 1(q) =q.SoC’=1(C’), which implies that C’CC,sothat Cisageodesic. Since there isagreat circle through anypoint ofS"inanygiven direction, these areallthegeodesics. Notice that aportion ofagreat circle which islarger than asemi-circle is definitely notofminimal length, evenamﬂng nearby jiatlzs. Antipodal points on Path Ofsmaller length thesphere have acontinuum ofgeodesics ofminimal length between them. All other pairs ofpoints have aunique geodesic ofminimal length between them butaninﬁnite family ofnon-minimal geodesics, depending onhow many times thegeodesic goes around thesphere andinwhich direction itstarts. Rieiizarznian 1l4eIrI'cs 341 The geodesics onaright circular cylinder Zarethegenerating lines, the © -\ \ 1 \- ._,_-____ .I -—-“ -1-.,- ‘\- I circles cutbyplanes perpendicular tothegenerating lines, andthehelices onZ.Infact, ifLisagenerating lineofZ,then wecansetupanisometry IIZ—L—>R2byrolling Zonto R2.The geodesics onZarejusttheimages ='ii..=.-1: 1-.2:_:3FI-II.--‘3:jg1r.-.1i 111:2}-.; ;-1 =z 5:.-.-F-,=:¢'_'-'.==_'-‘er; ;=::==._:-.=l';._:,:. -'. .-;>.1;:1;: I135‘{‘l‘_:T I?i.’-€.-‘.':-'-“- =-.1 ¢';;2?(lf». -.'I;i-5*; :i§';ii:'-2[ L512-..' ii’.-.-.‘i.-.-‘ i‘;:i'_";{‘.* —I§£i'~T--‘F'=;':1z..Y~. ‘-':-=13.-1'7 -__ ;__. _';_ _—— ' under 1-’ofthestraight lines inIR2. Two points onZhave inﬁnitely many geodesics between them. Wearenow inaposition towind upour discussion ofRiemannian met- ricsonMbyestablishing animportant connection between theRiemannian metric (,)andthemetric dzM><M—>Ritdetermines, d(p,q) =inf{L(y) Iyapiecewise smooth curve from ptoq}. Notice thatonboth thesphere andtheinﬁnite cylinder every geodesic ydeﬁned onaninterval [a,b]canbeextended toageodesic deﬁned onallofR.This is false onacylinder ofbounded height, abounded portion ofIR",orR“—{O}. lngeneral, amanifold Mwith aRiemannian metric (,)iscalled geodesically complete ifevery geodesic y:[a,b] —>Mcanbeextended toageodesic from R toM. 342 Chapter 9 18.THEOREM (HOPF—RINOW-DE RHAM). If(,)isaRiemannian metric onM,then Misgeodesically complete ifandonly ifMiscomplete inthemetric ddetermined by(,).Moreover, anytwopoints inageodesi- cally complete manifold canbejoined byageodesic ofminimal length. PROOF. Suppose Misgeodesically complete. Given p,qeMwith d(p,q)= r>0,choose UpasinTheorem 14.LetSCUpbethespherical shell ofradius 5<.9.There isapoint pg=expptiv, llvll=1 onSsuch that d(po,q) 3d(s,q) forallsGS.Weclaim that (*) @><l>a(rv) =9; thiswillshow thatthegeodesic }/(I) =expp(Iv) isageodesic ofminimal length between pandq.Toprove thisresult, wewillprove that (=t==l=) d()/(I), q)=I‘—I IG[5,r]. First ofall,since every curve from ptoqmust intersect S,weclearly have d(I>.9) =miI1(d(p.s) +d(-9.9))=5+d(po.9)-$65‘ Sod(P0>q) =I‘—5.This proves that(=l=*)holds forI=5. Now letI0G[5,r]betheleast upper bound ofa11Iforwhich (*=t=)holds. Then (**)holds forI0also, bycontinuity. Suppose I0<r.LetS’beaspherical shell ' -pg? IPo S ofradius 3’around }/(Io) and letpg’GS’beapoint closest toq.Thenq. d(}’(f0)»=?) =]1;l§}(d(l’(1’0)>-Y) +d(-Ylq)) =5’"l"d(P0’.'¥). Riemannian Mattias 343 SO (***) d(1>0", Q)=(F—lo)—5"- Hence d(P,110')24(1),?) —d(P0",=?) =10+5’- But thepath Cobtained byfollowing yfrom pto]/(lg) and then theminimal geodesic from 3/(I0) topg’haslength precisely I0+5’. Socisapath ofminimal length, and must therefore beageodesic, which means that itcoincides with y. Hence }’(l0+5')=110'- Hence (=l=>i==l=) gives dti/(Io+6’),q)=r—(I0+5’), showing that(>1-1*)holds forI0+5".This contradicts thechoice ofI0,soitmust bethatI0=I‘.Inother words, (*=|=)holds forI=r,which proves (*). From thisresult, itfollows easily thatMiscomplete with themetric d.In fact, ifACMhasdiameter D,and pGA,then themap expp: Mp —>M maps theclosed discofradius DinMponto acompact setcontaining A.In other words, bounded subsets ofMhave compact closure. From thisitisclear thatCauchy sequences converge. Conversely, suppose Miscomplete asametric space. Given anygeodesic y:(a,b) —>M,choose In—>b.Clearly 1/(In) isaCauchy sequence inM,soit converges tosome point pGM.Using Theorem I4,itisnotdiﬁicult toshow thatycanbeextended pastb.Consequently, byaleast upper bound argument, anygeodesic canbeextended to1R.'1' Asaparticular consequence ofTheorem I8,note thatthere isalways amin- imalgeodesic joining anytwopoints ofacompact manifold. 344 Chapter 9 ADDENDUM TUBULAR NEIGHBORHOODS LetM"CN"+k beasubmanifold ofN,with itM—>Ntheinclusion map, sothatforevery peMwehave i,,,(M,,) CNp.If(,)isaRiemannian metric forN,then wecandeﬁne Mpl CNpas Mpl={vGNp;(v,i',,w) =0forallwGMp}. Let E= UMp-L and wzE—> M take Mpl top. peM Itisnothard toseethat u=217:E—>Misal<—plane bundle over M,the normal bundle ofMinN. Forexample, thenormal bundle uofS"_1 (3IR"isthetrivial 1-plane bundle, forvhasasection consisting ofunit outwa1'd normal vectors. Ontheother hand ifMistheMobius strip andS1CMisacircle around thecenter, then itis nothard toseethatthenormal bundle vwillbeisomorphic tothe(non-trivial) bundle M—>S1. Ifweconsider S1CMCP2,then thenormal bundle ofS‘inP2isexactly thesame asthenormal bundle ofS‘inM,soittoois non-trivial. Riemannian Metrics 345 Our aimistoprove that forcompact Mthenormal bundle ofMinNis always equivalent toabundle If:U—>Mforwhich Uisanopen neighborhood ofMinN,andforwhich the0-section stM->Uisjusttheinclusion ofM into U.Inthecase where Nisthetotal space ofabundle over M,thisopen neighborhood canbetaken tobethewhole total space. Butingeneral the neighborhood cannot beallofN.Forexample, asanappropriate neighborhood ofS‘CR2wecanchoose R2—{O}. ~.. ,/// , l/I ‘-,\\ \\\\ “M. \\ \ \ / Abundle rr:U—>Mwith Uanopen neighborhood ofMinN,forwhich the 0—secti0n s:M—>Uistheinclusion ofMinU,iscalled atubular neighborhood ofMinN.Before proving theexistence oftubular neighborhoods, weaddsome remarks and aLemma. Ifrt:U—>Misatubular neighborhood, then clearly rros=identity ofM, sorr issmoothly homotopic totheidentity ofU, soIfisadeformation retraction, and Hi‘(U) RHk(M); thus Mhasthesame deRham cohomology asanopen neighborhood. Moreover, ifwechoose a Riemannian metric (,)forIIIU—>Manddeﬁne D={eGU:(e,e) 51}, then Disasubmanifold-with-boundary ofU,and themap IIIDI D—>Mis alsoadeformation retraction. SoMalsohasthesame deRham cohomology asaclosed neighborhood. 19.LEMMA. LetXbeacompact metric space andXoCXaclosed subset. Letf:X—>Ybealocal homeomorphism such thatfIXOisone-one. Then there isaneighborhood UofXosuch that f|Uisone-one. 346 Chapter 9 PROOFI LetCCX>< Xbe {(»\'>y)eX><X=x¢yandftxi=f(y)}- Then Cisclosed, forif(x,,,y,,) isasequence inCwith x8,—>xand yn—>y, then f(x) =limf(X,8) =limf(y,,) =f(y), and also X75ysince fislocally one-one. IfgrC->Risg(x,y) =d(x,X0) +d(y,Xo), then g>0onC.Since C iscompact, there ise>0such that g32.9onC.Then fisone-one onthe s-neighborhood ofXo.‘I’ 20.THEOREM. LetMCNbeacompact submanifold ofN.Then Mhas atubular neighborhood IIIU—>MinN,which isequivalent tothenormal bundle ofMinN. PROOF. Choose aRiemannian metric (,)forN,with thecorresponding norm |]I],andmetric dzNxN—>lR.Let E={v:veN8 andvGM,,J',forsomepeM} E8={veE:|Iv][<s} U8={qeN:d(q,M) <5}. Itfollows easily from Theorem 13,andcompactness ofM,that expisdeﬁned onE8forsufficiently small .9>0.Weclaim that forsufiiciently small 8,themap expisadiffeomorphism from E8onto U8.This willclearly prove thetheorem. LetVCEbethesetofanon-critical points forexp. Then V3M(consid- ered asasubset ofEviathe0-section), andV1=VF1E1iscompact; since exp isone~one onM(IV1,itfollows from Lemma 19that forsuﬁiciently small .9 themap expisadiffeomorphism onE8. Itisclear alsothatexp(E8) CU8.Toprove thatexpisonto U8,choose any qeU8,andapoint pGMclosest toq.Ify:[0,1] —>Nisthegeodesic of length <.9with }/(0) =pand }/(1) =q,itiseasy toseethat yisperpendicular toMatp(compare thesecond proof ofGauss’ Lemma). This means that q=exppdy/dr(0) where dy/dt(0) EE8.'2» One oftheinteresting features ofTheorem 20isthat alltheparaphernalia ofRiemannian metrics andgeodesics areused initsproof, while they donot even appear inthestatement. Theorem 20willbeneeded onlyinChapter ll, where wewillalsoneed thefollowing modiﬁcation. Rienzarznian 1l4ezrics 347 21.THEOREM. LetNbeamanifold-with-boundary, with compact bound- ary3N. Then 3Nhas(arbitrarily small) open [and closed] neighborhoods for which there aredeformation retractions onto 3N. PROOF. Exactly thesame astheproof ofTheorem 20,using only inward point- ingnormal vectors. *1‘ _\ "-’.\\ r,f:».1I’;, ////¢\ \\\ 4. .<si‘*‘§»" \:§~t‘.‘.‘.‘§I‘” \\til:153733;" %\\ /E//ta 348 Chapter 9 PROBLEMS 1.LetVbeavector space over aﬁeld Fofcharacteristic %2,andlethiV>< V—>Fbesymmetric and bilinear. (a)Deﬁne q:V—>Fbyq(v) =h(v,v). Show that if¢1,...,q5,, isabasis forV"‘,then q= §(1o'v*t-v*; 1..t=1forsome a,-_;. (b)Show that q(—v)=q(v) ho.v)=élqw+v)—Q01)—q(v)]- (c)Suppose q:V—>Fsatisﬁes q(—v) =v,andthatli(u, v)=q(u+v)—q(u)— q(v) isbilinear. Show that qtu+v+w)—r1(u)—q{v+w)= qtu+v)—q(u)—q(v)—q(u+w)—q(u)—q(w)- Conclude that q(0) =0,andq(2u) =4q(u). Then show that q(v) =h(v, v). 2.Let (,)beaEuclidean metric forV*. Suppose Q58-,tlr; GV*satisfy Q51A A(pk=tlr;A--- Atlrk750,and letW8)and Wu’,bethesubspaces ofV* spanned bytheqb;andti/8-. (a)Show thatweW8;ifandonly ifwAQ5;A~--A415;,=0.Conclude that W8;=W,;,. (b)Leto;,...,o;8 beanorthonormal basis ofW8)=W,;,. If¢,-=Z,a,-8-er,-, show that thesigned k-dimensional volume oftheparallelepiped spanned by ¢1,...,¢;<isdet(a,-J-). (The signis+if¢;,...,(pkhasthesame orientation as 01,...,o;,, and—otherwise.) (c)Using Problem 7-9,show that thisvolume isthesame fortlr1,.. .,tlrk. (d)Conversely, ifW¢=W,;,, and thesigned volumes oftheparallelepipeds are thesame, show that¢]A Aqﬁk =‘$1/\---A tlrk. Ifweidentify Vwith V""",sothatwehave awedge product v1A---A018 ofvectors v,-eV,then wehave ageometric condition forequality with wlA Awk. InLegons surlaGéoméirie desEs/it.-ces deRiernuim, E.Cartan uses thiscondition to dgfine S2,‘(V*) asformal sums ofequivalence classes ofkvectors; hededuces geometrically thecorresponding conditions onthecoordinates ofv8-,w,-. 3.LetVbeann—dimensional vector space, and (,)aninner product onV which isnotnecessarily positive deﬁnite. Abasis v1,...,v,., forViscalled orthonormal if(v,-,vj)=:l:¢i,-J,-. Riem annian /l4em'cs 349 (a)IfVsé{O},then there isavector vGVwith (v,v)#0. (b)ForW<:V,letwl={UGv1(v,w)=OforallwGW}.Provethat dimWl 3n—dimW.Hint? If{wg} isabasis forW,consider thelinear functionals A,-:V—>Rdeﬁned byA,-(v) =(v,w,-). (c)If(a)isnon-degenerate onW,then V=WEBW1-, and (,)isalso non-degenerate onl/VJ". (d)Vhasanorthonormal basis. Thus, there isanisomorphism f1R"—> Vwith f*( ,)=(,)8forsome 2'(the inner product (,)8isdeﬁned on page 30l). (e)The index of(,)isthelargest dimension ofasubspace WCVsuch that (,)|Wisnegative deﬁnite. Show thattheindex isn—r,thusshowing thatr isunique (“Sylvester‘s Law ofInertia”). 4.Let(,)bea(possibly non-positive deﬁnite) inner product onV,andlet vi,...,v8beanorthonormal basis (seeProblem 3).Deﬁne aninner product (.)"ons.2’<(v) byrequiring that U*,']/\-~-/\U*gk l§!'1<---<I'k§ﬂ beanorthonormal basis, with It(v*,-, /\---A11*,-,‘,, 11*,-1A---Av*,-,8) =det((v8-M 153)). ()Show that (,)"isindependent ofthe basis v1,...,v;8. (Use Problem 7-l6.) (b)Show thatN (Q51/\ /\¢1<=1l’1/\'" /\Wtlk =d@Y((¢t'=ll'jl*) =<1l@i((¢‘t', Will)- (c)If(,)hasindex i,then (U*1A---/\v*,,,U*1A---/\U*,,)" =(-1)‘. (d)Forthose who know about ®andAl‘. Using theisomorphisms 8)!‘V*"v (®k V)*andA"(V*) %(A"V)*, deﬁne inner products on®kV andA"V by using theisomorphism V—>V*given bytheinner product onV.Show that these inner products agree with theones deﬁned above. 5.Recall thedeﬁnition ofv;><---><v,,_; inProblem 7-26. (a)Show that (vi>< ><v,,_;,v,-) =0. (b)Show thatIv;><---><v,,_1| =s/det(g,-j), where g,-J;=(v,-,v,-). Hint: Apply theresult onpage 308toacertain (n—1)-dimensional subspace ofR“. 350 Chapter 9 6.LetE=rrtE—>Bbeavector bundle. Anindeﬁnite metric on§is acontinuous choice ofanon-positive deﬁnite inner product (,)poneach IF‘(p).Show thattheindex of(,)pisconstant oneach component ofB. 7.This problem requires alittle knowledge ofsimple~connectedness andcov- ering spaces. (a)There isnoway ofcontinuously choosing a1-dimensional subspace ofSip, foreach pGS2.(Consider thespace consisting ofthetwounitvectors ineach subspace.) (b)There isnoRiemannian metric ofindex 1onS2. 8.Let(,)and(,)'betwoRiemannian metrics onavector bundle E= rt:E—>B.LetSbethesetofeeEwith (e,e) =1,anddeﬁne S’similarly. Show thatSishomeomorphic toS’.IfEisasmooth bundle overamanifold M, show thatSisdiﬂeomorphic toS’. 9.Show byacomputation thatifthefunctions g;_;andg";_;arerelated by iixlEixi Stag =ggijmﬁt 1.1 with det(g;_;) 790,and thefunctions gl-l,g”'»" aredeﬁned by H H Z9”‘9t,t =5}. Z9’”‘9’t; =5}. Ifi /<1 thenmt .r_B-Bx 3x gm”=Zg"a—.vw~i..»" This, ofcourse, istheclassical wayofdeﬁning thetensor [having thecompo- nents]git’. 10.(a)Let(,)beaRiemannian metric onM,andAatensor oftype so that A(p): Mp—>Mp. Deﬁne atensor Boftype by B(Pl(UhU2l =(/1(P)(vt),v2)- Iftheexpression forAinacoordinate system is H ., 3_ 1.______ A—..Z;Ai dlr®3x-l’I..I= Rz'emam2z'a22 1l4e£i*ics 351 show that B=Z,-J, B,-1,dxl®dxk, where I7 But=Z/1;’git- i=* (b)Similarly, deﬁne atensor Coftype by C(Pl(l1>7\2l =(A(Pl*(5l1)=5\2l- Show thatifChascomponents CH, then Fl‘ 1"---l The tensors BandCaresaid tobeobtained from Aby“raising andlowering indices”. 11.(a)LetX1,...,X,, belinearly independent vector ﬁelds onamanifold M with aRiemannian metric (,).Show that theGram-Schmidt process canbe applied tothevector ﬁelds allatonce, sothatweobtain neverywhere orthonor- malvector ﬁelds Y1,...,Yn- (b)Forthecase ofanon-positive deﬁnite metric, ﬁndY1,...,Y"with (Y,-,I’)-)= :l:ti;j inaneighborhood ofanypoint. 12.(a)IffI[a,b] —>IRispositive, show that thearea ofthesurface obtained byrevolving thegraph offaround thex-axis is [ab2rrf,/1+(f")2. (b)Compute thearea ofS2. 13.LetMCR"bean(n—1)-dimensional submanifold with orientation ,t_1,. The outward unitnormal v(p)atpGMisdeﬁned tobethat vector inIR"p oflength 1such thatv(p), (v;)p, ...,(v,,_;)p ispositively oriented inR"pwhen (v1)p, ...,(v,,_1)p ispositively oriented inMp. (a)IfM=3Nforann-dimensional manifold-with-boundary NCR",then v(p) isoutward pointing inthesense ofChapter 8. (b)LetdV,,_; bethevolume element ofMdetermined bytheRiemannian metric itacquires asasubmanifold ofR".Show thatifwe consider v(p) asan element ofIR",then “(Pl di/n_1(p)((U1)p, ...,(IJ;;_1)p) =Cif2E( L11 ). 7-in-1 352 Chapter" 9 Conclude that dV,,_| (p)istherestriction toMpof Zt-1)‘-'v"(p)d>t'(p) /\A /\---Adx"(p). t'=| (e)Note that U]>< ><v,,_| =cxv(p) forsome orGIR(byProblem 5).Show thatforwGR”wehave (w,v(p)) .(vt>< ><v,,_;,v(p)) =(w,v| >< xv,,_|). Conclude that vi(p) -dVtt-1(Pl '-=restriction toMpof (--l)l_ldXl(p) A--- Adx"(p) A--- Adx”(p). (d)LetMCIR"beacompact n-dimensional manifold-with-boundary, with v theoutward unit normal onBM. Denote thevolume element ofMbydV,,, andthat of3MbyzlV,,..;. LetX=Z,a"8/Bx‘ beavector ﬁeld onM.Prove theDivergence Theorem: g/idivXdVn =[ (X,1J)dV;t-| M 3M (thefunction divXisdeﬁned inProblem 7-27). Hint: Consider theform w onMdeﬁned by H w=Z(-1)’ la'dx' A---Adx‘A---Adx". rl (e)LetMCR3beacompact 2—dimensional manifold-with-boundary, with orientation pt,and outward unit normal v.LetTbethevector ﬁeld onBM consisting ofpositively oriented unitvectors. Denote thevolume element ofM bydA,andthatof8Mbyds.LetXbeavector ﬁeldonM.Prove (theoriginal) Stokes’ Theorem: f(V>< X,v)dA=/ (X,T)ds M BM (V><Xisdeﬁned inProblem 7-27). 14.(a)Let V"bethevolume oftheunit ball inlR". Show that I V,,=I(1-.t—2)‘"")/21/,,-|t1>t-. — ---1 Riemannian 1l4e£rics 353 I (b)If1,,=f(1-x2)‘"-ll/2 dx,showthat-1 -11,,=5’-1,,_2.it (c)Using V1=2,V2=IT,show that Hr:/2 -i Heven V_tn/2)!,,._ 2(n-H)/2 (n—l)/2 —--i—-—1 3Z H nodd. 7rn)'2 r(1+n/2)‘l (d)LetA,,_| bethe(n-—-1)—volume ofS"-1. Using themethod ofproof in Corollary 8-8,butreversing theorder ofintegration, show that(Interms ofthe1"function, thiscanbewritten ' 1V”=f r"_'A,,_1dr =--A,,_|. 0 1'? (e)Obtain thissame result byapplying theDivergence Theorem (Problem 13), 15.(a)Letct[0,1] —>R"beadifferentiable curve, where IR"hastheusual Riemannian metric (,)=Z,-dxf ®dxi. Show that I H : 11' 2I-(9)fIf[tn(1)1dr-O i=l (b)Forthespecial case c‘:[0,1]—>R2given byc(r)=(I,f(1')), show that this length, I f./1+1r'm12 tit.0 istheleast upper bound ofthelengths ofinscribed polygonal curves. Hint: Iftheinscribed polygonal curve isdetermined bythepoints (r,-,c(r,-)) for 354 Chapter 9 apartition 0=10 < <In=1of[0,1], then wehave l¢'(l'r‘)""CU:-1)| =\/(F:-fs—1)2 +(fllil ""f(fi-1))2 =\/(rt-Qtfr.--1)2+f*(t}5>(r,--1,--1? forsome E;G[r,-_.|,r,-]. (c)Prove thesame result inthegeneral case. Hint; Use theresults ofProb- lem8-1,anduniform continuity ofV onacompact set. Itisnatural tosuppose that thearea ofasurface is,similarly, theleast upper bound oftheareas ofinscribed polygonal surfaces, butasH.Schwarz ﬁrst observed, thisleast upper bound isinﬁnite forabounded portion ofacylinder! Toillustrate Schwarz’s example Ihave plagiarized thefollowing picture from a book called Matrurmatnuuectmii Aucuius HaMuaeooﬁpasunx, written bysomeone called M.Cnmsax. Q if ill/l‘7l7l7lll7Top view Toincrease thenumber oftriangles, wemaintain thehexagonal arrangement, bnlmove theplanes ofthehexagons closer together, sothatthetriangles aremore nearly inaplane parallel tothebases oftheCylinder. Inthisway, wecanincrease thenumber oftriangles indeﬁnitely, while thearea ofeach approaches 111/2. The topic ofsurlace area fornon-clifferentiable surfaces isacomplex one, which wewillnotgointohere. Riemannian /l4eti"z'cs 355 16.Letct[0,l]—>Mbeacurve inamanifold Mwith aRiemannian metric (,).Ifp:[0,l]—>[0,l]isadiﬂeomorphism, show that I-(C)=I-(6OP)- 17.Show that themetric donMmay bedeﬁned using C°°, instead ofpiece- wise C°°curves. (Show how toround olTcorners ofapiecewise C°°path sothat thelength increases bylessthan anygiven 8>0;remember that theformula forlength involves only ﬁrstderivatives.) 18.(a)LetBCMbehomeomorphic totheball{pGIR":lpl51}andlet SCMbethesubset corresponding to{pGIR"I[pl=1}.Show that M-—-S isdisconnected, byshowing that M-—-BandB-—-Saredisjoint open subsets of M-—-S. (b)IfpGB-Sand qGM—B,show that d(p,q) 2,miI§d(p,q"). Use q’E thisfactandLemma 7"tocomplete theproof ofTheorem 7.(Inthetheory of inﬁnite dimensional manifolds, these details become quite important, forM-S cloes nothave tobedisconnected, andTheorem 7isfalse.) 19.(a)Byapplying integration byparts totheequation onpages 318-319, show that 41* ba2 8F , 2 0zf I)li$('f'.\ ./_(r)af u= I8i -§<:,r<1>,r'<:>>dr] dz; thisresult makes sense even iffisonly CI. (b)DuB021;Rqymondlv Lemma. Ifacontinuous function gon[a,b] satisﬁes b In’(r)g(r)dr =0 forallC°° functions 1]on[a,b] with r](a) =r](b) =0,then gisaconstant. Hint: The constant cmust be 1 b G Weclearly have b fn"(r)[g(r) -c1dr=0, soweneed toﬁnd asuitable 1]with r;’(r) =g(r)-—-c. (c)Conclude thatif theC'function fisacriticalpointof J,then fstillsatisﬁes theEuler equations (which arenotafm'0rz' meaningful iffisnotC2). 356 Chapter 9 20.The hyperbolic sine, hyperbolic cosine, and hyperbolic tangent functions sinh, cosh, andtanh aredeﬁned by _ ex-e“" ex+e"" sinhxs1nhx=i-, coshx=—---—, tanhx=i. 2 2 cosh X (a)Graph sinh, cosh, and tanh. (b)Show that cosh2- sinhz=1 tanhz +1/coshz =1 sinh(x +y)=sinhxcosh y+coshx sinhy cosh(x +y)=coshx cosh y+sinhxsinhy sinh’ =cosh cosh’ =sinh. (c)Forthose who l-tnow about complex power series: sinhx =E, cosh x=cosix. (d)The inverse functions ofsinh and tanh aredenoted bySlI1h_' and tanh_', respectively, while cosh“ denotes theinverse ofcosh |[0,00). Show that SlI1h(COSh—IX) =Vx2._1 (sinh—l)!(x) :# cosh(sinh"" I)=V+'2 ‘ 2 _ 1 (cosh"')"(x) : .cosh(tanh 'x)=L F/1__xg X--'l5' l’\J>—ll!-_-.>1 21.Consider theproblem ofﬁnding asurface ofrevolution joining twocircles ofradius l,situated, forconvenience, ataand --a. Wearelool-ting forafunction 1\ l --I"\ -u arrr| ~-_rW-r1rr \3 --....-L-rrrr|r rm Riemann tanMetrics 357 oftheform x f(x) =c‘cosh -E where cissupposed tosatisfy CCosh€-=l (c>0). (a)There isaunique yo>0with tanh yo=1/yo. Examine thesign of1/y-- tanhy fory>0. (b)Examine thesign ofcoshy -—-jisinhy fory:>0. (c)Let Aa(c) =ccosh g c>0. Show that A6hasaminimum ata/yo, ﬁnd thevalue ofA0there, andsketch thegraph, (d)There exists cwith ccosha/c =1ifandonly ifa5y0/cosh yo.Ifa= yofcosh yo,then there isaunique such c,namely c=a/yo =1/cosh yo.If at<y0/cosh yo,then there aretwo such c,with c|<a/yo <('2.Itturns out that thesurface for('2hassmaller area. ts —————-—__~_____ l1 _ :| 1_____W --yo/ cosh yo -0 l ayq/cosh yo l (e)Using Problem 20(d), show that -——J-)2-—- =Vyoz -—1. coshyo [yo~1.2,so~/yoz~I~.67.] 358 C/rapier 9 [These phenomena can bepictured more easily ifweusethenotion ofan envelope—c.f Volume III,pp.l76f’C The envelope ofthel—parameter family ofcurves X _/;(x) =ecosh -E isdetermined bysolving theequations 3jI,(x) x x x0:i =cosh-—-—--sinh-—.Ele e t‘ e W'eobtain _ _A Jt-—=ztyo, y=ecosh -—-=ecosh yo,e e sotheenvelope consists ofthestraight lines cosh yo y=IE-—----——x. yo The unique member ofthefamily through (y0/ cosh yo,l)istangent tothe envelope atthat point. Fora<y0/cosh yo,thegraph offeristangent to \‘ ” ‘ 4‘ 1“ . 1\ (I I: \ \\ - ,1 \..\\}\ \ ll 5/ \ / ‘lg _ 1 _ [71.A/[I. I.“-11,|F3 "‘>NI.t Ti.‘\‘\‘x \_ 1 “‘ -~. 1’K ‘“‘-____ '1 ‘ . I ‘\ '""‘<-..._____ ______,-// 1 .______H__- ‘--F'____- \ T“"'-—-_. 4-—-"”"T' 1\\ ~ ’ \ . ,/ \ \‘*-1 ._--/ ‘ 4‘ 1‘:\- ____. 1 ___ _ _ ..,.I ‘ .._L| _ 1'‘t Q I O5] :Cosh yo I”I \\\ C 1J0 y=—i—>.’. .\yo Ir \\ theenvelope atpoints P,QG(-—a,a), butthegraph ofﬁgistangent tothe envelope atpoints outside [-—a,a]_ The point Qiscalled conjugate toPalong theextremal fq,anditisshown inthecalculus ofvariations thattheexistence ofthisconjugate point implies that theportion ofj},from Pto(a,1)does no! Riemannian 1l4e£ii'c.s 359 'vealocal minimum for fx/1 +(f")2. Com arewith thediscussion ofBl P conjugate points ofageodesic inVolume IV,Chapter 8,andnote theremark onpg.V.396.)] 22.Allofourillustrations ofcalculus ofvariations problems involved anF which does notinvolve I,sothat theEuler equations areactually gum. mm~%(gt/<:>.r*<:>>) =0- (a)Show thatforanyfand F:R2—>Rwehave d ,BF ,BF dBF E(‘Hf ay)"flBxWatayl= andconclude thattheextremals forourproblem satisfy ,BFF-fa--0. (b)Apply thistoF(x, y)=XV1+yztoobtain directly theequation dy/dx = vcyz-—-Iwhich weeventually obtained inoursolution totheproblem. 23. (a)Letxand X’betwocoordinate systems, with corresponding g,-_;andg’,-_; fortheexpression ofaRiemannian metric. Show that ago,_iEBx,‘Bx‘Bxi Bx’? T Bxl‘Bx’? Bx"°‘Bx"5i,j,/(".::l n_Bxl Bzxj Bx} Bzxl +.218“ Bx'°‘ Bx"l5Bx"1; +Bxtﬁ Bx’°‘Bx’1; ':,_i--=-- (la)Forthecorresponding [z'j,k]and [ur,B,y]",show that ,, .. ,,_, __an3.1‘-la>.~’< axawlaﬂ.}’l=M‘[11,/<1ax,,7,,T,+_Z]T,i-ax,,,x,,,.1,], ..-:. I,j= sothat [ij,k] arenotthecomponents ofatensor. (clAlso show that N ‘ ‘ Naia_;a.ry a2Iary Figs=ZPiiii,si,@Li "l" Ukm Bx Bx Bx IHIBxBx Bx 24.Show thatanyC°°structure onRisdiffeomorphic totheusual C°°st1'uc— ture. (Consider thearclength function onageodesic forsome Riemannian metric onIR.) 360 Chapter 9 25.Let(,)=Z,-dxi ®dxl betheusual Riemannian metric onR",and let2:,-J g,-_;du" ®du-l beanother metric, where u',...,u“ again denotes the standard coordinate system onIR". Suppose wearetold that there isadiffeo- morphism f:R“—>IR"such that Z,-J g,-_;dui®du-l =f*( ,).How canwe goabout ﬁnding fP (a)LetBf/Bu? =e,-:IR"—>IR".Ifweconsider e,-asavector ﬁeld onIR",show that f,.(B/Bu") =e,-. (b)Show that g,-_;:(e,-,e_;). (c)Tosolve forfitis,intheory atleast, suﬁieient tosolve forthee,~,andto solve forthese wewant toﬁnd differential equations Be- H I an] 1% ljef satisﬁed bythee,-’s.Show thatwemust have H H 2 I I def 3f 3f BM=Zg’"’*‘i1 =Z€.nT'51F' ?'=l l'=l (d)Show that agij Wt 32f! 82f! Buk I,2‘BuiBu" +Bu-lBu" =ZgjrA€k "l"gi'rAr];_; r-=1 (e)Bycyclically permuting 1',j,k,deduce that Bij,k =lij,/fl, sothat AB=1";-'1. InLepons surlaGéornétrie desEtjmces deRiemann, Cartan uses thisapproach tomotivate theintroduction oftheP5. (f)Deduce theresult Afj=1";-3directly from ourequations forageodesic. (Note thatthecurves obtained bysetting allbutoneficonstant aregeodesics, since they correspond tolines parallel tothex‘—a.xis.) Rz'emannz'an Metrics 361 26.If(V",( ,)")and (V”,( ,)”)aretwovector spaces with inner products, we<1l@ﬁn@( ,lonV=I/'69V”by (Ur69Unawl69wif) =(vr,w:)r _+_(UN, wn)n' (a)Show that (,)isaninner product. (b)Given Riemannian metrics onMand N,itfollows that there isanatural way toputaRiemannian metric onMxN.Describe thegeodesics onM><N forthismetric. 27.(a)Lety:[a,b] —>Mbeageodesic, and letp:[ur,,B] —>[a,b] bea diffeomorphism. Show that e=y0psatisﬁes dzcl‘ H defdc-ll dcl‘p”(r) —+ 1".-"(tr ——=—-—-drz ig ICDd: dz dzp’(r) (b)Conversely, ifesatisﬁes thisequation, then yisageodesic. (c)Ifesatisfies dzek H deidc-ii dek Wﬁg! l"5(c'(t))IW =Tito) forjJ,IIR->IR, then eisareparameterization ofageodesic. (The equation p”(r) =p'(t),ri(t) canbesolved explicitly: p(r) =freM(‘l ds,where M’(.s')=;.t(s) 28.Letebeacurve inMwith dc/dr aé0everywhere, andconsider thehy- persurfaces {expcwv :||v||=constant, where vGMC“) with (v,de/dz) =0}. Show that forvGMC“) with (v,dc'/dr) =0,thegeodesic u|—>expcwu -v isperpendicular tothese hypersurfaces. (Gauss’ Lemma isthe“special case” where eisconstant.) *t"1*=-e 362 Chapter 9 29.Lety:[61,19] —>Mbeageodesic with 1/(ct) =p,and suppose that expp is adiffeomorphism onaneighborhood (9CMpof{1}/(0) I05r51}.Show that yisacurve ofminimal length between pandq=3/(b), among allcurves inexp((9). (Gauss’ Lemma stillworks onexp((9).) 30.If(,)isaRiemannian metric onMandd:M><M—>Risthecorre- sponding metric, then acurve y:[61,19] —>Mwith d()/(ct), y(b)) =L(y) isa geodesic. 31. Sclzwarzﬁ" inequality forcontinuous functions states that (M-it/.”~)(/.‘n»with equality ifand only if and garelinearly dependent (over IR). (a)Prove Schwarz‘s inequality byimitating theproof ofTheorem 1(2). (b)Foranycurve yshow that [1-Ztnr5tb~awﬁtn. with equality ifand only ifyisparameterized proportionally toarclength. (c)Letyi[a,b] —>Mbeageodesic with L3(y) =d(y(a),y(b)). Ife(a) = }/(a) ande(b) =}/(b), show that L2L2E(r)=%s3%sE(v)- Conclude that E(y)<E(C)unless eisalso ageodesic with Lﬁtv)=dtc<a).ctb>>- Inparticular, suﬁiciently small pieces ofageodesic minimize energy. 32.Letpbeapoint ofamanifold Mwith aRiemannian metric (,).Choose abasis v|,...,v,,ofMp, sothat wehave a“rectangular” coordinate system X onMpgiven byZ,aiv; |—>(al,...,ct”);letxbethecoordinate system X0exp_'. deﬁned inaneighborhood Uofp. (a)Show that inthis coordinate system wehave F5(p)=0.Hint: Recall theequations forageodesic, andnote that ageodesic ythrough pisjust exp composed with astraight linethrough OinM,,,sothateach ykislinear. Riemannian /l4e£i'ics 363 (b)Let I‘!U—>Rbel‘(q) =d(p,q), S0that I‘0y=zk(}/‘)2, Show that d2((rOi/)2) di/"2 2k<11/"dr’"VT dlxiwl "Z1’‘"~TiTil~ Ir r,j,k (c)Note that 2drldrj 2(drk Z7-F7?5"Z7;)- 1,] k Using part (a),conclude that ifll}/(0) IIissuﬁiciently small, then d%r9rV —TE?_*>0’ sothat d(r01/)2/dr isstrictly increasing inaneighborhood of0. (d)LetBS={vGMp: ||v||;<_s}andS,={vGMp: ||v||=s}.Show that thefollowing istrueforallsuﬁiciently small s>0:ifyisageodesic such that }/(0) Gexp(S,,-) andsuch that }/(0) istangent toexp(S£), then there is6:>0 (depending ony)such that y(r) 92e>.;p(B_,) for075IG(-5,5). Hint: If}/(0) is s.==\ ‘ tangent toexp(S,;), then d(r0y)/dr =0. (e)Letqandq’betwopoints with r(q),r(q") <sandletybetheunique geodesic oflength <2sjoining them. Show that forsuﬁiciently small sthe maximum of2'0yoccurs ateither qorq’. (f)AsetUCMisgeodesically convex ifevery pair q,q" GUhasaunique geodesic ofminimum length between them, and thisgeodesic liescompletely inU.Show thatexp({v GMpI||v||<s})isgeodesically convex forsuﬁiiciently small s>0. (g)Let f:U—>IR"beadilcfeomorphism ofaneighborhood UofOGIR" into lit”. Show that forsuﬁiciently small .9,theimage oftheopen s-ball is convex. 364 Chapter 9 33. (a)There isaneverywhere differentiable curve c'(z') =(I,f(1'))inR2such that_length ofel [0,h]1L--_- 1.til-‘l11cth>-(1911BHint: Make elook something likethefollowing picture. l "length from (go)to(1,0)is1 .\ _-' length from ,r ‘(J-,,0) to(§,0)is-'5a’ ‘- ' ll ~\ 1 1 1 1' ‘~ \ 2I I \ I \ \ \ (b)Consider thesituation inCorollary 15,except that c(z) =Qifandonly if 1'=a,andsuppose u’(z) >0forznear 0.IfeisC‘,then v(z) approaches a limit ast—>0(even though v(0) isundeﬁned). Show that ifcisC‘,then there issome K>0such thatforall1'near 0wehave %—?(u,r) glfu ?)—c:(l,z)| 05:151. Hint: InMpweclearly have _-|p,|. dz T dz ' Since expq islocally adiffeomorphism there are0<K]<K2such that K1llvll5ll@XPq»<vll SKzllvll foralltangent vectors vatpoints near q. (c)Conclude that "“/ Ba 1z 2 i 3 dHm“CH0, 11])=HmA u(I)+at(u(r)-’) z1-1‘ it->0 d(p, c(h)) h—»-0 u(h) Rz'einaizm'an 1l/Ietrics 365 (d)IfcisCI,show that L(c) istheleast upper bound ofinscribed piecewise geodesic curves. 34.(a)Using themethods ofProblem 33,show that ifcisthestraight line joining v,wGMp, then limLexi’ °C)=1.v,w—>0 (b)Similarly, ify,,,w istheunique geodesic joining exp(v) and exp(w), and J/-;_;,'w --: EXP OC-U,w., thﬁn Lil/v,wlim )=l.v,w—>0 L(Cv,-w) (c)Conclude that Hm =,_v-w—>9 llv—wll 35-Letf:M—>Nbeanisometry. Show that fisanisometry ofthemet- ricspace structures determined onMand Nbytheir respective Riemannian metrics. 36.LetMbeamanifold with Riemannian metric (,)and corresponding metric d.LetftM—>Mbeamap ofMonto itself which preserves the metric d. (a)Ifyisageodesic, then f0yisageodesic. (b)Deﬁne f’:Mp—>Mﬂp) asfollows: Foryageodesic with 1/(0) =p,let ﬁti/(9)) =‘”+),@ -{=0 Show that llf"(X) =ll/Yll, and that f’(cX) =cf"(X). (c)Given X,YGMp, useProblem 34toShow that 2<X.Y> 1X12+111/1211:X—zY|12llXll-llYll_ |lXl|-llYll IllIXll-IIIYII IIXH2-1-IIYII2 .[d(9><11=*X, <=><1>IY)]2 llXll'llYll I—*° llIXll'llIYll 366 Chapter 9 Conclude that (X,Y)p =(f'(X),f"(Y))f(p), and then that f’(X +Y)= f"(X) +f'(Y)- (d)Part (c)shows that f’:Mp—>Mjqp,isadiffeomorphism. Usethistoshow that fisitself adiffeomorphism, andhence anisometiy. 37.(a)Forv,wGlit"with waé0,show that .llv+rwlI—Ilvll (v.w)lim P—:1» r—»-9 r llvll The same result then holds inanyvector space with aEuclidean metric (,). Hint: Ifv:IR"—>IRisthenorm, then thelimit isDv(v)(w). Alternately, one can usetheequation (u,v)=llull- -cost? where Bistheangle between u andv. (b)Conclude that ifwislinearly independent ofv,then U) .l|v+fwl|-llvll-llfwllhm as0. v{-2-O I (c)Lety:[0,1]—>Mbeapiecewise C'critical point forlength, andsuppose that }/’(zo+) 56y’(t0_) forsome toG(0,1). Choose z|<toandconsider the variation ozfo1'which tir(u) isobtained byfollowing yuptoI1,then theunique geodesic from y(I|) toJ/(Io +u),ancl ﬁnally therestofy.Show that ifz|is }’(1) }/(to+zt) V00) YU1) J/(9) close enough toto,then dL(5z(u))/du|N=0 qé0,acontradiction. Thus, critical paths forlength cannot have kinks. Itiianzaiziiiaiz 1l4etric.s' 367 33-Consider acylinder ZCR3ofradius r.Find themetric dinduced bythe Riemannian metric itacquires asasubset ofR3. 39. Consider acone C(without thevertex), and letLbeagenerating line, Unfolding C—Lonto R2produces amap frC-—L—>R2which isalocal iSOl'l‘lCtl')‘l, butwhich isusually notone-one. Investigate thegeodesics onacone (thenumber ofgeodesics between twopoints depends ontheangle ofthecone, andsome geodesics may come back totheir initial point). 40.Letg:S”—>P"bethemap g(p) =[p]={p,-—p}. (a)Show that there isaunique Riemannian metric ((,))onP"such that g*(( ,))istheusual Riemannian metric onS"(theonethatmakes theinclusion ofS"into lli"+' anisometry). (b)Show that every geodesic y:lit—>P"isclosed (that is,there isanumber a such thaty(r—l-a) =1/(1') forallz‘),andthatevery twogeodesics intersect exactly once. (c)Show that there areisometries ofP”onto itself taking anytangent vector atonepoint toanytangent vector atanyother point. These results show that P"provides amodel for“elliptical” non-Euclidean geometry. The sum oftheangles inany triangle is>rt. 41.ThePoincare upper half-plane 3t’2isthemanifold {(x,y)GR2:y>0} with theRiemannian metric ()ﬂdx®dx;l- dy®dy9 J)2 (a)Compute that 2 1 1 1 2 1 k ass Chapter 9 (b)LetCbeasemi-circle inJfzwith center at(0,c)andradius R.Considering itasacurve z1~>(t,y(t)), show that gizi/(1)), _-J/(I), _}/(3)2 dd z-c 1/(z)' (c)Using Problem 27,show that allthegeodesics inH2arethe(suitably pa- rameterized) semi-circles with center onthex-axis, together with thestraight lines parallel tothey-axis. _ (d)Show thatthese geodesics have inﬁnite length ineither direction, sothatthe upper half-plane iscomplete. (e)Show that ifyisageodesic and p921/,then there areinﬁnitely many geodesics through pwhich donotintersect y. (f)Forthose whoknow alittle about conformal mapping (compare with Prob- LemIV7-6). Consider theupper half-plane asasubset ofthecomplex num- bers C.Show that themaps z+bfa):-jig a,b,c,dG]R, ad-bc>0 areisometries, and that wecan take any tangent vector atone point toany tangent vector atanyother point bysome f...Conclude thatiflength AB= length A’B" andlength AC=length A’C" andtheangle between thetangent vectors of,8and 1/atAequals theangle between thetangent vectors ofB’ and y’atA’,then length BC=length B'C' andtheangles atBand B’and atCandC’areequal (“side-angle-side”). These results show thatthePoincare B 13 1/ B’5'A C cﬂ Y Ar upper halﬁplane isamodel forLobachevskian non-Euclidean geometry. The sumoftheangles inanytriangle is<rt. Riezzzanniarz 1l4etric.s 369 42. LetMbeaRiemannian manifold such that every twopoints ofMcanbe joined byaunique geodesic ofminimal length, Does itnecessarily follow that theRiemannian manifold Miscomplete? 43.LetMbeamanifold with aRiemannian metric (,),andchoose afixed point pGM.Suppose thatevery geodesic y:[a,b] —>Mwith initial value 3/(a) =pcanbeextended toallofIR.Show thattheRiemannian manifold M isgeodesically complete. 44.Letpbeapoint inacomplete non-compact Riemannian manifold M.Prove thatthere isageodesic y:[0,oo) —>Mwith theinitial value 1/(0) =p,having theproperty thatyisaminimal geodesic between anytwoofitspoints. 45.LetMand Nbegeodesically complete Riemannian manifolds, andgive M><NtheRiemannian metric described inProblem 26.Show thattheRie- mannian manifold M><Nisalsocomplete. 46.This problem presupposes knowledge ofcovering spaces. Letg:M—>N beacovering space, where NisaC°°manifold. Then there isaunique C°° structure onMwhich makes ganimmersion. If(,)isaRiemannian metric onN,then g*( ,)isaRiemannian metric onM,and(M,g*( ,))iscomplete ifandonly if(N,(,))iscomplete. 47.(a)IfM"CN""'kisasubmanifold ofN,show that thenormal bundle v isindeed ak-plane bundle. (b)Using thenotion ofWhitney sum EBintroduced inProblem 3-52, show that vGBTM 2(TN)|M. 48.(a)Show that thenormal bundles 1.11,1.1;ofM"CN"""‘ deﬁned fortwo different Riemannian metrics areequivalent. (b)Ifif=rt:E—>Misasmooth Ic-plane bundle over M",show thatthe normal bundle ofM(IEisequivalent to5 49.(a)Given anexact sequence ofbundle maps ft0 >E1 >E;->E3 >0 asinProblem 3-28, where thebundles areover asmooth manifold M[or,more generally, over aparacompact space], show that E22E1EBE3. (13)If1;=IIIE_>Misasmooth bundle, conclude thatTE22r*($)es rr*(TM). 370 Chapter 9 50.(a)LetMbeanon-orientable manifold. According toProblem 3-22 there isS1CMsothat (TM )|SIisnotorientable (theProblem deals with thecase where (TM)|S1isalways trivial, butthesame conclusions willhold ifeach (TM)|S' isorientable; infact, itisnothard toshow that abundle over S]is trivial ifandonly ifitisorientable). Using Problem 47,show that thenormal bundle vofS1CMisnotorientable. (b)UseProblem 3-29toconclude thatthere isaneighborhood ofsome S1CM which isnotorientable. (Thus, anynon-orientable manifold contains a“fairly small” non-orientable open submanifold.) CHAPTER 10 LIE GROUPS Tl1l$ chapter uses, andilluminates, many oftheresults andconcepts ofthe preceding chapters. Itwillalsoplay animportant role inlater Volumes, where weareconcerned with geometric problems, because inthestudy ofthese problems thegroups ofautomorphisms ofvarious structures play acentral role, andthese groups canbestudied bythemethods nowatourdisposal. Atopological group isaspace Gwhich alsohasagroup structure (theproduct of0:,bEGbeing denoted byab)such that themaps (a,b) t—->ab from G><GtoG rtt->a‘"' from GtoG arecontinuous. Itclearly suflices toassume instead that thesingle map (a,b)|-> ab"l iscontinuous. VVewill mainly beinterested inavery special kind of topological group. ALiegroup isagroup Gwhich isalsoamanifold with a C°°structure such that (IJ)»~>Xi’ xt—->Jr“! areC°°functions. Itclearly sufﬁces toassume that themap (x,y)t->xy‘l isC°°.Asamatte1' offact(Problem l),iteven sufﬁces toassume thatthemap (x,y)t—->xyisC°°. The simplest example ofaLiegroup isIR",with theoperation +.The circle S1isalso aLieg1'0l.1p. One way toputagroup structure onSIisto consider itasthequotient group R/Z, where ZCIRdenotes thesubgroup ofintegers. The functions xt—->cosZrrx and xt—->sinZrrx areC°°functions onIR/Z, andateach point atleast oneofthem isacoordinate system. Thus themap (xy)t—->x—-y I-->xv"! m m m R><R—> 1R—> S]=1R/Z. which canbeexpressed incoordinates asoneofthetwomaps (x,y)t—->cos2rr(x —y)=cosZrrxcosZrry+sinZrrxsinZrry (x,y)i—->sin2rr(x —y)=sinZrrx cosZrry -—-cosZrrx sinZrry, isC°°: consequently themap (x,y)t—->xy"' from S]><S1toS]isalsoC°°. 371 372 C7zap!m' J0 IfGand HareLiegroups, then G><H,with theproduct C°°structure, and thedirect product group structure, iseasily seen tobeaLiegroup. In particular. thetorus S‘><SIisaLiegroup. Thetorus may alsobedescribed asthequotient group itmilltHitthepairs (a.b) and(a’,b') represent thesame element ofS‘><SIifandonly ifa’-a eZand b"—beZ. Many important Liegroups arematrix groups. The general linear group GL(n,]R) isthegroup ofallnon-singular real:2><nmatrices, considered asa subset ofIR":. Since thefunction det: IR":—>IRiscontinuous (itisapolynomial map), thesetGL(n,IR) =det“l(IR —{O}) isopen, andhence canbegiven the C°°structure which makes itanopen submanifold ofIR"2. Multiplication of matrices isC°°,since theentries ofABarepolynomials intheentries ofA andB.Smoothness oftheinverse map follows similarly from Cramer’s Rule: (A“I),.~,- =detA"j/detA, where Allisthematrix obtained from Abydeleting rowiandcolumn j. One ofthemost important examples ofaLiegroup istheorthogonal group O(n), consisting ofallAGGL(n,IlR) with A-A‘=I,where A‘isthetranspose ofA.This condition isequivalent tothecondition thattherows [and columns] ofAareorthonormal, which isequivalent tothecondition that, with respect totheusual basis ofIR",thematrix Arepresents alinear transformation which isan“isometry”, i.e.,isnorm preserving, and thus inner product preserving. Problem 2-33 presents aproof tliatO(n) isaclosed submanifold ofGL(n,IR), ofdimension nln—1)/2.Toshow that O(n) isaLiegroup wemust show that themap (x,_1=) t->x,1'“' which isC°° onGL(n,IR), isalso C°°asamap from O(n) ><O(n) toO(n). ByProposition 2-ll,itsuﬁices tosliow that itiscontinuous; butthisistruebecause theinclusion ofO(n) —>GL(n, IR)isahomeomorphism (since O(n) isasubmanifold ofGL(n, IR)). Later inthechapter wewillhavc another way ofproving that O(n) isaLiegroup, andinparticular, amanifold. Tlie argument intheprevious paragraph shows, generally, that ifHCGisa subgroup ofGandalsoasubmanifold ofG,then HisaLiegroup. (This gives another proof thatSIisaLiegroup, forSlCR2canbeconsidered asthegroup LieGroups 373 ofcomplex numbers ofnorm 1.Similarly, S3istheLiegroup ofquaternions ofnorm 1.Itisknow that these aretheonly spheres which admit aLiegroup structure.) Itispossible forasubgroup HofGtobeLiegroup with respect to aC°°structure that makes itmerely animmersed submanifold. Forexample, ifLCIR><IRisasubgroup consisting ofail(x,Cx]I forCirrational, then the iII1/'asimage ofLinS‘><S‘=IR><IR/(Z ><Z)isadense subgroup. Wedeﬁne aLie subgroup HofGtobeasubset HofGwhich isasubgroup ofG,andalsoa Liegroup forsome C°°structure which makes theinclusion map 1':H—>Gan immersion. Aswehave seen, asubgroup which isan(imbedded) submanifold isalways aLiesubgroup. Iteven turns out,after some work (Problem 18),that asubgroup which isanimmersed submanifold isalways aLiesubgroup, butwe willnotneed thisfact. The group O(n) isdisconnected; thetwocomponents consist ofallAeO(n) with detA =+1and detA =-1,respectively Clearly SO(n) ={AEO(n) : detA=1},thecomponent containing theidentity I,isasubgroup. This isnot accidental. l.PROPOSITION. IfGisatopological group, t11en thecomponent Kcon- taining theidentity eEGisaclosed normal subgroup ofG.IfGisaLie group, then Kisanopen Liesubgroup. PROOF. IfaeK,then a“'K isconnected, since bt->a‘"‘b isahomeomor- phism ofKtoitself Since e:a"a Ga""K,wehave 0"KCK.Since this istrue forallaEK,wehave K“‘KCK,which proves that Kisasubgroup. ForanybEG,itfollows similarly thatbKb“‘ isconnected. Since eEbKb“', wehave bKb“' (IK,soKisnormal. Moreover, Kisclosed since components arealways closed. IfGisaLiegroup, then Kisalsoopen, since Gislocally connected, soK isasubmanifold andasubgroup ofG.Hence KisaLiesubgroup. '3' Thegroup SO(2) isjustSI,which wehave already seen isaLiegroup. As aﬁnal example ofaLiegroup, wemention E(n), thegroup ofallEuclidean 374 Chapter 10 motions, i.e.,isometries ofIR".Alittle argument shows (Problem 5]that every element ofE(n) canbewritten uniquely asA-1"where ACO(n), and 1*isa translation, r(x) =r,,(x) =x+a. WecangiveE(n) theC°°structure which makes itdiﬁeomorphic toO(n) ><IR". Now E(n) isnotthedirect product O(n) ><IR"asagroup, since translations and orthogonal transformations donotgenerally commute. Infact, Ar..A*'(:<) =At/1*‘:-+0) =X+/1<a)=1'/1(a)(-Y), 5O Z I’/1(3), Ara 1 Consequently, Ara(Brg,)“‘ =Ar,,rg,B”' =Ar_,,_g,B”‘ =/1B‘]1'sta-b), which shows that E(n) isaLiegroup. Clearly thecomponent ofeEE(n) is thesubgroup ofallArwith AeSO(n). For any Liegroup G,ifaGGwedeﬁne theleftand right translations. Lat G—>Gand Ra: G—> G,by L_,,(b) =ab Ra(b) =ba. Notice that Laand Raareboth difieomorphisms, with inverses La-1 and Ra-1. respectively. Consequently, themaps Lag: é Gab Rm-I Gs—>Gba areisomorphisms. Avector ﬁeld XonGiscalled leftinvariant if La.,,X=X forallareG. Recall thismeans that L,,.,Xb =Xab foralla,b GG. Itiseasytoseethatthisistrueifwemerely have L,,,,X,. =Xa forallaEG. Consequently, given X9GGe.there isaunique leftinvariant vector ﬁeld X onGwhich hasthevalue X9ate. LieG?'O2‘1f).'- 375 2.PROPOSITION. Every leftinvariant vector ﬁeld XonaLiegroup G isC°°. PROOF. Itsuflices toprove that XisC°° inaneighborhood ofe,since the diffeomorphism Lathen takes XtotheC°°vector ﬁeld La,,.X around 0(Prob- lem5-I). Let(x,U) beacoordinate system around e.Choose aneighbor- hood Vofesothat a,beVimplies ab"' EU.Then foraeVwehave Xxl(a) :La*Xe(-ti) =X.o-"0La)- Since themap (a,b) t—->abisC°°onV><Vwecanwrite ﬂab) =xiLa(b) =fi(x'(a),...,x"(a),x'(b),...,x”(b)) forsome C°°function ffonx(V) ><x(V). Then Xi-‘((1) =Xeti-"0La) fl ‘ fl _ -3(x’ 0La) _ -3_gel M---8)”, where Xe_ggl E =i:cjDs+1.f‘(A'(@),X(@))- f'~=l which shows that XxiisC°°.This implies that XisC°°.*1‘ 3.COROLLARY. ALiegroup Galways hasatrivial tangent bundle (and is consequently orientable). PROOF. Choose abasis X18, ...,X,,eforGe.LetX1,...,X,,betheleftinvari- antvector ﬁelds with these values ate.Then X1,...,X,,areclearly everywhere linearly independent, sowecandefine anequivalence fITG—>G><IR" 1)‘. .f(Zc'”X.-tall =<<1.c‘.....c"). ~:~j=I 376 Chapter 10 Aleftinvariant vector ﬁeld Xisjustonethat isLa-related toitself foralla. Consequently, Proposition 6-3shows that [X,Y]isleﬁinvariant ifXandYare. Henceforth wewilluseX,Y,etc., todenote elements ofGe,andX,Y,etc.,to denote theleftinvariant vector ﬁelds with X(e)=X,Y(e) =Y,etc.Wecan then deﬁne anoperation [,]onGeby rnaﬂu [X,Y]=[X,Y](e). Thevector space Ge,together withthis[,]operation, iscalled theLiealgebra ofG,andwillbedenoted by£(G).(Sometimes theLiealgebra ofGisdeﬁned instead tobethesetofleftinvariant vector ﬁelds.) Wewillalsousethemore customary notation g(aGerman Fraktur g)for£(G). This notation requires some conventions forparticular groups; wewrite qI(n, IR) fortheLiealgebra ofGL(n,IR) o(n) fortheLiealgebra ofO(n). Ingeneral, aLiealgebra isaﬁnite dimensional vector space V,with abilinear operation [,]satisfying [X,X]=0 [[X,Y],Z]+[[Y,Z],X]+[[Z,X], Y]=0 ‘jacobi identity” forallX,Y,Z EV. Since the[,]operation isassumed alternating, itisalso skew-symmetric, [X,Y]=-[Y, X].Consequently, wecallaLiealgebra abelian orcommutative if[X,Y] =0forallX,Y. TheLiealgebra ofIR"isisomorphic asavector space toIR".Clearly .£(IR") isabelian, since thevector ﬁelds 8/Bxi areleftinvariant and[3/8x", 8/iixj] =0. The Liealgebra .£(S') ofS‘is1-dimensional, and consequently must be abelian. IfV;areLiealgebras with bracket operations [,],-fori=1,2,then wecandeﬁne anoperation [,]onthedirect sum V=V1€BV;(=V1xV;as aset)by [(X1,X2), (Y1,Y2)l =(IX1, Y1l1,[X2,Y2l2)- Itiseasy tocheck that this makes Vinto aLiealgebra, and that .,C(G ><H) isisomorphic to£(G)><.£(H) with thisbracket operation. Consequently, the Liealgebra £(SI >< ><S1)isalsoabelian. The structure ofgl(n,IR) ismore complicated. Since GL(n,IR) isanopen submanifold ofIR"2, thetangent space ofGL(n,IR) attheidentity Icanbe LieGroups 377 identiﬁed with IR”2. Ifweusethestandard coordinates xiionIR"2, then ann><n (possibly singular) matrix M=(M,-_,=) canbeidentified with 8M] . l-J ru LetMbetheleftinvariant vector ﬁeld onGL(n,IR) corresponding toM.‘We compute thefunction Mxki onGL(n,IR) asfollows. Forevery AGGL(n,IR), F?x"’<A) =ﬂit»-*1) =LA*MJo-*1)=M1o-*10LA)- Now thefunction xki0LA: GL(n,lR) —>GL(n,IR) isthelinear function ff (W0ants)=MAB) =ZAt..B...».cx=l with (constant) partial derivatives 3 A-'=I T-(KM 0LA)={klBx” 0 _]7*’:I. So ~ aMi-’“’(,4) =Mm-"’ 0LA)=M,-,-Ftﬁ’ 0LA) ha‘ H‘ H =Z MilAk:' =z MalAka- t=1 u=l Thus, _ i“MXki={Mji kzr Bx” 0 k;é1'. SoifNisanother n><nmatrix, wehave ~Jr!_ __3~klN_:(MJt )_ZN,,—ax,j (Mx) 1.1 H‘ =ZNkjM,-1=(NM)kl-j=] From thisweseethat ' [17.Iv];=Zuwv -NM»;i,;1<,1 ax-’ 378 Chapter 10 thus, ifweidentify gI(n,IR) with IR"2, thebracket operation isjust [M,N] =MN —-NM. Notice that inany ring, ifwedeﬁne [a,b] =ab—ba,then [,]satisﬁes the _]acobi identity. Since O(n) isasubmanifold ofGL(n, IR)wecanconsider O(n)_i asasubspace ofGL(n,lR);, andthusidentify 0(n)withacertain subspace ofnit’?This subspace may bedetermined asfollows. IfA:(—s,s) —>O(n) isacurve with 14(0) =I,and W6Cl€I1OI€(/l(I)),-_,; /-l;_,:(I),Il1€fl Z/4u<(l)/ljt<(*') =5:15 t<=1 diiierentiating gives /l:t<'(0)5;t +5fK/l;t'(0) =0, which shows that 1455(0) ==—»4jt'(0)- Thus O(n),t CIR""cancontain only matrices Mwhich areskew—symmetric, 0M]; M13 M1” —M|g 0 M: —M]3 0 ‘MIR 0 This subspace hasclimension n(n—1)/2,which iscxactly thedimension ofO(n), soO(n); must consist exactly ofskew—symmet1ic matrices. Ifwedidnotknow thedimension ofO(n), wecould usethefollowing lineofreasoning. Foreach 1',jwith 1'<j,wecandeﬁne acurve A:IR—>O(n) by I. J. 1 cost sin! 1' A(t)= (rotation inthe(i,j)—plane) —sinI cosI j I LieGraujis 379 with sin! and—sin! at(i,j)and (j,1'),1'sonthediagonal except at(t',t') and (j,j),and0'selsewhere. Then thesetofallA’(0) span theskew-symmetric ma- trices. Hence O(n)1 must consist exactly ofskew-symmetric matrices, andO(n) must have dimension n(n—1)/2. Wedonotneed anynew calculations todetermine thebracket operation ino(n). Infact, consider aLiesubgroup HofanyLiegroup G,andleti:H—> Gbetheinclusion. Since 2},:He—>Geisanisomorphism into, wecanidentify Hewith asubspace ofGe. Any XeHecanbeextended toaleftinvariant vector ﬁeld XonHandaleftinvariant vector ﬁeld XonG.Foreach aEHC G,wehave lefitranslations La:H—>H, La:G—>G and La of 11.0 La. So 7-=tt)?(a) :7-=t=La*X =La*(7.*X) : In_other words, Xand Xaref—related. Consequently, ifYEHe,then [X,Y] and [X,Y]aref—related, which means that [X1 =7.*(lX: Thus, HeCGe=gisasubalgebra ofg,thatis,Heisasubspace ofgwhich is closed under the[,]operation; moreover, Hewith thisinduced [,]operation isjustE)=.£(H). This correspondence between Liesubgroups ofGandsubalgebras ofgturns outtowork intheother direction also. 4-.THEOREM. LetGbeaLiegroup, and I]asubalgebra ofQ.Then there isaunique connected Liesubgroup HofGwhose Liealgebra isI]. PROOF. ForaEG,letAgbethesubspace ofGaconsisting ofallX(a) for XeI].The factthat I]isasubalgebra ofgimplies thatAisanintegrable distribution. LetHbethemaximal integral manifold ofAcontaining e.If beG,then clearly L,_t,,,.(A,,) =Ab“, soLb,leaves thedistribution Ainvariant. Itfollows immediately thatLbpermutes thevarious maximal integral manifolds ofAamong themselves. Inparticular, ifbGH,then Li,-t takes Htothe maximal integral manifold containing Lb-i(b) =e,soLb-t(H) =H.This implies that Hisasubgroup ofG.Toprove that itisaLiesubgroup wejust need toshow that (a,b)t—->ab"' isC°°. Now thismap isclearly C°°asamap intoG.Using Theorem 6-7,itfollows thatitisC°°asamap intoH. Theproof ofuniqueness islefttothereader. *3’ 380 Chapter 10 There isavery diﬁicult theorem ofAdo which states that every Liealgebra isisomorphic toasubalgebra ofGL(N,IR) forsome N.Itthen follows from Theorem 4thateveryLiealgebra iszltomorphic toI/zeLiea(gebra tyfsome Liegroup. Later onwewillbeable toobtain a“local” version ofthisresult. Wewillsoon seeto what extent theLiealgebra ofGdetermines G. Vifecontinue thestudy ofLiegroups along thesame route used inthestudy ofgroups. Having considered subgroups ofLiegroups (andsubalgebras oftheir Liealgebras), wenext consider, more generally, homomorphisms between Lie groups. If¢:G—>HisaC°°homomorphism, then <,i'>,,,,._.: Ge—>He.Forany aEGweclearly have (POLa=L¢:(a] °¢: Iv /it-I soifXEGe,and X=<,i'>.,.,._.X istheleftinvariant vector ﬁeld onHwith value ¢..._.,X ate,then ¢=|:a/?((1) =¢*aLa*X =L¢(a]*¢*eX =1?<¢<a))- Thus Xand XareQ5-ielated. Consequently, themap ¢,,.,._,: g—>I]isaLie algebra homomorphism, that is, ¢*e(aX "I"by) :a¢*eX “I”b¢=t=eY ¢=1=elX=Yl =l¢*eX:¢=reYl- Usually, wewilldenote ¢...,simply byqt‘)...:g—>I]. Forexample, suppose that G=H=IR.There areanenormous number ofhomomorphisms ¢:IR—>IR,because IRisavector space ofuncountable dimension overQ,andevery linear transformation isagroup homomorphism. Butif¢isC°°,then thecondition 4>(S+1)=¢(S)+¢>(r) implies that d¢(! +8) _d¢'($)_ ds —ds' evaluating ats=0gives 4/(I)=¢'(0), which means that¢(:)=ctforsome c(=¢>’(0)). Itisnothard toseethateven acontinuous ¢must beofthisform (oneﬁrstshows that¢isofthisform onthe LieGroups 381 rational numbers). Wecanidentify .£(IR) with IR.Clearly themap ¢,,.:IR—>IR isjustmultiplication byc. Now suppose thatG=IR,butH=S1=IR/Z. Aneighborhood ofthe identity eeS1canbeidentiﬁed with aneighborhood of0eIR,giving risetoan identiﬁcation of“C(S1)with IR.The continuous homomorphisms Q5:IR—>S1 areclearly oftheform XC n_>n-_>n/Z; once again, ¢,,.:IR—>IRismultiplication byc. Notice that theonly continuous homomorphism 45:S1—>IRisthe0map (since {0}istheonlycompact subgroup ofIR).Consequently, aLiealgebra ho- momorphism g—>I)may notcome from anyC°°homomorphism ¢:G—>H. However, wedohave alocal result. 5.THEOREM. LetGand HbeLiegroups, and (D:g—>IE)aLiealgebra homomorphism. Then there isaneighborhood UofeEGandaC°°map ¢:U—>Hsuch that ¢(ab) =¢(a)¢(b) when a,b,ab EU, andsuch thatforevery Xegwehave ¢*eX = Moreover, ifthere aretwoC°°homomorphisms ¢,1lr: G—>Hwith ¢,,,.= tlrie=<1),andGisconnected, then ¢=11/. PROOF. LetT(German Fraktur k)bethesubset TCg><If)ofall (X,(X)), for XeQ.Since (Disahomomorphism, iisasubalgebra ofQ><I]=£(G ><H).By Theorem 4,there isaunique connected Liesubgroup KofG><Hwhose Lie algebra isf.Ifrt]:G><H—>Gisprojection ontheﬁrstfactor, andto=31']|K, then to:K—>GisaC°°homomorphism. ForXGgwehave w*(X,<1>(X)) =X, soan:K(8,8)—>Geisanisomorphism. Consequently, there isanopen neigh- borhood Vof(e,e) EKsuch that totakes Vdiﬁeomorphically onto anopen neighborhood UofeEG.IfI21G><H—>Hisprojection onthesecond factor, wecandeﬁne ¢=.-H2o(0_] on U. 382 C/iapter 10 The ﬁrstcondition on¢isobvious. Asforthesecond, ifXEg,then w=t(X,‘1’(X)) =X, so Given ,i!/I G—>H,deﬁne theone-one map 6:G—>G><Hby 9(0) =(6111!/((1)) Theimage G’of6isaLiesubgroup ofG><HandforXetiweclearly have 9*/Y=(X,‘i’(X))= so.£(G’) -:f.Thus G’=K,which implies that11r(a) =¢(a) forallaeG.'Z~ 6.COROLLARY. IftwoLiegroups GandHhave isomorphic Liealgebras. then they arelocally isomorphic. PROOF. Given anisomorphism <1):gt—>I],let4'»bethemap given byTheo- rem5_Since ¢,,.e=(I)isanisomorphism, ¢isadiffeomorphism inaneighbor- hood ofeeG.'3' Remark: Forthose who know about Simply-Cot1neCteCl spaces itisfairly easy (Problem 8)toconclude thattwosimply-connected Liegroups withisomorphic Liealgebras areactually isomorphic, andthat allconnected Liegroups with a given Liealgebra arecovered bythesame simply—connected Liegroup. 7.COROLLARY. Aconnected Liegroup Gwith anabelian Liealgebra is itself abelian. PROOF. ByCorollary 6,Gislocally isomorphic toIR",soab=bufora,b inaneighborhood ofe.Itfollows that Gisabelian, since (Problem 4)any neighborhood ofegenerates G.'2' 8.COROLLARY. Forevery XeGe,there isaunique C°°homomorphism ¢:IR—>Gsuch that d¢-— nX. dz,=,, LieGroups 383 FIRSTPROOF. Deﬁne (D:IR—>£(G) by (lJ(a:) 2o:X. Clearly (DisaLiealgebra homomorphism. ByTheorem 5,onsome neighbor- hood (—s,1:)of0EIRthere isamap Q5:(—s,t:) —>Gwith ¢>(S+=‘)=¢(S)¢(I) |S|,lI|,|S+='| <8 and 61¢ d d:,0 dI,0 Toextend ¢toIRwewrite every Iwith |I|3suniquely as Ixk(e/2) +r kaninteger, |r|<s/2 (pm =¢(s/2) ---¢(s/2) -¢(r) [¢(s/2) appears ktimes] k30 ¢(—e/2) ---¢(—e/2) -¢(r) [¢(—e/2) appears —ktimes] k<0. Uniqueness alsofollows from Theorem 5. SECOND (DIRECT) PROOF. IfftG—>IRisC°°, and¢:IR—>GisaC°° homomorphism, thenand deﬁne d¢ ___../(¢(!+/1)) —f(¢(-')) Eur) WIll-I-iii) It -f(¢(l)¢(/1)) —f(¢(!))=l1m it-->0 L? d¢-6-5 u=ofoL¢U)o¢ d¢ . $1-¢(r)*E _o<1) =L¢<.>.X<f) =>?<¢<n)o").ru Thus ¢must beanintegral curve ofX,which proves uniqueness. Conversely, ifQ5:IR—>Gisanintegral curve ofX,then It"->95(8)'¢(f) isanintegral curve ofXwhich passes through ¢(s) attime Ix0.The same is clearly true for I*~>¢(s+I), so¢isahomomorphism. Weknow thatintegral curves ofXexist locally; they canbeextended toallofIRusing themethod oftheﬁrstproof. "I' 334 Chapter" I0 Ahomomorphism ¢:IR—>Giscalled a1-parameter subgroup ofG.We thus seethatthere isaunique 1-parameter subgroup ¢ofGwith given tangent vector dd)/d!(0) GGe.VVehave already examined theI-parameter subgroups ul More interesting things happen when wetake GtobeIR-{O},with multiplication asthegroup operation. Then allC°°homomorphisms Q5:IR—> IR—{O},with ¢(S+1’)=¢(S)¢(I), must satisfy ¢'(I)=¢'(0)¢(t’) ¢>(0)=1- The solutions ofthisequation are 4,0») _..:e¢t'(0)f_ Notice thatIR—{0} isjustGL(1,IR). AllC°°homomorphisms ¢:IR—>GL(n,IR) must satisfy theanalogous differential equation W ¢'(*')=<1/(0)'¢(f)= ¢(0)=1, where -now denotes matrix multiplication. The solutions ofthese equations canbewritten formally inthesame way (**I ¢(t’)=¢XP(I¢’(0)). where exponentiation ofmatrices isdefined A_IAA2A3€Xp()— ~l"'1""'"-I"-2T~l-""37-I--'-. . I 0 This follows from thefacts inProblem 5-6,some ofwhich willbebriefly reca- pitulated here. IfA=(a,-_;) and |A|:max |a,-_;|, then clearly IA+Bl5|/1|+lBl IABI 5HI/ll-IBII hence |A|,‘ 5n""‘|A|" 5n"|A|". Consequently, A”, +AM" <<~|/1|)”, <~|A|)”+" 0NN! (N4-K)! rNF "'+(N+1<)t T as_’°°‘ LieGroups 385 sotheseries forexp(A) converges (the(i,j)‘hentry ofthepartial sums converge), and convergence isabsolute and uniform inanybounded set. Moreover (see Problem 5-6), ifAB=BA, then exp(A +B)=(exp A)(exp B). Hence, if¢(t) isdeﬁned by(>t<>t<), then ¢,(1)= §,§,p(I¢’(0) +/1¢>’£0)) —@Xp(I¢’(0)) Zjig)t@xp<1=¢;f0>> —11c,,,(,,,.(0,, I 21 2 _/~i>%,t%<t>2_. ,... =lfllii ' LII' ‘*"P(""’!(°)I =¢’(0)¢(r). so¢does satisfy (>t<). ForanyLiegroup G,wenow deﬁne the“exponential map" exp: g—> G asfollows. Given XGct,let¢:IR—>Gbetheunique C°°homomorphism withd¢/dz(0)=X.Then exp(X) =¢(1). Weclearly have exp(I1 +t;)X =(expt‘1X)(expt2X) exp(—IX) =(exp t‘X)"'. 9.PROPOSITION. The map exp: G,—>GisC°°(note that Ge’»¥IR"hasa natural C°°structure), and0isaregular point, sothatexptakes aneighborhood of0EGediffeomorphically onto aneighborhood ofeeG.Iftit:G—>His anyC°°homomorphism, then gxp 01!/* =‘(If0¢Xp_ €Xpl lﬁxp 11/ 386 C/topler I0 PROOF The tangent space (Ge ><G)(X_,,) oftheC°° manifold Ge><Gatthe point (X,a) canbeidentiﬁed with Ge69Ga. Wedeﬁne avector ﬁeld Yon Ge><Gby Y(X,a) =0EBX(o). O O ts1);>is' IIT Then Yhasaflow or:IR><(Ge><G)—>Ge><G,which weknow isC°°. Since expX =projection onGoftx(1,0 EBX), itfollows that expisC°°. Ifweidentify avector vE(G,,.)0 with Ge,then thecurve c(t)=ivinGehas tangent vector vat0.So dexp(c(t)) dexp,,0(v) - dr 0-81-I0exp(tv) I: = =1)‘. Soexp,,0 istheidentity; andhence one-one. Therefore expisadiffeomorphism inaneighborhood of0. Given tutG—>H,andXEGe,letti):IR—>Gbeahomomorphism with 5.12=Xdr,=0 Then 11/o¢:IR—>Hisahomomorphism with d(il/°¢)i—--- —,,X.dt if [=0 Consequently, @XP(i!/=tX) =if045(1)=11/(¢XP X)-'1' IO.COROLLARY. Every one-one C°° homomorphism ¢:G—>Hisan immersion (so¢(G) isaLiesubgroup ofH). PROOF. If¢.,.,,(X(p)) =0forsome non-zero XGg,then also<,t5,,.,,(X) =0. Butthen e=exp¢,,,,(rX) =¢(exp(rX)), contradicting thefactthat ¢isone-one. Q2» LieGroups 387 ll.COROLLARY. Every continuous homomorphism ¢:IR—>GisC°°. PROOF. LetUbeastar-shaped open neighborhood of0GGeonwhich exp isone-one. Forany toEexp(-§_;U), ifto=exp(X/2) forXGU,then G=exp(X/2) =[exp(X/4)]2, expX/4eexp(%-U). Soahasasquare rootinexp(§-U). Moreover, ifa=b2forbecxp(%U), then b=exp(Y/2) forYeU,so exp(X/2) =61=b2=[exp(Y/2)]2 =expY. Since X/2, YeUitfollows that X/2=Y,soX/4 =Y/2. This shows that every aEexp(%U) hasaunique square root inthesetexp(%-U). Nowchoose t~>0sothat¢(t)Eexp(%U) for|t|5t~.Let¢(t~)=expX, XEexp(-%U). Since t¢<t/2)? =¢<t:>=[expX/212.itfollows from theabove that ¢(s/2) =exp(X/2). Byinduction wehave ¢(e/2") =exp(X/2”). Hence ¢("1/2" '8)=¢(8/2")”' =[@XP(X/2")I’" =¢XP(m/2" 'X)- Bycontinuity, Q0O0.0 ¢(se) =expsX forallse[—l,1 I2.COROLLARY. Every continuous homomorphism ¢:G—>HisC°°. PROOF. Choose abasis X1,..., X,,forGe. The map It—->¢(exptX,-) isa continuous homomorphism ofIRtoH,sothere isY;eHesuch that ¢(exp IX,-) =exp!Y,-. Thus, (*I ¢((¢><P!1XiI '--(QXP t'ttXe)) =(QXP 1'1Y1)''-(EXP (nYn)~ Now themap 11/:IR“—>Ggiven by 11/(t1,. ..,r,,) =(expI1X1) ---(expt,,X,,) isC°°andclearly 8 Tl/at )=XI, 0 sotlrisadiffeomorphism ofaneighborhood Uof0GIR"onto aneighbor- hood VofeeG.Then onV, ¢=(¢°11/)°I1/-1, and(=t=)shows that¢011/isC°°.So¢isC°°ate,andthuseverywhere. '2' 388 Chapter" I0 I3.COROLLARY. IfGandG’areLiegroups which areisomorphic astopo- logical groups, then they areisomorphic asLiegroups, that is,there isadiffeo- morphism between them which isalsoagroup isomorphism. PROOF. Apply Corollary lltothecontinuous isomorphism anditsinverse. '2' The properties oftheparticular exponential map exp:n"2(=gl(H,IR)] ->o1.(tt,n) maynowbeused toshow thatO(n) isaLiegroup. Itiseasytoseethat exp(M‘) =(exp M)‘. Moreover, since exp(M +N)=(exp M)(exp N)when MN =NM, wehave (exp M)(exp —M) :I. SoifMisskew~symmetric, M=—Ml,then (expM)(exp M)‘=1, i.e.,expM EO(n). Conversely, anyAEO(n) sufﬁciently close toIcanbe written A=expM forsome M. LetA‘=expN.Then I=A-Al= (cxpM)(exp N),soexpN =(expM)"" =exp(—M). Forsuﬁiciently small M and Nthisimplies that N=—M. SoexpM‘ =A‘=exp(—M); hence M‘=-M. Itfollows thataneighborhood ofIinO(n) isann(n-—1)/2 dimensional submanifold ofGL(n,IR). Since O(n) isasubgroup, O(n) isitself asubmanifold ofGL(n,IR). _]ust asinGL(n, IR),theequation exp(X +Y)=expXexpYholds whenever [X,Y]=0(Problem l3). Ingeneral, [X,Y]measures, uptoﬁrst order, the cxtent towhich thisequation fails tohold. Inthefollowing Theorem, andin itsptoof, toindicate thatafunction e:IR—>Gehastheproperty thatc(r)/I3 is bounded forsmall r,wewilldenote itby0(r3). Thus 0(r3) willdenote diﬁerent functions atdifferent times. I4-.THEOREM. IfGisaLiegroup and X,YGGe,then 2 (I)expIXexp!Y =exp(r(X +Y)+%[X, Y]+003)} (2)exp(—rX) exp(—!Y) exprX exprY =exp{!2[X, Y]-|-0(r3)} (3)exp!XexprYexp(—!X) =txp{tr +r2[X,Y]+003)}. LieGroups PROOF. Wehave X/df(a)=-fa(f):La*X(f)=X(f°La)=‘c}6% f(a‘3XPuX) u=0 Similarly, .. -7 d(11) lf(a) =2-1; Of(a-expuY). I1: Forﬁxed s,let ¢(r) =f(expsX exprY). Then , d d(tn) cp(r)=-G-,-;f(exp sXexprY) =H; f(exp sXexprY expuY) u=0 =(Yf)(exp sXexprY) by(ii). Applying (iii)toYfinstead offgives r-1.4:-1.4 (iv) ¢”(!) _[Y(Yf)](exp sXexprY). Now Taylor's Theorem says that ¢(t)=¢(0)+¢’(o)t+$12 +otﬁ). Suppose thatf(e) =0.Then wehave (v) f(exp.tXexptr)=f(exp.tX)+1'(Yf)(exp .tX) +€[Y(Yf)](exp.s'X) +003). Similarly, forany F. d ...-J;-F(expsX) =(XF)(expsX) 2 1-... 1-... ;L§F(exp SX)=[X(XF)I(exp SX) F(expsX) =F(e)+s(XF)(e) +%2[2?()'?F)](@) +0(.t3). 390 C/zapzer 10 Substituting in(v)forF=f,F= and F=l7(I7f) gives r-4 nu (vi)f(¢><pSXexptl’) _s(Xf)(e) +:(Yf)(e) +‘;2[>?<1'?f)1<e>+§[?<?"f>1<e>+ Slj;(?f)(e) +0(9)+003)+0(s2:)+ooﬁ). Inparticular‘, /u ru (vii) f(exptX exprl’) -t[(X +Y)f](e) +:2 +2??+ fl(e)+0(9). Now forsmall twecanwrite exp¢‘X exptl’ =expZ(r) forsome C°°function Zwith values inGe.Applying Taylor’s formula toZ gives 2(1)=12,+1222 +0(9), forsome Z1,Z2GGe. Iff(e) =0,then clearly f(A(!) +O(!3)) =f(A(t)) -|- 0(!3), soby(vi)wehave (viii) f(expZ(t)) =f(exp(!Zi +!2Z;)) +0(r3) =42,fut»)+r1(22f)(@) +§[2"1(Zf)](e) +003). Since wecantake thef’stobecoordinate functions, comparison of(vii)and (viii) gives j(H+?=2; H-4 H-4 Hahn ﬂung ZZ ~ XX ~~ YY %+Z=*=T+’“’+T~ which gives l Zl:X+Y: thus proving (1). Equation (2)follows immediately from (1). LieGroups 391 Toprove (3),again choose fwith f(e) -=0.Then similar calculations give (ix) f(exptX exptl’ exp(-—tX))~ M MM ~m =,[<r+ r~r);1(@)+.~2[(¥ +§+%+ri>'-rrui>'r)] (E) +0(9). Ifwe write explX expr)’ exp(-:X) =exp(£S1 +1252 +0(:3)), then wealsohave (X) f(expt‘X exprl’exp(-—!X)) =f(exp(t‘S1 +952))+0(9) =its]me)+r2(§2f)(e)2 +%[$1(51f)](@) +003). Comparing (ix)and(x)gives thedesired result. '1' Notice thatformula (2)isaspecial caseofTheorem 5-l6(compare alsowith Problems 5-l6 and 5-l8). The work involved inproving Theorem 14isjustiﬁed byitsroleinthefol- lowing beautiful theorem. l5.TI-IEOREM. IfGisaLiegroup andHCGisaclosed subset which is alsoasubgroup (algebraically), then HisaLiesubgroup ofG.More precisely, there isaC°°structure onH,withlfterelative !0]20l0gy, thatmakes itaLiesubgroup ofG. PROOF Weattempt toreconstruct theLiealgebra ofHasfollows. LetI)CGe bethesetofallXGGesuch that expIX GHforallI. Assertion J.LetX,-GGewith X,-—>Xandlett,-—>0with each I;;é0.Suppose expt,-X; EHforall1'.Then XEI]. Progﬁ Wecanassume I;>0,since exp(-I,-X,-) =(expr,-X,-)_' EH.ForI>0, let k,-(r) =largest integer 5 I Then r r _— -—- 1< kl"(!) S _5 Ii is 392 Chapter 10 SO :,-k,-(z)->r. Now kill)exp(k,-(¢‘)!,-X,-) =[exp(t,-X,-)] EH, k,-(:)z,-X,- ->IX. Thus exp:X6H,since Hisclosed andexpiscontinuous. Weclearly alsohave exprX EHfor! <0,soXEI).Q.E.D. Wenow claim that I]CGeisavector subspace. Clearly XeI}implies sXeI)forallse]i?..IfX,YeI],wecanwrite by(l)ofTheorem l4 exp:X exprl’ -=exp{!(X +Y)+rZ(r)} where Z(r) —>0asr—>0.Choose positive I;—>0andletX;=X+Y+Z(r,-). Then Asse"r£z'0n Jimplies that X+YEI].Alternatively, wecanwrite, forﬁxed r. IX FYH-— r(X+Y)+!2[X Y]+O(l/n2)l'expnexpn Tcxp 2n ’ I’ taking limits asn—>oogives expr(X +Y)GH. (Similarly, using (2)ofTheorem l4weseethat [X,Y]eI),sothat I]isa subalgebra, butwewillnoteven usethis(act.] Now letUbeanopen neighborhood of0eGeonwhich expisadiiTeomo1'- phism. Then exp(I) OU)isasubmanifold ofG.Itclearly suﬁices toshow that ifUissmall enough, then H('1exp(U) =exp(I] ('1U). Choose asubspace I)’CGecomplementary toI],sothat Ge2I)EBI)’. As.rem'on 2.The map ¢:Ge—>Gdeﬁned by ¢(X+X')=expXexpX’ XeI),X'eI]' isadilleomorphism insome neighborhood of0. Proqy’. Choose 21basis X1,...,X;e,...,X,, ofGewith X;,...,X;e abasis forI]. Then ¢isgiven by n it n 0,-X5) =exp(Za,-Xe) exp( 2 0,-X,-). i=1 i=1 l'=ff-I-I LieGr0up.s 393 Since themap 217:, a,-X; i->(01,. ..,a,,) isadiffeomorphism ofGeonto IR”. itsuiiices toshow that /4 n 11/(a;,...,a,,) =.exp(Za,-Xe) exp( Z 0,-X,-) isadiffeomorphism inaneighborhood of0eR”.This isclear, since 81]/,,, 0)=X,-. Q.E.D. Asserﬁon 3.Thcre isaneighborhood V’of0inI)’such that expX’¢Hif 07’:X’eV’. Proqf Choose aninner product onI)’andletKCI)’bethecompact setofall X’eI)’with l5|X’| 52.Ifthe assertion were false, there would beX,-’6I)’ with X,-’—>0andexpX,-’eH.Choose integers 11,-with n,-X,-’ EK. Choosing asubsequence ifnecessary wecanassume X,-’-»>X’EK.Since l/ne—>0, exp(l/n,-)(n,-X,-’) EH, itfollows from A.r.ser!z'07i Jthat X’EI],acontradiction. QED. Wecannow complete theproof ofthetheorem. Choose aneighborhood U=W><W’ofGeonwhich expisadiffeomorphism, with Waneighborhood of0eI) W’aneighborhood of0Eh’ such that W’iscontained inV’ofAssertion 3,and¢ofAssertion 2isadiffeomor- phism onW><W’.Clearly exp(I] OU)CHOexp(U). Toprove thereverse inclusion, leta6HOexp(U). Then a=e>;pXe;<pX’ XeW,X’eW’. Since a,expXeHweobtain expX’ GH,so0=X’,andaGexp(I) QU).¢I¢ 394 C/zapter 10 Uptonow, wehave concentrated ontheleftinvariant vector ﬁelds, butman} properties ofLiegroups arebetter expressed interms offorms. Aform wis called Ieftinvariant ifLe”w _-=toforallaeG.This means that w(b) =Le*[w(ab)]. Clearly, aleftinvariant k-form wisdetermined byitsvalue w(e) ES2"(Ge). Hence, ifwl,...,w”areleftinvariant l-forms such that w‘(e), ...,w”(e) span Ge"‘. then every leftinvariant k-form is . . IZ a,-,...i,,.w’1/_\---/\w”‘ =Z:/11w 1'1<---<:';,- 1 forcertain constamfs 0;.Ifw’(e), ...,w”(e) isthedual basis toX1,... ,XeEGe. then anyC°°vector ﬁeld Xcanbewritten H X= forC°°functions f’. j=I Then w’(X)==f’, so0)’isC°°.Itfollows thatanyleftinvariant form isC°°. Iftoisleftinvariant, then foraeGwehave Le*dw =d(Le*w) =dw, so(la)isalso leftinvariant. The formula onpage Q15 implies that foraleft invariant l—form toandleftinvariant vector ﬁelds Xand Ywehave In-urn-1 ru ru ru ru n-urn dw(X,Y)_X(w(Y)) -Y(w(X)) -w([X,Y]) =—w([X,Y]). Hence (*l dw(¢’)(X, Y)=—w(@)([X, Y1), thebracket being theoperation inQ. Thc interplay between leftinvariant and right invariant vector ﬁelds isthe subject ofProblem ll.Here weconsider thecase offorms. Le»Groups 395 16.PROPOSITION. Letti:0_>0be,1/(3)=3-1. Aform toisleftinvariant ifandonly ifiZr*w isright invariant. (IrweEs2*(0e), then1[r*we=(—l)"we. (Iftoisleftandright invariant, then dw--=0. -P~L:Ql\DZ\-\—»/\-\—»/\-\—»/\-\—»/(IfGisabelian, then gisabelian (converse ofCorollary T). PROOF. (l)Clearly ‘I/°Rb-"=Lb—1 ‘*1!/= so Rb*¢¥ Z ,¢,*Lb_]*. Iftoisleftinvariant, then R),*(ilr*w) =(1/*L,,_1*w =i!r"‘w. so(Z/"’w isright invariant. The converse issimilar. (2)Itclearly suﬁices toprove thisfork=l.Soitisenough toshow that (!r.,e(X) =—X forXEGe.Now Xisthetangent vector atI=0ofthe curve Il->ex-prX. Soi0,..eX isthetangent vector att=0ofr i->(expt‘X)_’ = exp(-—rX); thistangent vector isjust-X. (3)If0)isaleftandright invariant k-form, then ¢*(we)=<-1)"we. Since 1!/*a) andtoareboth leftinvariant, wehave i!r*w =(—l)"w. The form dwisalsoleftandright invariant, so 11/*(dw)=(—l)”+'dw. But mam) =d(11/*w) =d((—l)kw) =(—l)”dw. Sodw=0. (4)IfGisabelian, then allleftinvariant l-forms toarealso right invariant. So dw=0forallleftinvariant l-forms. Itfollows from (>z<)that [X,Y]=0forall X,Yeg. 396 C/zapter 10 Allemate prnqfty’ (4).ByTheorem l4,ifGisabelian, then forX,YEGewe have£2 3 :2 3 5[X,Y]+ 0(1): 3[Y,X]+ on). Hence §[X,Y]+om)/:1 .-=§[Y,X] +0(9)/:2. Letting I—>0,weobtain [X,Y]=[Y,X].#9 Since do)isleftinvariant foranyleftinvariant 0),itfollows thatforabasis 0)‘, ...,w”ofinvariant l-forms wecanexpress each dwk interms ofthe(vi/\er’. First choose X,,_..,X,, EGedual tow’(e), ...,w”(e). There areconstants C)’; such thatN rnm=Z%ml’¢'=l clearly wealsohave T7 tn%=Z%hi=1 The numbers Ci’;arecallcd theconstants ofstructure ofG(with respect tothe basis X1,...,XeofQ).From skew-symmetry of[,]andthejacobi identity we obtain 0M%=—dN (2)Z(c;}c,§, +c,;';,c,§,+c,;'*,c,=j,.) =0. l£=l From (>z<)onpage 394- weobtain dwk=—X:C,’} cu’Aor’.-=—%ZZC,-’,‘, w’Aw’. i<j i.j Itturns outthat(2)isexactly what weobtain from therelation d2w"' =0.Con- dition (2)isthus anintegrability condition. Infact, wecanprove (Problem 30) that ifC,-’jareconstants satisfying (l)and (2),then wecanﬁnd eveiywhere linearly independent l-forms col,...,cu”inaneighborhood of0ER"such that . 1 - -dw’ =—-2-Z:C,~’§ w’Aw‘. 1'-I LieG?'0I1fJ'.S 397 Moreover, theexistence ofsuch w’implies (Problem 29)that wecandeﬁne a multiplication (0,b)i~>abinaneighborhood of0which isagroup asfaras itcanbeandwhich hasthew"asleftinvariant l-forms. From thislatter fact and(asuitable local version of)Theorem 5wecould immediately deduce the following Theorem, forwhich wesupply anindependent proof. I7.THEOREM. LetGbeaLiegroup with abasis ofleftinvariant l-forms w‘,...,w”andconstants ofstructure C,-’}.LetM”beadifferentiable manifold andlet9’,...,6” beeverywhere linearly independent l-forms onMsatisfying 40*=-Zcgel /\9j.i<j Then forevery peMthere isaneighborhood Uandadiffeomorphism f:U—>Gsuchthat 6’=f*w’. PROOF. Letrt]:M><G—>Mand1:2:M><G—>Gbetheprojections. Let Elk=rr1*9", (Bk=rr;_~*w". Then ¢/(ék-03*)=-20,’;-([é'” AG’)-[JM311)£'<j 2_Zcj;-[é*' /\(51-@1’)+(é" -3")/\(i)"]. i<j ByProposition 7-l4, M><Gisfoliated byn-dimensional manifolds whose tangent spaces ateach point areannihilated byallGk—a3". Choose 0GG andletT‘bethefolium through (p,a). Now til,..,Ei",(D’,. ..,5)"arelinearly independent everywhere; soonI"(,,,e], which isthesetofvectors in(M><G)(,,,e] where ti"-(Bk=0,thesetsPl,...,3”anda3l,...,z3" areeach linearly inde- pendent. Hence Ir):I"—>MandI-T22I"—>Gareeach diffeomorphisms insome neighborhood of(p,a). This means that Fcontains thegraph of adiffeomorphism ffrom aneighborhood Uofptoaneighborhood ofa. %G M 398 Chapter 10 Letf_:U—>M><Gbethemap fin=tqeftqn <:1". Since Elk—(Bk=0onT‘,wehave 0__=j“-=1=(gl'< _(Bk) 2j"-*R_l=1=6k ___j'-'=t=n_2=1=wk :2-..(J1'] 0f-)*9k -—(J1'2 0_;;)*w"' =9"-f*w". »:¢ Itisalsopossible tosaybyhow much anytwosuch maps differ: 18.THEOREM. LetMbeaconnected manifold, letGbeaLiegroup, and letj},jj:M—>GbetwoC°°maps such that f1*(w) =f2*(w) forallleftinvariant l-forms w.Then flandfzdiffer byalefttranslation, that is,there isa(unique) aGGsuch that f2=Laoj]. PEDESTRI/1N PROOF. Case J.M=IRandtheIwomaps yl,1/2:IR—>Gsatisﬁ 3/1(0) =:)/2(0). Wemust show that yl=3/2.Forevery leftinvariant l—form w wel1El\-'1' d d ("(1/2(!))= 72*") )= Yliiw I =wll/1(0) ___( * dY1 —(LY2il'lY1(Il"’) °=’(Y2(*’))j 7 I =WU/2(1))([Li»wm<:)-1],_ 1%)- Itfollows that we[L ]Q Ch "' Y3(!l)’](?l"’ *(yr‘ Ifweregard 1/,asgiven, andwrite thisequation outinacoordinate system. thcn itbecomes anordinaiy diﬁerential equation for}/2(ofthetype considered LieG?'0Z£f).s 399 intheAddendum toChapter 5),soithas aunique solution with theinitial condition }/2(0) =1/,(0). Butthissolution isclearly )/2=1/1. Case 2.M=IR,butthemaps y,,)/2area2'bz'tra:§)i. Choose aGGsothat J/2(0) =Q'14(0)- Ifa)isaleftinvariant l—form, then (La91/i)*(w) =1/)*(Le*w) =1/1*(w) =1/z*(w)» Since Le03/1(0) =3/2(0), itfollows from Case Jthat Le0y,=3/2. Case 3.Genera! ease. Let[J0GM.Choose aGGsothat fi(P0) =Q'fI(P0)- ForanypEMthere isaC°°curve c:IR—>Mwith 0(0) =pgandc(I) =p. Lety;= oc.Then r2*(w) =v*fz*(w) =v*f1"‘(w) =1*1*(w)- ByCase 2,wehave 1/2(1): a-1/((3) forallr. inparticular forI=l,sof2(p) =a-f,(p). ELEGANT PROOF LetIr,-:G><G->Gbeprojection ontheFl‘factor. Choose abasis w’,...,w” fortheleftinvariant l-forms. For(0,b)EGxG,let N A(e,,g,] =nker(Ir;*w’ —:rr;;*a)’). £'=l Then Aisanintegrable distribution onG><G.Infact, ifA(G) CG><Gisthe diagonal subgroup {(0,0) IerGG},then themaximal integral manifolds ofA aretheleftcosets ofA(G). Now deﬁne hrM—>G><Gby /itp)=(Mp), 12(3))- Byassumption, Iz*(rr;*w’ -rr2*w’) :=f,*w’ -—ﬁ*w’ --=0. Since Misconnected, itfollows that lz(M) iscontained insome leftcoset ofA(G). Inother words, there area,bEGwith af;(p)==bﬂ(p) forallpeM. ¢$~ 400 C/zapter 10 I9.COROLLARY. IfGisaconnected Liegroup and ftG—>GisaC°° map preserving leftinvariant forms, then f=Leforaunique aGG. W’hile leftinvariant I-forms play afundamental role inthestudy ofG,the leftinvariant f2—fOI‘ITlS arealsovery important. Clearly, allleftinvariant n-forms areaconstant multiple ofanynon-zero one. IfU”isaleftinvariant n—form. then 0"determines anorientation onG,andiff:G—>IRisaC°° function with compact support, wecandeﬁne f faflo c Since 0”isusually keptfixed inanydiscussion, thisisoften abbreviated to _/éf or /Gf(a)da. The latter notation hasadvantages incertain cases. Forexample, leftinvariance of0”implies that /f(a) da=_/C f(ba) c/cz. 6 G /ifa” =/igo”, where g(a) 2-f(ba); G Ginother words. |note that Lbisanorientation preserving diffeomorphism, so fafo"-=LLg,*(j'o”)=_L(foL,=,)L,=,*cr"=--L(foL;,)o”. which proves theformula]. Wecan, ofcourse, also consider right invariant n-forms. These generally turn outtobequite different from theleftinvariant n-forms (seetheexample inProblem 25).Butinonecasetheycoincide. 20.PROPOSITION. IfGiscompact andconnected and0)isaleftinvariant )2-form, then toisalso right invariant. PROOF. Suppose to7E0.Foreach aEG,theform Re*w isleftinvariant. so there isaunique realnumber f(a) with Re*w =f(a)w. Since Re* 0R[,* 2(Reg,)*, wehave f(@b) =f(b@) ==f(-<1)- f(b)- Sof(G) CIRisacompact connected subgroup ofIR-{O}. Hence f(G) ={I}.92+ LieGroups 401 Vilecanalso consider Riemannian metrics onG.Inthecase ofa compact group Gthere isalways aRiemannian metric onGwhich isboth leftand right invariant. Infact, if(,)isanyRiemannian metric wecanchoose a bi-invariant n-form 0"anddeﬁne abi-invariant ((,onG (<1/.W»---[GG<L..Re.<v). L..Re.<W)) dadb. Weareﬁnally ready toaccount forsome terminology from Chapter 9. 2].PROPOSITION. LetGbeaLiegroup with abi-invariant metric. (l)Forany0EG,themap Ia:G—>Ggiven by]e(b) =ab_]a isanisometry which reverses geodesics through a,i.e.,ifyisageodesic and1/(0) =0,then fen/(=')) =r(—I)- (2)The geodesics ywith 1/(0) :-_-eareprecisely theI-parameter subgroups ofG,i.e.,themaps r|~—>cxp(IX) forsome X6gt. PROOF (I)Since If Z b—I. themap lee:Ge—>Geisjustmultiplication by-I(seetheproof ofProposi- tion l6(2)), soitisanisometry onGe.Since 1..=.-R,,_1IeL,,_1 foranyaEG,themap lee: Ge—>Ge-1 isalsoanisometry. Clearly Iereverses geodesics through e. Since Ia5:RaIeRa_]: itisclear that Ieisanisometry reversing geodesic through a. (2)Lety:IR—>Gbeageodesic with 3/(0)-=e.Forﬁxed t,let J7(u)=J/(I+M)- Then )7isageodesic and17(0) =1/(I). So I)/(t]Ie(Y(u)) =I)/(r)(}"("'l~*’)) =Iy(r)(l7(“‘u *0) =)7(z+:1):)/(u+2:). Butalso IY(IIIe(b) =1/(r)b1/(1). 402 Chapter 10 SO J/(‘)1/(u)J/(I) =J/(M+21‘)- ItFollows byinduction that y(nt) =3/(t)” foranyinteger n. If1’=11’: andI”=n”:forintegers n’andn”,then 14:’+:")=i»m"’+"” ==i»(:')i»<:”>= soyisahomomorphism onQ.Bycontinuity yisal—parameter subgroup. These aretheonly geodesics, since there arel—parameter subgroups with anytangent vector atI=0,andgeodesics through earedetermined bytheir tangent vectors att=0.0:0 Weconclude thischapter byintroducing some neat formalism which allows ustowrite theexpression fordwk inaninvariant way that does notusethe constants ofstructure ofG.IfVisad-dimensional vector space, wedeﬁne a V-valued k-form onMtobeafunction (0such thateach w(p)isanalternating map w(p): Mpx---><Mp—> V.\i........_-,_,i_.-I ktimcs Ifv1,...,vdisabasis forV,then there areordinary k-forms to‘,...,wd such that ForX1,...,Xk GMPwehave d . w(p)(X1,...,X;¢) =Zw'(p)(X;,...,Xk)v,-: :'=l wewillwrite simply d (L)-‘=5 wi°U|‘. II Forany V-valued k-form towedeﬁne aV-valued (k+1)-form dwby d dw=Zdwl -v,-; i=1 asimple calculation shows thatthisdeﬁnition does notdepend onthechoice of basis 11;,...vd forV. LieGroups 403 Similarly, suppose p:U><V—>Wisabilinear map, where UandVhave bases u;,.. .,ucand111,...,vd, respectively. IfwisaU-valued k-form C (UZE (1):-‘LII. i=l and T}isaV-valued I-form d n=_Zn”-vi, J=1 then c d Z:Z211)"/\ Tlj'P(I1;,v;) i=1 j=] isaW-valued (k+1)-form; acalculation shows thatthisdoes notdepend onthe choice ofbases u1,...,u,,or111,...,vd.Wewilldenote thisW-valued (k+1)- form byp(u)/\17). These concepts have anatural place inthestudy ofaLiegroup G.Although there isnonatural way tochoose abasis ofleftinvariant l-forms onG,there is anatural g1-valued l—form onG,namely theform todeﬁned by nu (=i=) a)(a)(X(a)) =XGg. Using thebilinear map [,]IQ><Q—>g,wehave, forany Q-valued k-form 27 andanyg-valued l—form XonG,anewg-valued (k+1)-form [1]/\A]onG. Now suppose that X;,...,X,, EGe=gisabasis, andthat w‘,...,w" isa dual basis ofleftinvariant l-forms. The form todeﬁned by(*)canclearly be written n w=Zwk -Xi. )i.'=] Then (1) dwzzdwk-Xk R-=1 =i:(Z:C,-ljwi /\wj) -Xi. ,l¢=] i<j 404 Chapter 10 Ontheother hand, It tnm=Zqnkal S0 It II II (2) [w/\w]=Z(ZZc,-’j-w’/\w1-X,,). R-=1 i=1 j-=1 Comparing (l)and(2),weobtain theequations ofstructure ofG: The equations ofstructure ofaLiegroup willplay animportant role in Volume III.Forthepresent wemerely wish topoint outthattheterms dwand [to/\cu]appearing inthisequation canalsobedeﬁned inaninvariant way. For theterm do)wejustmodify theformula inTheorem 7-l3: IfUisavector ﬁeld onGand fisag-valued function onG,then (Problem 20)wecandeﬁne a Q-valued function U(f) onG.Ontheother hand, w(U) isaQ-valued function onG.Forvector ﬁelds UandVwecanthen deﬁne dw(U, V)=U(w(‘/)) -V(w(U)) -—w([U= V1)- Recall thatthevalue ataeGofthe right sidedepends only onthevalues Ua and VaofUand Vata.Ifwe choose U=X,V2Yforsome X,Y EGe, then nun“ nun“ dw(a)(X,,, 11,)h0-0-w(a)([X, 1/1,) =—-w(e)([)?, ﬁe) since[52,?]isleftinvariant =—-w(e)([X, Y]) bydeﬁnition of[,1in0,. =‘[X,Y] } ..... -_ bydeﬁnition ofa). =~[w(¢1)(Xa),w(0)(Ya)] Itfollows thatforanyvector ﬁelds UandVwehave Ida)fU, V)=. i/-[t.-)(U),ti)(V)]. i Problem 20gives aninvariant deﬁnition ofp(w/xn) andshows thatthisequation isequivalent totheequations ofstructure. LieG?'0i!1j).s 405 WARN INGIInsome books theequation which wehave justdeduced appears asdw(U, V)=——-%[w(U),w(V)]. The appearance ofthefactor -£5here has norlzing todowith the%intheother form ofthestructure equations. Itcomes about because some books donotusethefactor (k+1)!/kl 1!inthedeﬁnition of/\.This makes their AAqequal to-%ofours forl-forms Aandq.Then the deﬁnition ofd(Z to,-dxl) asZdw; Adxlmakes their dwequal to-Eofours forl-forms to. 406 Chapter 10 PROBLEMS 1.LetGbeagroup which isalsoaC°°manifold, andsuppose that(x,y)i->x_r isC°°. Find f_l when ftG><G—>G><Gisf(x,y) =(x,xy). Show that (e,e)isaregular point off. (Conclude thatGisaLiegroup. ..9,@€ 2.LetGbeatopological group, andHCGasubgroup. Show thatthe closure HofHisalsoasubgroup. 3.LetGbeatopological group and HCGasubgroup. N()IfHisopen, then soisevery coset gH. (b)IfHisopen, then Hisclosed. 4.LetGbeaconnected topological group, andUaneighborhood ofeeG. LetU”denote allproducts 0;---0,, fora,-eU. ()Show that U""" isaneighborhood ofU". ()Conclude that U"U"=G.(Use Problem 3.) ()IfGislocally compact andconnected, then Gistr-compact. “UN 5.Letf:IR"—>IR"bedistance preserving, with f(0) =0. Q-@@@Show that ftakes straight lines tostraight lines. Show thatftakes planes toplanes. Show thatfisalinear transformation, andhence anelement ofO(n). ()Show thatanyelement ofE(n) canbewritten A-rforAeO(n) and ra translation. 6.Show thatthetangent bundle TGofaLiegroup Gcanalways bemade into aLiegroup. 7.Wehave computed that forMEgI(n,]R) wehave ... ... 3 ... "M=ZMi-"*'-W. whereMXWA) =ZM,,A,,,,. kI Ot=l (a)Show thatthismeans that M(A) =A-MGR-GL(n,]R),;. nu (Itisactually clear afmiori that Mdeﬁned inthisway isleftinvariant, forL,;,,. = LAsince LAislinear.) (b)Find theright invariant vector fieldwith value MatJ. LieGroups 407 8.LetGandHbetopological groups and¢:U—>Hamap onaconnected open neighborhood UofeeGsuch that ¢(ab) =¢(a)¢(b) when a,b,ab EU. (a)Foreach ceG,consider pairs (V,(Zr),where VCGisanopen neighbor- hoodofcwithV-v-IcU,andwhere11»;v_>Hsatisﬁes 11/(0)-u/(1>)~' = ¢(ab_') fora,b GV.Deﬁne (V;,(lr1) Q»(V2,(lr2) iftr,.-=1//2onsome smaller neighborhood ofc.Show thatthesetofall '2,»equivalence classes, forallc6G, canbemade intoacovering space ofG. (b)Conclude that ifGissimply-connected, then Q5can beextended uniquely toahomomorphism ofGintoH. 9.InTheorem 5,show that ¢and 11/areequal even ifthey aredeﬁned only onaneighborhood UofeEG,provided that Uisconnected. 10.Show thatCorollary 7isfalse ifGisnotassumed connected. 11.IfGisagroup, wedeﬁne theopposite group G°tobethesame setwith themultiplication ~defined bya»b=b-a.IfQisaLiealgebra, with operation [,],wedeﬁne theopposite Liealgebra 41°tobethesame setwith theoperation [X,Y]°=-[X, Y]. (a)G°isagroup, and iftl/:G—>Gisai~—>0-], then (Irisanisomorphism from GtoG°. (b)g°isaLiealgebra, andXI->——Xisanisomorphism ofgonto g°. (c).£(G°) isisomorphic to[.£(G)]° =g°. (d)Let[,]betheoperation onGeobtained byusing right invariant vector ﬁelds instead ofleft invariant ones. Then (g,[ ,])isisomorphic to£(G°), and hence tog°. (e)Use thistogive another proof that gisabelian when Gisabelian. exp = . .""' ""'SID G COS G12.(a)Show that 0 a cosa sina a0 (b)Usethematrices Aand Bbelow toshow that exp(A +B)isnotgenerally equal to(exp A)(exp B). 01 00 A400)B»-(10)13.LetX,YeGewith[X,Y]=0. (a)UseLemma 5-13 toshow that (exp sX)(exp!Y) =(exp!Y)(exp sX). (b)More generally, useTheorem 5toshow that exp isahomomorphism onthesubspace ofGespanned byXand Y.Inparticular, exp(X +Y)= (expX)(exp Y). 408 Chapter 10 14.Problem l3implies that exp!(X +Y)=(exptX)(exptY) if[X,Y]=0.A more general result holds. LetXand Ybevector ﬁelds onaC°° manifold M with corresponding local l—parameter families oflocal diffeomorphisms {tin}, {ti/5}. Suppose that [X,Y]-.=0,and let1],=qt‘),Oti/;=(Zr,0¢,-. (a)Show that %=Xm.-om +¢,..tYu1o<p))- (b)Using Corollary 5-l2, show that %=Xmm) +Y(m(P))- Inother words, {1],} isgenerated byX+Y. 15.(a)IfMisadiagonal matrix with complex entries, show that detexpM=etracc M (b)Show thatthesame equation holds foralldiagonalizable Mwith complex entries. (c)Conclude that itholds forallMwith complex entries. (The diagonalizable matrices aredense, compare Problem 7-l5.) (d)Using Proposition 9,show thatforthehomomorphism det: GL(n,R) —> R-{O},themap det,,.: gI(n,R) —>.-.C(R -{O}) -=Risjust Ml->trace M. (e)Use this fact togive afancy proof that trace MN =trace NM. (Look at trace(MN -—-NM) -=trace[M, (f)Prove theresult inpart (d)directly, without using (c).(Since det... andtrace arehomomorphisms, itsuﬁices tolook atmatrices with only onenon-zero entry.) (g)Now usetliisresult andProposition 9togive afancy proof of(c). 16.(a)LetUbeaneighborhood oftheidentity (1,0) ofSI(considered asa subsct ofR2). Show that nomatter how small Uis,there areelements aeU which have square roots outside Uinaddition totheir square root inU. (b)Show thatforeach n?_l,there isaneighborhood UofeeGsuch that eveiy element inUhasaunique nil‘root inU. (c)ForG=S1,show thatthere isnoneighborhood Uwhich hasthisproperty foralln. 17.(a)Let(x,V)beacoordinate system around e6Gwith X'l(€) =0.Let xttob) =f='(it»‘(o),...,1-"(o),x‘(o),...,,\-"(on LieGroups 409 forC°°functions fl.Show that D,-f"(0) =o.+.-fro) =6;! (b)Ifoz,,B: (-—s,e) —>Garedifferentiable, show that (¢1'l5’)"(0)= t1"(0)+ l3'(0)- (c)Also deduce thisresult from Theorem l4-(l). (Not even thefullstrength of(l) isneeded; itsuﬁices toknow that exptX expi'Y =exp{I(X +Y)+0(1)}. The argument ofpart (a)isessentially equivalent totheinitial part ofthededuction of(l).) 18.LetGbeaLiegroup, andletHCGbeasubgroup ofG(algebraically), such thatevery aEHcanbejoined toebyaC°°path lying inH.LetI]CG_._. bethesetoftangent vectors toallC°° paths lying inH. (a)Show that I]isasubalgebra ofGe.(Use Theorem l4.) (b)LetKCGbetheconnected Liesubgroup ofGwith Liealgebra I].Show that HCK.Hint: _]oin anyaEHtoebyaC°°curve c,andshow that the tangent vectors ofclieinthedistribution constructed intheproof ofTheorem 4. (c)Letc-;,...,c;, becurves inHwith {c,-"(0)} abasis forX].Byconsidering themap f(tl,...,t*) =c;(!')---c;,(tk), show that KCH.Thus, HisaLie subgroup ofG.Itiseven truethat HCGisaLiesubgroup ifHispath connected (bynotnecessarily C°°paths); seeYamabe, Onanarcwise connected .mbg2'oup ofaLiegroup, Osaka Math._]. 2(1950), l3——l4-. (d)IfHCGisasubgroup andanimmersed submanifold, then HisaLie subgroup. 19.Fora6G,consider themap bt~—>aba_' =L,,R,,_'(b). The map (LaRa_])#3 Q_*ii isdenoted byAd(a); usually Ad(a)(X) isdenoted simply byAd(a)X. (a)Ad(ab) -=Ad(a)oAd(b). Thus wehave ahomomorphism Ad:G—>Aut(g), where Aul(t1), theautomorphism group ofg,isthesetofallnon-singular linear transformations ofthevector space Qonto itself (thus, isomorphic toGL(n,R) iftihasdimension n).The map Adiscalled theadjoint representation. (b)Show thal exp(Ad(0)X) =(I(CK]JX)o-‘. Hint: This follows immediately from oneofourpropositions. 410 Chapter 10 (c)ForAEGL(n,R) andMEgI(n,R) show that Ad(A)M =AMA-‘. (Itsufﬁces toshow thisforMinaneighborhood of0.) (d)Show that Ad(exp:X)Y =Y+t[X,Y]+0(8). (e)Since Ad: G—>g,wehave themap ____ tangent space ofAut(g) atthe AdeQ(rGe)"*identity map1,,oftitoitself. This tangent space isisomorphic toEnd(§1), where End(g) isthevector space of alllinear transformations ofgintoitself: Ifcisacurve inAut(g) with 0(0)=lg, then toregard c"(0) asanelement ofAut(g), weletitoperate onYeQby c’(0)(Y) =5; c(Y).{=0 (Compare with thecase gt=R",Azu!(t1) =GL(n,R), End(g) =12><nmatrices.) Use(d)toshow that Ad,.,.(X)(Y) =[X,Y]. (Aproof may also begiven using thefact that [X,Y] -=Li:-Y.) The map Yl-->[X,Y]isdenoted byadXGEnd(g). (f)Conclude that dX2Ad(expX)=exp(adX)==lg+adX+-{if-+-~. (g)LetGbeaConnected Liegroup and HCGaLiesubgroup. Show that H isanormal subgroup ofGifandonly ifI]=.£(H) isanideal ofti=.£(G), thatis,ifandonly if[X,Y]EI]forallXGQ,YeI]. 20.(a)Letf:M—>V,where Visaﬁnite dimensional vector space, with basis v;,...,vd. ForXpEMp, deﬁne X,,(f) GVby d X(f)=Z/r,.tf‘) -v.-.I'-=1 where_/=3;,fl-v,-forf‘;M_>ts.Showthatthisdeﬁnition isindepen- dent ofthe choice ofbasis v1,...,vdforV. LieGroups 411 (b)Ifo)isaV-valued k-form, show that do)may bedeﬁned invariantly bythe formula inTheorem 7-l3 (using thedeﬁnition inpart (a)). (c)Forp:U><V—>W,show that p(o)/\1))may bedeﬁned invariantly by p(w A ''>Xk>Xk+1: '°'IXk+f) l Z 2 58715 'p(w(XU(i]: ''°9XU(»iC])I 7l(Xo'(it+1)> °°'>XU(»iC+f]))' O'ESj,-_|__i Conclude, inparticular, that toA<»1<X.Y)=2tw<X).t»tY)1- (d)Deduce thestructure equations from (b)and(c). 21.(a)Ifo)isaU-valued k-form and27isaV-valued I-form, andp:U><V—> W,then dUMvAm)=oMwAn%+P4VmwAdm- (b)ForaQ-valued k-form o)andI-form 1)wehave toAto=<-1)’*’+‘[11Aw]- (c)Moreover, ifKisag-valued m-form, then (—l)l""[w /\[T7/\ll]+(—l)“[H /\ll/\wll+(—l)""' [1/\[H/\wll=0- 22.LetGCGL(n,R) beaLiesubgroup. Theinclusion map G—>GL(n,R) —> R"2willbedenoted byP(for“point”). Then dPisanR"2-valued l—form (it corresponds totheidentity map ofthetangent space ofGinto itself Wecan also consider dPasamatrix ofl-forms; itisjust thematrix (dxlj), where each dxij isrestricted tothetangent bundle ofG.Wealso have theR"2-valued l—form (ormatrix ofl-forms) P4 -dP, where -denotes matrix multiplication, and P_1 denotes themap AI-->A-1 onG. -'-dP=p(P_‘AdP),wherepixi“><n"’_>n"*ismatrixmultiplication. dP=A-dP.(Use f*d =df*.) dPisleftinvariant; and (dP) -P‘! isright invariant. dPisthenatural (1-valued l—form o)onG.(Itsuﬁices tocheck that F*oP=oo1)(e)Using dP=P-o),show that 0=dP-o)+P-do), where thematrix of 2-forms P-do)iscomputed byformally multiplying thematrices ofl-forms dP ando).Deduce thatI§@@€wiqw do)+o)-o)..-=0. 412 Chapter I0 Ifo)isthematrix ofl-forms o)=(o)"j), thissays that do)” =—Z10)“ Ao/‘j. k Check thatthese equations areequivalent totheequations ofstructure (usethe form do)(X, Y)=—[o)(X),o)(Y)].) 23.LetGCGL(2,R) consist ofallmatrices ('3 with a#0.Forconve- nience, denote thecoordinates xi‘andX12onGL(2,R) byxand)2 (a)Show that forthenatural g-valued form o)onGwehave ldx dy 0’"'Z(00 sothat dx/x and dy/x areleflinvariant l-forms onG,and aleftinvariant 2-form is(dxAdy)/xi. (b)Find thestructure constants forthese forms. (c)Show that MP) _P_, 2 —ydX0+xdy) andﬁnd theright invariant 2-forms. 24.(a)Show thatthenatural gl(n, R)-valued l—form o)onGL(n, R)isgiven by .. 1 ". .:1: ticdJr;(U Ldet(xaﬁ) A > where tr”)=dowel’) -tee)". (b)Show thatboth theleftandright invariant n2-forms aremultiples of l_.......__....ﬁ.(d-1' d"1 (d-1" d-"1" _ (dct(xaﬁ)) itA Ax)A AAA Ait) 25.The special linear group SL(n,R) CGL(n,R) isthesetofallmatrices of determinant l. (a)Using Problem l5,show thatitsLiealgebra -3I(n,R) consists ofallmatrices with trace =0. LieGr0up.s 413 (b)Forthecase ofSL(2, R),Show that _;_ __ vdx—ydu vdy~—ydv P dP—(*—LldJt‘+IdLl —-udy+xdv ’ where weusex,y,u,vforxi], x12, x2‘, X22. Check that thetrace is0by dillcrcntiating theequation xv—yu=l. (c)Show that aleftinvariant 3-form is vdx/xdu/\dy-ydx/xdu/\dv. 26. ForM,N E13(12): £(O(n)) ={MIM=—M‘}, deﬁne (N,M)=—traceM-N‘. ((,)isapositive deﬁnite inner product on0(n). (b)IfAEO(n), thenI-‘J\-_/ (Ad(A_)M,Ad(A)N) =(M,N). (Ad(A) isdeﬁned inProblem l9.) (c)The leftinvariant metric onO(n) with value (,)atO(n) ,1isalso right invariant. 27.(a)IfGisacompact Liegroup, then exp: g—>Gisonto. Hint: Use Proposition Ql. (b)LetAESL(2,]R). Recall that Asatisﬁes itscharacteristic polynomial, so A2—(trace A)A +I=0.Conclude that trace A22—2. (c)Show that thcfollowing element ofSL(2, R)isnotA2forany A.Conclude thatitisnotintheimage ofexp. -2 0 (0-I/2)(d)SL(2,]R) does nothave abi-invariant metric. 28.Letxbeacoordinate system around einaLiegroup G,letrrj:G><G—>G betheprojections, andlet(y,z) bethecoordinate system around (e,e) given byJr‘=xi031'], :5=x’orrg. Deﬁne Q5":G><G—>Rby ¢i(a,b)==X’_(r1b), 414 Chapter 10 andletX,-betheleftinvariant vector ﬁeld onGwith 3 Xi(£’) == (a)Show that H _ 8 Xi‘*Z11’?_ BA11:2] where _ 111;‘(<1)=33%-ta,e). (b)Using L,,Lb =Lab, show that '[La*Xi(b)l(xI) =tXiwb>1<x’). Deduce that X,-(b)(x" 0La)=11/;(ab). andthen that -_ _ Hj 34>" 1Zr,<1»)-gum») -=it-(ab).i=1 “ Letting 1}==(#21:) betheinverse matrix of1!!=(diff), wecanwrite 34"! H1 w" I= This equation (oranyofnumerous things equivalent toit)isknown asLz'e’s_ﬁrs1 fundamental Z/zearem. The associativity ofGisimplicitly contained init,since we used thefactthat LaLb=Lab. (c)Prove theconverse ofLiesﬁrstfundamental 1/zearem, which states thefollowing. Let¢=(qﬁl,...,¢") beadifferentiable function inaneighborhood of0eR2" [with standard coordinate system yl,...,_1=",:1,...,::"] such that ¢(a,0) --=a foraGR". Suppose there aredifferentiable functions glrjinaneighborhood of0eIR"[with standard coordinate system xi,...,x"]such that 1!/}(0) =5} 3¢'i __H; ~; for(a,b)inaneighborhood(*>‘ﬁ(a>b)"-i%¢;'(¢((?,b))°¢j(b) OMGR2, Lie(,-'r0uj).\ 415 Then (0,b)|—->¢(a, b)isalocal Liegroup structure onaneighborhood of0eR" (itisassociative andhasinverses forpoints close enough to0,which serves as theidentity]; thecorresponding leftinvariant vector ﬁelds are U Ia X;= -J-—..12¢’ 32:1 [Toprove associativity, note that a*'((,b,;- " -,.-‘L-3ia_i’,%) =Z11/l<¢<¢<a.b>.z>>-v,»<z) hr(*1!=l andthen show that ¢(a, ¢(b, 2))satisﬁes thesame equation.] 29.Lie’:secorzdfmdamenial tfceorem states that theleftinvariant vector ﬁelds X5of aLiegroup Gsatisfy H [X,-,X,-1=ZCf;-Xi k=l forcertain eonstarzts C,-‘5—in other words, thebracket oftwoleftinvariant vector ﬁelds isleftinvariant. The aim ofthisproblem istoprove theconverse qfLie’s secondﬁmdanzental theorem, which states thefollowing: ALiealgebra ofvector ﬁelds onaneighborhood of0eIR",which isofdimension :2over IRandcontains a basis forlR”@, isthesetofleftinvariant vector ﬁelds forsome local Liegroup structure onaneighborhood of0eIR". (a)Choose X},...,X”intheLiealgebra sothat X,-(0) =8/Bxilo andset n _8 I1,2“ 8x1 lfN (Oi=Z dA'j, j=1 then thewfarethedual forms, andconsequently dwk=~—ZC5»cu’/\w’ Cf‘;constants. :'<j (b)LetIr,-:R"><IR"—>R"betheprojections. Then H I1 Irfwj -—I-n*w" =Z:(ihj °F2)[d(-Y’ 0rm)-Z30!/i 0Hz)--7Tl*(9i:|- 1:1 f=I 416 Clzopter 10 Consequently, theideal generated bytheforms d(x" orrz)-— XL, (tlrforrg)-rr1*to" IF isthesame astheideal J3generated bytheforms rt;to}-—rr;*wj. Using thefact thattheC15,,areconstants, show thatd(1)C1.Hence 11?."xR”isfoliated by I?-dimensional manifolds onwhich theforms d(x"orrg)—2;, (115?01:2) -rr;*w" allvanish. (c)Conclude, asintheproof ofTheorem l7,thatforﬁxed a,there isafunction (Dal R"—>R"satisfying (1),,(0)=aand N d:l<b)=Zvi<<1>..<b))-Mb).II orequivalently, 9-i1= r/lieon-rm) aA’j I a J I Now set¢(a,b)=a(b), andusetheconverse ofLie’s ﬁrstfundamental theo- rem. 30.Lie’: tho-dﬁmdanzerzlal tizeormr states that theCJlksatisfy equations (l)and (2) onpage 396, i.e.,that theleftinvariant vector ﬁelds form aLiealgebra under [,].The aim ofthisproblem istoprove theconverse ofliefs" t/zirdfundomentaf .‘,l;ri0!'el??, which states that anyn—dimensional Liealgebra istheLiealgebra for some local Liegroup inaneighborhood of0eR". LetCgbeconstants satisfying equations (1)and(2)onpage 396. Wewould liketoﬁnd vector ﬁelds X;,...,X,,onaI1Clgl1bO1“l1OOCl of0eR"such that [X,-.X,-]=22:, C,-'5Xi.Equivalently, wewant toﬁndforms to‘with dwk =-ZC5to‘/\wj. I‘<1 Then theresult willfollow ﬁom theconverse ofLie’s second fundamental the- orem. (a)Let/Ifbefunctions onIR><IR“such that PL’:-5*-Zak W 8! _ I‘ -- 1" I-J /iffto,A‘)=0. These areequations “depending ontheparameters x”(seeProblem 5-5(b]]. Note that hf(r,0) =rift,sothathf(l,0) =5;‘.Letoi‘bethel—form on1R><R" deﬁned by Gk=231,1.‘ dx’. LieGz'0up.~ 417 and write dak =ll‘+(drAdk). Where ll‘andarkdonotinvolve dz.Show that ahl Bhl ..k_ _____._!_ _______5_ . .It_ W.)axAdi! ak =dxk ___ xl'0.j_ I1] (b)Show that av=at/\(-Z0};dxi/ta)‘-20,-’;xklj). HJ I-J (c)Let 6*=v<+%Zq§s="/isa:',j Show that d6"=an/\(_cf;i-W‘-Zcf}cj,i-we Aoi):,_) :,_; r.. +terms notinvolving dr. Using . . l . . ZZC.-’;-em‘ A<1’=5ZZ<e.-’;-Ci. ~e;i‘.e:,->0‘ MS 1"S I-aj r‘ l-Ij ' : + AU;_; rs andequation (2)onpage 396, show that d9"=d:/\(-21¢“,-'j-x"l" +%ZZC',-’f.C_,l-xros/xol)J is]. lirj as +terms notinvolving dr. Finally deduce that eel=dfA-Zc,’;x1o’ +termsnotinvolving dl. j,I 418 Clzapter 10 (d)Wecanwrite 6*=Egg‘,-dx' /\dXJ. i<j where gf‘J.(0. x)=0(Why-?). Using (c),show that agi‘. k "TIL="ZCr:“gir-r,s Conclude that 6*=0. (e)\'\"enow have ] . .It_ k}\ _III l - . ea‘-=-52c,-'j-0' Ac’+(drmi‘) iii Show that theIorms to/"(.\') =ok(l,x) satisfy dwl‘=——%ZC,-‘lim’./\wj. 5,] CHAPTER ll EXCURSION INTHE REALM OFALGEBRAIC TOPOLOGY Tl1lS chapter explores further properties ofthedeRham cohomology vector spaces ofamanifold. Ourmain results willberestatements, interms ofthe deRham cohomology, offundamental properties oftheordinary cohomology which isstudied inalgebraic topology. Because wedeal only with manifolds. many oftheproofs become signiﬁcantly easier. Ontheother hand, wewillbe using some oftheinain tools ofalgebraic topology, thusretaining much ofthe flavor ofthatsubject. Along theway wewilldeduce allsorts ofinteresting con- sequences, including atheorem about thepossibility ofimbedding n-manifolds in]R"’+‘. LetMbeamanifold with M=UUVforopen setsU,V CM.Before examining thecohomology ofMwewillSimply lookatthevector space Ck(Ml ofk~forms onM.Let iU:U—>M IV:I/—>M jU:UﬁV—>U j;/:UﬁV—>V betheinclusions. Then wehave twolinear maps orand15. ='*a9'-* ="*-'-=' C"(M)-————-M i”"c'<<u)eec’<(v)i>5 J”JPC"(UnV) deﬁned by =I(w)=(1't/*(w),iv*(w)) ﬁ{Al1A‘2) =ju*(M) -J'v*(l2)- Here iU*(w) isjust therestriction oftotoU.etc. Clearly /3oor=01lnother words, iinageor Cker;5’.Moreover, theconverse holds: kerb’ Cimage or.For, if;B(M,l;) =0,then it;=lgonUHV.sowecandeﬁne toonMtobeM onUandkgonV.andthen a(w) =(}.;,}tg). Theequation imagea =kerﬁ is expressed bysaying thattheabove diagram isexact atthemiddle vector space. Wecanextend thisdiagram byputting thevector space containing only 0atthe 419 420 Chapter 1] ends; thearrows ateither endofthefollowing sequence aretheonly pOSSiblt- linear maps. 1.LEMMA. Thesequence . 50+CHMj£+@ﬂH$CH@—+CMUﬁV}+0 isexact atallplaces. PROOF. ltisclear thatorisone-one. This isequivalent toexactness atCl‘(M). since theimage ofthe ﬁrstmap is{0}CC"(M). Similarly, exactness atCk(UO V)isequivalent toﬁbeing onto. Toprove that ﬁisonto, let{¢u,¢V} bea partition ofunity subordinate to{U,V}.Then toGCk(U PlV)is w=l3(¢vw, -¢uw)- where ¢|/to denotes theIorm equal togbvw onUF1V,and equal to0on U-(Un1/).+:~ Byptittin ginthemaps (1.wecanexpand ourdiagram asfollows, r r ‘~ -Q wl oi» C’*'(U)e|aC"(v) C"(Unv)-_>0 le leee Q w -1 w sothat therows areallexact. ltiseasy tocheck that thisdiagram commutes. thatis,anytwocompositions from onevector space toanother areequal: in (c/&Bd)oa=aod jdead =dl a J1.= I = (foﬁ fj'o(dEB£) d easel 5 5 * ];'.t*cur.n'02? intheRealm of/l{_gt*b?'0r'r 7opol0g_3' 421 Our ﬁrstmain theorem depends only onthesimple algebraic structure in- herent inthisdiagram. Toisolate thispurely algebraic structure, wemake thefollowing deﬁnitions. Acomplex Cisasequence ofvector spaces Cl‘. k=0,1,2,... ,together with asequence oflinear maps e*=cl->c"+‘ Satisfying dk‘H oelf‘=0,orbriefly, c/2=0.Amap or:C;—>Cgbetween complexes isasequence oflinear maps 0/<1c,'<->cf such that thefollowing diagram commutes forall1:. 1: C3!‘ t eel |e-*I.C-‘k+I 0/‘J’ C-_,k+1 Themost important examples ofcomplexes areobtained bychoosing C‘: Ci‘(M) forsome manifold M,with dl‘theoperator donk-forms. Another example, implicit inour discussion, isthedirect sum C=C1€BC;oftwo complexes, deﬁned by ct‘=c,’*'esC;"‘. e’*=eﬁ‘@e2’~'. Foranycomplex Cwecandeﬁne thecohomology vector spaces ofCby . kerell‘HA =L". (C) image dbl Nz1tu1'ally, ifC={C"'(M)}, then Hk(C) isjustHl‘(M). lfor:C;—>C;isa map between complexes. then wehave amap, alsodenoted byor. or:H"(C1)-> H"(C3). Todeﬁne orwenotc that every clement ofH"'(C1) isdetermined bysome xeCii‘with d/‘(xl =0.Commutativity oftheabove diagram shows that d2"(or"‘(x)) =0z""'la';"'(.r) =0,soo/‘(x) determines anelement ofH"‘(C2), which wedeﬁne tobeor(the class determined byx).This map iswell-deﬁned. 422 (,7zap£er 11 forifwe change xtox+d;""" (y)forsome yEC/‘“', then a*(x) ischanged to <1"‘<>t-+di’<"‘o)) -=aw)+a"<di""‘om =am+d2"*‘<<>/‘"‘o)>. which dctermines thesame element ofHk(C;). W'hen C1,‘ .-=Ck(M), C2)‘ = cmv). and<1;C"(M) _>ckov) isf*forf;N->M,thenthismapisjust _{*¢H"(M) _>Hmv). Now suppose thatwehave anexact sequence ofcomplexes :1 I50>C1 >C2 >C3 >0. which really means avastcommutative diagram inwhich allrows areexact. J_t_,JR‘--I i--I ﬁk Cglk-'i 0 ‘d‘k-I “dgk-I d3!»--I ~. A -. A, (?5‘f e s >C1A —’-Q e—)C2k —'-g e IfI"‘+ 13+ C-3k+l 0 t I ~ ~; 1 \'\"haIdoesthisimplyaboutthemaps=1;H’~'(c,) _>H’*'(c2) and5;H"tc2) _> H""(C3)? The nicest thing that could happen would beforthefollowing dia- gram tobcexact: It ‘Y k '5 t-0->H(C1)—> H(C2)—> H(C3)->0. This isnottrue. Forexample, ifUand Vareovei'lappi11g portions ofS2for which there isadeformation retraction ofUOVintoS],then wehave anexact sequence 0—>c’*<s2> —>c"e1s»c’“<v> ->C’“<voV)—>0- ‘ﬁg;~51»“cg-in as-.33¢k,-ﬁﬁ-".1,§~;‘r<'C"! F E.xcum'm2 intheRealm qf./1Zgebmic 'Ttip0}0_g_3~ 423 butnotanexact sequence '(s1) HI(U)oH‘(v) H‘(t1nv)?>0. 22 22 22 0 0 R Nevertheless, something very nice istrue: I52.THEOREM. lf0>CIa>C; >C3>0isashort exact sequence ol complexes, then there arelinear maps 6"‘:H"<c3)—>H"+‘(ci> sothat thefollowing inﬁnitely long sequence isexact (everywhere): 0 Q‘ o H 0 5 10—> H(Ci)—> H(C2)—> H(Cs)—>H (Ci)—>--' 5 —>H"<c1)LH"‘<c1)LHk(C3)—>H"+‘<ci) —> PROOF. Throughout theprooﬂ diagram (>1=)should bekept athand. LetxE Cgkwith cl3"(x) =0.Byexactness ofthemiddle row of(*),there isyeC31‘ with ﬁk(y)=x.Then 0=613*to=613*ﬁ"o)=ﬁ"‘*‘di*<y)- Sod2"(y) eker)3"+I =iITl21gCCt/k+I§ thus d;"‘(_v) =0/‘+I(z) forsome (unique) :eC1""'I. Moreover. __= d2k+]ak+I(z) Z __: Since a"+] isone-one, thisimplies that d1"+I(::) =0,so2determines anele- ment ofHk+](C1): thiselement isdeﬁned tobe5!‘ofthe element ofHk(C3) determined byx. lnorder toprove that 5"iswell-deﬁned, wemust check that theresult does notdcpend onthcchoice ofxECgkrepresenting theelement ofH"(C3). So wehave toshow that weobtain 0eH"+I(C1) ifwestart with anelement of thelorm d3k'I(x’) forx’eC3"“I. lnthiscase, letx’=ﬁ"“I(y’)_ Then A,:d3k—I(x:) =d3k—-Iﬁk—-I(y.') :ﬁkd2k—-I(yr)’ sowechoose dgk—](_]") as_1'.This means thatd;"'(_1-') =0,andhence :=0. 424 Chapter 1] ltisalso necessary tocheck that ourdeﬁnition isindependent ofthechoice ofywith ﬁ"‘(y) =x;thisislefttothereader. Theproof thatthesequence isexact consists of6similar diagram chases. We willsupplv theproof that kercr (Iimage 5.LetxECiksatisfy di"(x) =0,and suppose that cri"(x) ECf‘represcnts 0EHk(Cg). This means that a"‘(x) .-= d2"‘I(y) Iorsome yECgk—I. Now d3k--Iﬁk—-I(J,) :ﬁkd2f<.'—-I(y) :ﬁkaI¢'(x) : So;5’i"“I(_1’) represents anelement ofH"“I(C3). Moreover. thedeﬁnition of5 immediately shows thattheimage ofthiselement under 5isprecisely theclass represented byX.Q‘ ltisaworthwhile exercise tocheck thatthemain stepintheproof ofTheo- rem 8-l6 ispreciselv theproof thatkeror Cimage 5,together with theﬁrstpart oftheproof that 5iswell-deﬁned. AllofTheorem 8-l6 canbederived directly from thefollowing corollanj ofLemma land Theorem Q. 3.THEOREM (THE MAYER-VlETORlS SEQUENCE). lfM=UUV, where UandVareopen. then wehave anexact sequence (eventually ending in0’s): u-=H°tM> -—~H"(M) -H"(U)EB H"(V) ->H"(unviQH"+'(M) ->. Asseveral oftheProblems show. thecohomology ofnearly everything can becomputed byasuitable application oftheMayer-Vietoris sequence. Asa simple cxamplc. weconsider thetorus T=Si><S',and theopen sets U and Villttstratcd below. Since there isadeformation retraction ofUand V U ...-A,.,-- 1/ - £2‘?-5' ’“s§“§§§\' ‘J.'=_-, ‘-5' “*“ _‘*.xi“‘'=;-,. £5 1.-». ‘iviix--=-=»».~\ g,_-' »-- ,--.-.-- .. ,» '<_»\_---. -’z§t£‘!z> i’_,..1:-“‘ .‘, Q l"_""\’-‘.3? if __"1-.._ >33, ' -iii-5.=" ‘~t .-:¢-.-.- ~?"*"I:»¢;=."’.¢ 1,qi._. .~?,;‘_\-,_(.;;~i-.1 is ,‘_._ _§;;;5¥.,,§,y- #1, mi ""“‘q=:' 1-.lg:.’.' TI“"$3.'“~".-.‘}"<. ..r.:-- :{ ,,,_-:1-' 2"wt -51‘-z..- onto circles. andadeformation retraction ofUOVonto 2circles. thel\'Iayer- Vietoris sequence lr~ E.1'cttm'0?i intheHeals: if/llgebra2'c Ybpologt 425 —>H°(T) —>H°(U) esH°(V} —>H°(Unvi—>H‘m —>H'tu) esH‘(V) -_» Rt 2? ts rt -~»»H‘(Uovi—~+H2(T) —>0. €= BE lR<€BlR [R The map HI(U HV)—>H2(T) isnot0(itisonto H2(T)), soitskernel is l~dimensional. ThustheimageofthemapH1(U)EB H‘(v) _>H‘(unv) isl-dimensional. Sothekernel ofmamap isl—dimensional. andconsequently themap HI(T) —>HI(U) EBHI(V) hasal~dimensional image. Similar rea- soning shows that thismap also hasa1-dimensional kernel. ltfollows that climH1(T)=2.The reasoning used here canfortunately besystematized. 4-.PROPOSlTlON. lfthesequence or 0>V1 >1/2 >--- >Vk_1—>V,1.—>O isexact, then 0=dimV1—diml/2 +dimv3----+ r-1)""1 dimv,.. PROOF. Byinduction onk.Fork=lwehave thesequence O—>V1—>0. Exactness means that {0}CV1isthekernel ofthemap V1—>0,which implies that V1= Assume thetheorem fork-—I.Since themap V2—>V3haskernel a(V1), it induces amap V2/cr(V1) —>V3.Moreover, thismap isone-one. Sowehave an exact sequence ofk—-Ivector spaces O—> V2/a(V1) —>V3—> —>Vk_.1—> V1,—>0: hence 0-=dlITlV2/tI(V1)"'C'liITlV3+--> 7?‘ 0:0 which proves thetheorem for 426 Chapter 1] Rather than compute thecohomology ofother manifolds, wewillusethe Mayer-Vietoris sequence torelate thedimensions ofH"(M) toanentirely dillerent setofnumbers, arising from a“triangulation” ofM,anewstructure which wewillnow deﬁne. The standard n-simplex A,,isdeﬁned astheset A1,,={XelRI"+l :0_*Exi§ land Z;:1lx“=l}. A2 A A0 ‘/I / \AF i ll 1 (lnProblem 8-5. A1,,isdeﬁned tobeadiﬁ'ei'ent_ although homeomorphic, set.) Thesubset ofA,,obtained bysetting :2—kofthecoordinates xiequal to0 ishomeomorphic toA1.andiscalled aA"-face ofA,,. lfACMisadif- Ieomorphic image oI'some Am, then theimage ofak~face ofAm iscalled a k-face ofA.Now byatriangulation ofacompact n~manifold Mwemean a ﬁnite collection lo”,-} ofdilfeomorphic images ofAnwhich cover Mandwhich satisfy thefollowing condition: lftr";Flo”; 72ll).thcn forsome ktheintersection tr";no"; isak~Iace ofboth ct”;and0”,-. tee 4»4» L1 _ . . I'Q 1hestandard lI‘1H1'l!1'L1lEll]0li ixahd . OofS2(aftc1 Stembem nﬂersiimomof3~s1mplcxc C ii I. .. _ 2 i % lnvalicl '\A“triangulations” v (n =: 21' ltisadiﬁicult theorem that every C°° manifold hasatriangulation; fora proof seeMunkres. EIcmcm‘a2_"1-Dgjjlémzlzial Y?ij)0!r)gJ'. or\’\*’hitney, Geonretnic Inlegratiazz4% Excw'sz'0i? intheRealm of/lZgebraic Ybpologj" 427 Tlteotjn Assuming that our manifold Mhasatriangulation {o"1-} wewill call each or";an??~$impleX ofthetriangulation; anyk-face ofanytr”;willbecalled ak-simplex ofthetriangulation, Weletarkbethenumber ofthese k-simplexes. Now letUbethedisjoint union ofopen balls, onewithin each it-simplex J",-1 andletV,,_1 bethecomplement ofthesetconsisting ofthecenters ofthese balls, sothat V,,_.1 isaneighborhood ofthe union ofall (n-—l)—simplexes ofM.Then Q- 0 <.' M=UUV,,_1 where UHV,,_.1 hasthesame cohomology asadisjoint union ofancopies ofS"*1. Consider ﬁrstthecase where n>2.The Ma_ver~Vieto1-is sequence breaks intopieces: (1) 0—>H°(M) —>H°(U) oH°(V,,_1)—> H°(Unv,,_,1_> H'(M) —>H'w)eH](Vﬂ—I) —>H'wnVu-I)ll ll 0 0 (2) I-'or1<:/<<n-1. H’<*'ru nv,,_.,)_>H"(M) _>H""(U) asH’*(v,,_,) _>H*(Un1/,,_,) ll ll 1'. U U ll (3) H"-1w nV;:—-I)—>H“*'tM> —»H"-‘wt esH""tv.,_.> —-il ll U U ->H"_'(U nVn—])—>H"(M>—>H"w>esH"(Vn—l)ll U 428 Chapter 1] Applying Proposition 4tothese pieces yields dimH"(V,,..1)=dimH*(M) 05k5»-2 dimH"—‘t1/,,_t) =dimH"-1(M) -dimH"(M) +=1... Forthecase I?=2weeasily obtain thesame result without splitting upthe sequence. Wenow introduce theEuler characteristic X(M) ofM,deﬁned by X(M)=dimH°tM) -dimHHM) +dimH2(M) _+1-1)"dimH”(M). This makes sense foranymanifold inwhich allHk(M) areﬁnite dimensional; weanticipate here alater result that Hi‘(M) isﬁnite dimensional whe11ever M iscompact. Theabove equations then imply that n—l Xtt/,,-1) =Z:(-1);‘ dimH"(v,,__,) k=0 n—2 =Zt-_1)'< dimH"(M) k=0 +(-1)"_1[dim H""‘(M)-dimH"(M) +ct,,] =;((M)-(—1)”a,,. 01' X(M) =X(Vn-l)“l"("'1)nan- 5.THEOREM. Foranytriangulation ofacompact manifold Mwehave X(M)=¢Y0—0t1+0f2 "-'-+(—1)”=Yn- PROOF. Inthemanifold V,,_, wedeﬁne anew open setUwhich consists of adisjoint union ofsetsdiffeomorphic toIR",oneforeach (n—1)-face, joining theballs oftheoldU. ii“‘(W (‘oITlP0ne11ts of ‘ new U(H==2) v ___,_, /V . components of 1.1 r\iii lalt"rut"si01i 2'17I/tr.’Rmlnz qf./1lge/Hair Ybpologi 429 WewillletV,,_2 bethecomplement ofarcs, inthenew U,joining thecenters oftheballs intheoldU. i§\ VI;-1_2 iSthl? complement ofL, (n.-::2‘) (nx3'}V,,_2 isthe complement of L. Anargument precisely likethat which proves theequation =X0/n—l) "l-'("'1)n05n alsoshows that X(Vn—I) =><tv.._2) +t»-1)"-‘=1.._1. Similarly, weintroduce V,,_3, ...,V11;thelastofthese isadisjoint union ofan setseach ofwhich issmoothly contractible toapoint. Hence X(V0) =org,while inallother cases wehave X(l/it) =X(Vt'-1) +("-1)"'=1t<- Combining these equations, wehave X(M) =X(Vn—1)+(_1l”art =><tv,._2)+[<~1)"-‘=n_1 +t~1)"=1..1 =x(V0)+[(-1)‘a1+---+(-—1)”m=] ==1o—in+---+(—1)"0ta- OO.’ 6.COROLLARY (DESCARTES-EULER). Ifaconvex polyhedron has V vertices, Eedges, and Ffaces, then V--E-I-F=2. 430 Chapter 1] Ifwe turn from Hi‘toHfweencounter avery diﬁ"erent situation. IfUCM isopen, aform towith compact support CMmaynotrestrict toaform with compact support CU:theinclusion map ofUinto Misnotproper. Onthe ,ta;:J£:s,, Support £0‘w"I;“;’+>(:1-i'.' U other hand, iftoisalorm with compact support CU,then tocanbeextended toMbyletting itbe0outside U;wewilldenote thisextended form by _’ support to to(to). IfCf(M) denotes thevector space ofA--fornis with compact support onM,we candeﬁne anew sequence. 7.LEMMA. The sequence -;$"_ .1- -I; .1. o_>cfttmv)15'»-iscftu)ocftv1l‘i'._Jl>cftM)_>o 1SCXZICL PROOF. Itisclear that jg’EB-~j1/’isone-one; infact, each map jg’andjv’ ISone-one. Toprove thatt'U’+i;»’ isonto, lettobeak-form withcompact support onM. andlet{¢u,¢;/} beapartition ofunity forthecover {U,V}.Then w=¢uw+<tn/w isclearly theimage of(¢Uw,¢1/w) 6C'(’."(U) EBCg‘(V). Itisclear thatimage (jU'EB-—j1/’)Ckertiy’ +1'1/). Toprove thecoliyersc. suppose that (A.1.,A2) 6C'f{U) EBCf(V) satisﬁes t'U’(lt) +i1/'(lg) =0. This means that It.=-kg. Since supportl1 (IUand support lgCU.this shows that support ktCUOVand support lgCUOV.So(l|,lg) isthe itnage ofM6Cf(U HV).~1~ Errzutsion intheReal»: Q/‘fl[gs/Jiraic Yizjiri/0_g_1' 431 8.THEOREM (MAYER-VIETORIS FOR COM PACT SUPPORTS). Ifthe tnanifold M=UUVforU,Vopen inM,then there isalong exact sequence 5_>Hfwnt/1_> H§(t1) oH§(v) _>H§(M) __>H§+‘(Un t/1_> . PROOF. Apply Theorem 2totheshort exact sequence ofcomplexes given by theLemma. *I* This sequence ismuch harder towork with than theMayer-Vietoris sequence. Forexample, suppose wewant toﬁndHfforR"-{O}, which isdilfeomorphic to S"“‘ xlli. Ifwe write S"=ULJV intheusual way, sothatUHV isdiffeomorphic toS"“‘ ><R,then S”><IR=(U><IR)U(V><R),where (U><R)O(V><R) isdiffeomorphic toS”_'><R2.The only waytouseinduction istoﬁndHf forallS"><lR'",starting with S1><Rm. Thedetails willbelefttothereader; wewillmerely record onefurther result, forlater use.andthen proceed toyet another application ofTheorem 2. 9.COROLLARY. IfM=UUVforU,Vopen inM,then there isadual long exact sequence _>Hj‘+‘(tmv1* ->Hf(M)* _>[Hj‘(t11oH§tv1]* _>Hftunvy _> PROOF. Wejusthave toshow thatifthesequence oflinear maps =1 I5l’Vt"—->W2-—->W3t isexact atW2.then soisthesequence ofdual maps andspaces * =t =1! 5 1|‘ a 1|‘ W3 -—> W3 ~—-> W1 . ForanyA6W3*wehave ot*1B*(l)=0t*(lo1B)=lo(1B00t)=l00=0. SO0*0ﬁ*=0. Now suppose it6W;*satisﬁes 0z*(l) =0.Thcn Aoor=0.Weclaim that W1-°'_> W;it as tr It lli there isl:W3->Rwith A=1t5’*(l), i.e.,It=160i.Given aw6W3which isa 432 Chapter 1] oftheform B(w’), wedeﬁne itwl=M13’)- This makes sense, forifﬁ(w') =ﬁ(w"), then w—w"=a(:) forsome 2,so ltw) —Atw") =lac) =0.This deﬁnes 1onﬁ(W2) (IW3. Now choost W<3ll/3withW3=]3(l'l/2)oW,anddeﬁneittobeoonw.+:~ Wenow consider arather different situation. LetNCMbeacotnpacr sub- manifold ofM.Then M-Nisalsoamanifold. Wetherefore have thesequence -» C.5‘tM-N)i>CHM) '-—>cttm. where eis“extension”. This sequence isnotexact atCf(M): thekernel oft"‘ contains allto6C,‘."(M) which are0onN,while theimage ofecontains all to6Cf(M)which are0inaneighborhood ofN. Tocircumvent thisclifﬁculty. wewillhave touseatechnical device. We appeal ﬁrsttoaresult ﬁ-om theAddendum toChapter 9.There isacompact neighborhood VofNand amap rt:V—>Nsuch that Visamanifold-with- boundary, andifj:N—>Vistheinclusion, then rt0jistheidentity ofN. while jortissmoothly homotopic totheidentity ofV.Wenow construct a sequence ofsuch neighborhoods V=V13V23V33 with Q,V;=N V2X W_ N Vt //.5 i ii J?‘ 5;£g*;é.m"_| ,,_:_,:-27".-75.?-s.< _< Now consider twoforms to;6CI‘(V,-), to;6Ck(V,,-)- Wewillcallto;and to; equiztaltznl ifthere isI>1',jsuch that to,-IV;=w,|v,_ Itisclear thatwecanmake thesetofallequivalence classes intoavector space 9"(N), the“germs ofk-forms inaneighborhood ofM”. Moreover, itiseasy todeﬁne cl:§<k(N) ->Q"-"‘(N), sothat weobtain acomplex Q.Finally. we cleﬁne amap ofcomplexes -=t=k I c,(M)—~>§\"(N) intheobvious way: tot—->theequivalence class ofanyw|V,-. Excuiition inf/zeRealm o]"AZgebrat'c 75110503 433 10.LEMMA. The sequence 1; e1; llisO—> C,(M-—N)—->Cc(M)—->9 (N)—>O isexact. PROOF. Clearly eisone-one. Ifcu6Cf(M -N).then cu=0insome neighborhood UofN.Since Nis compact andQ;V;=N,there issome l‘such thatV;CU,andconsequently cu=0onV}.This means tl1at l'*e(w) =0.Conversely, suppose P».ECck(M) satisﬁes l'*(}t) =0.Bydeﬁnition ofQ"(N), thismeans that llV;=0forsome ll Hence MM -—-Nhascompact support CM-—-N,andA=e(}t|M -——N). Finally, anyelement of§k(N) isrepresented byaform nonsome V}.Let f:M—>[0,1] beaC°° function which is1onV}-+1, having support fC interior V}.Then ff]ECf(M), andfT)represents thesame element of9"‘(N) as1;;consequently thiselement isl'*(flq). '3' ll.LEMMA. TheCOl10mOlOgy vector spaces H"($0 ofthecomplex {E-,1k(N )} areisomorphic toHk(N)forallk. PROOF. Thisfollows easilyfromthefactthat1*:H"‘(V;) _>H"(N) is2111 isomorphism foreach V,-.Details arelefttothereader. *1‘ 12.THEOREM (THE EXACT SEQUENCE OFAPAIR). IfNCMisa compact submanifold ofM,then there isanexact sequence (ll.-._>Hf(M--N)—> Hfliw)-> H"(N)-> H§+‘(M-N)-> PROOF. Apply Theorem 2totheexact sequence ofcomplexes given byLemma 10,andthen useLemma ll.4* Intheproof ofthistheorem, thedeRham cohomology ofthemanifold—with— boundary V}entered only asanintermediary (and wecould have replaced theV;bytheir interiors). Butinthenext theorem, which wewillneed later, it istheobject ofprimary interest. 13.THEOREM. Let Mbeamanifold—with-boundary; with compact bound- ary3M. Then there isanexact sequence ls_>Hflm -aim ->Hf(M)—> H"(8M)—-> H,§*+‘(M --8M)-> 434 Chapter 11 PROOF. _]ust liketheproof ofTheorem 12,using tubular neighborhoods V}of 3M inM.'1‘ 1/,V2 M 1/ '3\:,vT=?€"-ii’? Anf‘Asasimple application ofTheorem 13,wecanrederive Hc'(]R )rom a l~:nowledge ofH"(S"_'), bychoosing Mtobetheclosed ballBinR",with Hf(B) %Hk(B) =0forkqé0.The reader may useTheorem 12tocompute Hf(S" ><Rm), byconsidering thepair (S"><lR’",{p} ><Rm). Then Theo- rem l3may beused tocompute thecohomology ofS"><S’"_' .-=3(S" >< closed ballinRm). Forournextapplication wewillseekbigger game. LetMCR""" beacompact n—dimensional submanifold oflR"+‘ (acom- pact “hyperslirface” oflR”"'l). Using Theorem 8-17, thesequence ofthepair (n"+', M)gives H;l(R!J+l) Z} H‘fl+l (Rn+l __ Z) H;I+l (]Rn+l) i) Hn+I ll Z? ll O R O Itfollows thal (>l=) number ofcomponents of]R”+' -—M=dimH"(M)+1. Butwealsoknow (Problem 8-25) tlial (>l=>l=) number ofcomponents of]R"+' ~M32. 14-.THEOREM. IfMCR""" isacompact hypersurface, then Misori- entable, andlR""" -M hasexactly 2components. Moreover, Mistheboundary ofeach component. PROOF. From (=-l=)and (=l==-l=)weobtain dimH"(M)+13 2. ]j'_i'cui:l‘t'lril inI/zcRealm qfAZge/n'a2'r' Trips/0_gj' 435 Since dimH"(M) iseither 0or1,weconclude that dimH"(M) =1,soMis orientable; then (=l=)shows that llR”"" -—Mhasexactly twocomponents. The proof inProblem 8-25 shows thatevery point ofMisarbitrarily close topoints indiﬂerent components ofllR"+' —M,soevery point ofMisintheboundary ofeach ofthetwocomponents. '1' 15.COROLLARY (GENERALIZED [C°°]_]ORDAN CURVE THEOREM). IfMCllR"+' isasubmanifold homeomorphic toS",then lR"+' -—Mhastwo components, andMistheboundary ofeach. 16.COROLLARY. Neither theprojective plane northeKlein bottle canbe imbedded inR3. Our next main result willcombine some ofthetheorems wealready have. However, there areanumber oftechnicalities involved, which wewillhave to dispose ofﬁrst. Consider abounded open setUCR”which isstal-—shaped with respect to0. Then Ucanbedescribed as ' U={IX:x6S"'"' and 0gr<p(x)} foracertain function pt.S‘”_' —>R.Wewillcallptheradial function ofU. Q\‘I \ IfpisC°°, then wecan prove that UisClilTCOIT10l‘pl1lC totheopen ball Bof radius 1inR".Thebasic ideaoftheproof istotakeIxeBtop(x)r -xeU. This produces difficulties at0,soamodiﬁcation isnecessary. 17.LEMMA. Ifthe radial function pofastar—shaped open setUCR”isC°°, then Uisdiﬂeomorphic totheopen ballBofradius 1inR”. 436 C/zajlier 11 PROOF. Wecanassume, without lossofgenerality, that p31onS"_'. Let fl[0,1] —>[0,1]beaC°°function with I f--=Oinaneighborhood of0 1‘E f’20 f f(1)==1- Deﬁne /1:B—>Uby I /:(r.v)= [1‘+(p(.X')— l)f(r)]x, xeS"_', 05:<1. Clearly Itisaone-one map ofBonto U.Itistheidentity inaneighborhood of0,soitisC°°.with anon-zero _]acobian, at0.Atanyother point thesame conclusion follows from thefactthatIl->I+lplx) -—l)f(r) isaC°°function with strictly positive derivative. *2» Ingeneral. thefunction pneed notbeC°°;itmight noteven becontinuous. However, thediscontinuities ofpcanbeofacertain form only. l8.LEMMA. Ateach point xeS"_'. theradial function pofastar—shaped open setUCR”is“lower semi-continuous": forevery 8>0there isaneigh- borhood Wof.vinS”_' such thatp(y) >p(x) -—sforally6W. PROOF. Choose Ix6Uwith p(x) ~I<s.Since Uisopen. tl1ere isanopen ball Bwith .vEBCU.There isclearly aneighborhood Wof.\'with the property thatfor_l'eWthepoint 13'isinB,andhence inU.This means that for3'6Wwehave p(_l') 3I>p(x) -—-8.Q? E.r0m'st'0i2 zittheRealm ofA(gebraic Yiljlology 437 Even when pisdiscontinuous, itlooks asifUshould bediffeomorphic toR". Proving thisturns outtobequite afeat, andwewillbecontent with proving thefollowing. 19.LEMMA. IfUisanopenstar—shaped setinR",thenH"‘(U) isH"‘(R") andHftul ~H_,_l‘(R”) forallk. PROOF. The proof forHl‘isclear, since Uissmoothly contractible toapoint. Wealso know that H,_f’(U) #11RwH,.f."(R"). ByTheorem 8—l7, wejusthave to show that H,;l‘(U) =0for035k<n. Letcubeaclosed k-form withcompact support KCU.Weclaim thatthere isaC°°function p:S"_‘ —>Rsuch that ,5<pand KCV={rxrx eS"_' and0_-5! <,5(x)}. This willprove theLemma, forthen Visdi1’l'eomorphic toR",andconsequently cu=digwhere 1’)hascompact support contained inV,andhence inU. Foreach xeS"_', choose I,<p(x) such thatallpoints inKofthe form ux for05u5p(x) actually have u<1,.Since Kisclosed andpislower semi- -»r.v /’ﬁ \,1 ._ I . _l // ‘\.,_.r continuous, there isaneighborhood W,ofxinS"_' such thatI,may also beused asr_,.forallyEW.LetWx,,..., Wx,cover .S'”_', let¢;,...,¢; bea partition ofunity subordinate tothiscover, and deﬁne l5=I.v,¢i+-~-+ts-,¢r- Any point xES"_' isinacertain subcollection ofthel/1",.-,., sayWx,,...,Wx, forconvenience. Then p;+;(x),...,p;(x) are0.Each r,;,,...,t,,., is<p(x). Since ¢l(X)+- --+¢;(x) =1.itfollows that,5(x) <p(x). Similarly, KCV.'1' 438 C/zapter 11 ‘Wecanapply thislastLemma inthefollowing way LetMbeacompact manifold, andchoose aRiemannian metric forM.According toProblem 9-32, every point hasaneighborhood Uwhich isgeodesically convex; wecanalso choose Usothat forany p6Uthemap expp takes anopen subset ofMp diileomorphically onto U.Let{U1,...,Ur}beaﬁnite cover bysuch open sets. IfanyV=U,-,f"l---OU;-,isnon-empty, then Visclearly geodesically convex. If p6V,then expp establishes adiﬂeomorphism ofVwith anopen star-shaped setinMp. Itfollows from Lemma 19that Vhasthesame Hf‘andHfasR". Ingeneral, amanifold Mwillbecalled ofﬁnite type ifthere isaﬁnite cover {U;,...,U,}such that each non—empty intersection hasthesame H1‘and Hg‘ asR";such acover willbecalled nice. Itisfairly clear thatifwe consider N={1,2, 3,...}asasubset ofR2,then M=R2-—-Nisnotofﬁnite type. Toprove thisrigorously, weﬁrst usethe Mayer-Vietoris sequence forR2=MUV,where Visadisjoint union ofballs around 1,2,3, ....Weobtain ct 01 H‘(R2) H'(M)eBH'(V) H'(Mnl/llsﬂltnl), ll ll Ii 0 0 0 where MOVhasthesame H'asadisjoint union ofinﬁnitely many copies ofS';thisshows thatH‘(M) isinﬁnite dimensional (seeProblem 7formore information about thecohomology ofM).Ontheother hand, 20.PROPOSITION. IfMhasﬁnitetype,thenHt(M)andH§tM) EIFCﬁnite dimensional forallk- PROOF. Byinduction onthenumber ofopen setsrinanice cover. Itis clear forr=1.Suppose itistrueforacertain 2-,andconsider anicecover {U;,...,U,,U} ofM.Then thetheorem istrueforV=U;U UU,and Excursion intheRealm ofALgebrazlr Ybjlologr 439 forU.ItisalsotrueforUt"lV,since thishasthenicecover {U|'“lU|,...,U|'“lU;"}. Now consider theMayer—Vietoris sequence .--_>H""(Unv) isH’*tM) -3->H"(U)e9H"‘(V)—> Themapamaps Hl‘(M)onto afinite dimensional vector space, andthekernel of0'isalso ﬁnite dimensional. SoHk(M) must beﬁnite dimensional. The proof forH,;"(M) issimilar. '1' Foranymanifold Mwecandeﬁne (seeProblem 8-31) thecupproduct map H’<(M) ><H"(M) -51>H’*+"(M) by tlwl,[tillt—>[wAti]- Wecanalsodeﬁne H"(M)><Hi<Ml3->H§‘+’<Ml bythesame formula, since cu/\r;hascompact support if1)does. Now suppose thatM"isconnected andoriented, with orientation /.1..There isthen aunique element ofHf(M) represented byanyT)EC,j."(M) with j r)=1. (Mm) Itisconvenient toalso useittodenote both thiselement ofH,;"(M) andthe isomorphism H,‘,"(M) —>Rwhich takes thiselement to1ER.Now every cr6H"(M) determines anelement ofthedual space HC”‘l‘(M)* by F-L ﬂl—>cru}3€H§(M)—->R. Wedenote thiselement ofHc"_’* (M)*byPD(cr), the“Poincare dual” ofcr,so thatwehave amap PD:H"‘(M)+H;*‘*tMl*. PD(<r)tﬁ) =tttrrU5)- One ofthefundamental theorems ofmanifold theory states thatPDisalways anisomorphism. Weareallsetuptoprove thisfact, butweshall restrict thetheorem tomanifolds ofﬁnite type, inorder nottoplague ourselves with additional technical details. Aswith most bigtheorems ofalgebraic topology, themain partoftheproof iscalled aLemma, andthetheorem itself isasimple corollary. 440 C/2Gj)l€’)' 1] 21.LEMMA. IfM=UUV foropen setsUandVandPDisanisomorphism forallkonU,V,and UPlV,then PDisalso anisomorphism forallkonM. PROOF. LetI:n-k.Consider thefollowing diagram, inwhich thetoprow istheMayer-Vietoris sequence, andthebottom rowisthedual oftheMayer- Vietoris sequence forcompact supports. H""(u) tsH""'tv; —»11*-‘(U nV)-1»H"(M) —>H"(u) eaH"(V) _»Hk(Unv) PDEBPD lPD ]PD lPDEBPD lPD [Hj+‘(U> eaHj+'(v)}' _>H§+'tu nV)‘_>Hjtmy _>[larjtuy eaHjtt/;}* _>Hjtunvr Byassumption, allvertical maps, except possibly themiddle one, areisomor- phisms. Itisnothard tocheck (Problem 8)thatevery square inthisdiagram commutes uptosign, sothat bychanging some ofthevertical isomorphisms totheir negatives, weobtain acommutative diagram. \'Venow forget allabout ourmanifold anduseapurely algebraic result. “THE FIVE LEMMA”. Consider thefollowing commutative diagram ofvec- torspaces andlinear lnaps. Suppose that therows areexact, andthat qbt,¢p_. tgba,¢5areisomorphisms. Then tgb;-,isalsoanisomorphism. VI"">V2‘*2>V32’>1/4°“‘>Vs1l¢l F12 l¢3 l¢4 l¢s W]_,_____§l 5W2 52>W3 153,W4 54>W5 PROOF. Suppose ¢3(x) =0forsome xEV3.Then B3¢3(x) =0,so¢4t'13(..\') = 0.Hence <13(x)=0,since Q54isanisomorphism. Byexactness atV3,there is )‘6V;with Jt'=0tp_(_l‘). Thus 0=¢3(Jt') =¢3a2(y) =ﬂ2¢;(y). Hence ¢2(y) =/3;(:) forsome 2eW1.Moreover, :=¢;(w) forsome wEVt.Then ¢z(J’) =161(5) =Bl¢l(w)=¢aotl(w), Wl1lCl1implies thaty=a,(w). Hence X=Q20’) =vlzttntwll =0- Soat;isone-onc. Theproof that¢3isonto issimilar, andislefttothereader. This proves the original Lemma. '1' E.l'cm'sz'm2 ini/reRealm qfAlgebraic Yripology 441 22.THEOREM (THE POINCARE DUALITY THEOREM). IfMisa connected oriented n-manifold ofﬁnite type, then themap PD:H"(M) _>H;-*(M)* isanisomorphism forallk. PROOF. Byinduction onthenumber rofopen setsinanice cover ofM.The theorem isclearly true forr=1.Suppose itistrue foracertain r,andconsider anicecover {U;,...,U,,U} ofM.LetV=U;U---UU,. Thetheorem istrue forU,V,andforUPlV(asintheproof ofProposition 19).BytheLemma, it istrue forM.This completes theinduction step. '3' 23.COROLLARY. IfMisaconnected oriented n-manifold ofﬁnite type, then Hl‘(M) andH,’."_k(M) have thesame dimension. PROOF. Use theTheorem and Proposition l9,noting that V*isisomorphic toVifVisﬁnite dimensional. '3' Even though thePoincare Duality Theorem holds formanifolds which arenot ofﬁnite type, Corollary 23does not. Infact, Problem 7shows that H1(R2-N) andHQ(R2-—-N)have different (inﬁnite) dimensions. 24-.COROLLARY. IfMisacompact connected orientable n-manifold, then H*"(M) andH""‘(M) havethesamedimension. 25.COROLLARY, IfMisacompact orientable odd-dimensional manifold, then )((M) =0. PROOF. Intheexpression for;((M), theterms (—1)k dimHk(M) and t-1)"-" dimH"_"‘(M) .-=(-l)"+‘ dimH"-’*(M) cancel inpairs. '1' Amore involved useofPoincare duality willeventually allow ustosaymuch more about theEuler characteristic ofanycompact connected oriented man- ifold M".Webegin byconsidering asmooth k-dimensional orientable vector bundle E=rt:E—>Mover M.Orientations p.forMand vforEgive an orientation p.69vforthe(n+k)-manifold E.since Eislocally aproduct. If {U1,...,U,}isanicecover ofMbygeodesically convex setssosmall thateach bundle $|U; istrivial, then aslight modification oftheproof forLemma l9 442 Chapter 1] shows that {rr_' (U1), ...,rr‘"' (U,)} isanice cover ofE,soEisamanifold of ﬁnite type. Notice alsothatforthemaps s=0-section M ’E wehave rt0s=identity ofM sort issmoothly homotopic toidentity ofE, sorr*: Hl(M) —>H"(E) isanisomorphism forallI.The Poincare duality theorem shows thatthere isaunique class UEH§‘(E) such that rr*).tu U=teesUeH;*+*(E). This class Uiscalled theThom class ofE.Our ﬁrstgoal willbetoﬁnda simpler property tocharacterize U. LetFp=rr_‘(p) betheﬁbre ofEoveranypoint peM,andletjp:Fp—>E betheinclusion map. Since jpisproper, there isanelement j_;,*U 6H,;"(F_,,). Ontheother hand, theorientation vfor§ determines anorientation upforFP, andhence anelement up6H,l‘(F_,,). 26.THEOREM. Let(M,/.1.)beacompact connected oriented manifold, and 5=rt:E—>Manoriented k-plane bundle over Mwith orientation v.Then theThom class Uistheunique element ofH,f.l‘(E) with theproperty that for allp6Mwehave j_;,*U =Up.(This condition means that .[ 19*” =1» (FPIIUIP) where Uistheclass oftheclosed form cu.) PROOF. Picksome closed form weC,.f."(E) representing U,andlet1]EC"(M) beaform representing p.,sothat f(M,,,) T]=1.Our deﬁnition ofUstates that (l) ‘[rr*n/\w=1. I-I LetACMbeanopen setwhich isdiﬁeomorphic toR",sothatAissmoothly contractible toanypoint pEA.Also choose Asothatthere isanequivalence f:at-‘t/1) ->A><Rk. 15.1-cm'si0n intheRealm of/llgebraic Yilpology 443 This equivalence allows ustoidentify IF‘(A)with A><Fp.Under thisidentiﬁca- tion, themap jp:Fp—>rr_'(A)corresponds tothemap el—>(p,e) fore6Fp, which wewillcontinue todenote byjp.WewillalsouseH21A><F_,,—>Fpto denote projection onthesecond factor. LetIIllbeanorm onFp. Bychoosing asmaller Aifnecessary, wecan assume thatthere issome K>0such that, under theidentiﬁcation ofrt‘!(A) with AxF_,,,thesupport ofw|;rr‘”' (A)iscontained in{(q,e) :qeA,||e||<K}. l:: support w 1A ll A/. Using thefactthatAissmoothly contractible top,itiseasytoseethatthere isasmooth homotopy H:(A><Fp)><[0,1] —>A><Fpsuch that H(e,0) =e Hles1)=(P=Tf2(@)) =Ji=(1't2(@)); wejustpulltheﬁbres along thesmooth homotopy which makes Acontractible éeiﬁ_.__ ‘l1llll"'toe.FortheHconstructed inthiswayitfollows that H(e,l') ¢support wif||e||3K. Consequently; theform H*w on(A><Fp)><[0,1] hassupport contained in {(q,e, r):||e||<K}.Aglance atthedeﬁnition ofI(page 224) shows that the 444 Chapter 11 form ]H*w onA><Fphassupport contained in{(q,e) I||e||<K}. Theo- rem 7-l4 shows that (jporrg)*w -w=r';*(H*w) -r';;*(H*w) =d(]H*w) +](dH*w) =d(]H*w). Thus (2) rr2*j_;,*w -cu=d)., support itC{(q,e) :||e||<K}. So (3) f rr*nAw=/ rr*lqA2r2*jp*w -/ rr*r;lAd)t. AXFI7 AXFI: AXFI: Now, ontheonehand wehave (Problem 8-l7) (4) f rr*1)Arrg*j_;,*w =f2r*l1 j_,,*w. A><F,, A r}, Ontheother hand, weclaim that thelastintegral in(3)is0.Toprove this, it clearly suﬁices toprove that theintegral is0over A’><Fpforany closed ball A’CA.Since rr*p. A:1}.=:l:d(rr*p. Adlt). wehave where rr*p. AAhas (5) f rr*p. Aall=:l:/i d(J'r*p. Alt) compact support on "”‘Ftt "'*Ft= A’><Faby(2) =:1;f rr*p.AA byStokes’ Theorem 3A‘lXFp :O, because theform Jr*p. AA.isclearly OonBA’ ><Fp(since 3A’ is(n-—-1)-dimen- sional). Combining (3),(4),(5)weseethat f J'f*?]/\(1):‘[J'{*l’)-l/i j,,*w. A:-:F,; ,4 F, Excursioii intheRealm qfA(gt?b?'aic Yilpology 445 This shows thatff,-Pj,,*w isindependent ofp,forpeA.Using connectedness. itiseasy toseethat itisindependent ofpforallpEM,sowewilldenote it simply byff,j*w. Thus f ir*r;Aw=[rr*r;-[j*w. Jr""(A) A F Comparing with equation (l),andutilizing partitions ofunity, weconclude that /j*w= 1, F which proves theﬁrstpartofthetheorem. Now suppose wehave another class U’EHf(E).Since Hfte) isH”(E) 4.».H"(M) R:R. itfollows thatU’=cUforsome cER.Consequently. 1'a*U’=1'a*<-“U=C~va- Hence U’hasthesalne property asUonly ifc=1.'1' The Thom class UofE=rt:E—>Mcannow beused todetermine an element ofH"(M). Lets:M—>Ebeanysection; there alwaysis one(namely the0-section) andanytwoareclearly smoothly homotopic. Wedeﬁne theEuler class )((E) 6H"(M) ofE xté)=8*!!- Notice that ifEhasanon-zero section .9:M—>E,and cu6C_,j."(E) rep- resents U,then asuitable multiple c-sofstakes Mtothecomplement of support cu.Hence, inthiscase x(E)=(C-S)*U=0- Theterminology “Euler class” isconnected withthespecial caseofthebundle TM, whose sections are,ofcourse, vector ﬁelds onM.IfXisavector ﬁeld onMwhich hasanisolated 0atsome point p(that is,X(p) =0,butX(q) 540 forq¢pinalieighborhood ofp),then, quite independently ofourprevious considerations, wecandeﬁne an“index” ofXatp.Consider ﬁrstavector 446 Chapter H ﬁeld Xonanopen setUCR"with anisolated zero at0EU.Wecandeﬁne afunction fy; U-—-{O}—>S"_I byf,y(p) =X(p)/|X(p)|. If1°:S”_l —>U ist'(p)=sp,mapping S”_' intoU,then themap fxoi:S"_' —>S“_‘ hasa certain degree; itisindependent ofe,forsmalls, since themaps it,1'2:S"_' —> Ucorresponding tos;and2;willbesmoothly homotopic. This degree iscalled theindex ofXat0. index 0 index 0 index l index I index --I index 2 index --2 index IinR" index (--1)” inR"xltsAsNow consider adiffeomorphism /itU—>VCR"with 11(0) =O.Recall that h,,.X isthevector ﬁeldonVwith (h,.X)(y) =]1*(Xh—I(y)). Clearly 0isalsoanisolated zeroofh,,.X. 27.LEMMA. IfhtU—>VCR"isadilleomorpliism with h(0) =0,andX hasanisolated 0at0,then theindex ofl:,,.X at0equals theindex ofXat0. Excursion intheRealm ty’Algebraic Topology 447 PROOF. Suppose ﬁrstthathisorientation preserving. Deﬁne H:R"><[0,1]—> R” by h(rx) 0<r51 ommu)l=o This isasmooth homotopy; toprove thatitissmooth at0weuseLemma 3-2 (compare Problem 3-32). Each map H;=xl—>H(x,l)isclearly adiffeomor- phism, 05I51.Note thatH;eSO(n), since flisorientation preserving. There isalsoasmooth homotopy {I-1,}, 15I52with each H,ESO(n) and H;=identity, since SO(n) isconnected. So(seeProblem 8-25), themap I1is smoothly homotopic totheidentity, viamaps which arediffeomorphisms. This shows that ﬁn; issmoothly homotopic tofy,»onasuﬁiciently small region of R"—{O}.Hence thedegree ofj},_X o1'isthesame asthedegree offyo1'. Todeal with non-orientation preserving h,itobviously sufﬁces tocheck the theorem forh(x) =(xl,..,,x”_', —x”). Inthiscase ﬂay =l10_/lyohpl, which shows thatdegree ﬁ,,,,y oi=degree fyo1'.'1'H(x,l')={ <- Asaconsequence ofLemma 27,wecannowdeﬁne theindex ofavector ﬁeld onamanifold. IfXisavector ﬁeld onamanifold M,with anisolated zero at pEM,wechoose acoordinate system (x,U)with x(p) =0,anddeﬁne the index ofXatptobetheindex ofx,,.X at0. 28.THEOREM. LetMbeacompact connected manifold with anorien- tation p.,which is,bydeﬁnition, also anorientation forthetangent bundle E=rt:TM —>M. LetX:M—>TM beavector ﬁeld with only aﬁnite number ofzeros, andletobethesum oftheindices ofXatthese zeros. Then M®=v-MEHWM1 PROOF. Letpt,...,prbethezeros ofX.Choose disjoint coordinate systems (Ul>xl)> -''1lufaxf) xflpf) Z0:and let Br=Xtpllll? 6R”I|P|.E1l)- IfwECf(E) isaclosed form representing theThom class UofE,then we aretrying toprove that / X*(w) =o. (Mali) 448 C/zapzer 1] Wecanclearly suppose that X(q)¢support wforq¢U;B,-.So [MX*<w>=LX*<w>; thusitsuﬁ-ices toprove that (=i<) fX*(w) =index ofX atp,-. 5', Itwillbeconvenient todrop thesubscript ifrom now on. Wecanassume that TM istrivial over B,sothat JT'_1(B) canbeidentiﬁed with B><Mp. Letj,,andHghave thesame meaning asintheproof ofThe- orem 26. Also clioose anorm ||llonMp. Wecanassume that under the identiﬁcation ofir"'(B) with B><Mp; thesupport ofw|ir"'(B) iscontained in{(q,v):qeA,||v||51}.Recall from theproof ofTheorem 26that J'rg*j,,"'w -—-to=dl support }-.C{(q,v) I||v||51}. Since wecanassume that X(q)¢support Aforqe3B,wehave (1)IX*(w) =fX*i'rg*(j_,,*w) -fX*(d}.)B B B =fX*i'r2*( j_,,*w) -fX*0.) byStokes’ Theorem B aB =fX*J"r2*(j,,*w). B Onthemanifold Mpwehave . _ pan(n—1)—form onMhpw _6/,0 (with non-compact suppltirt). IfDCMpistheunil disc (with respect tothenorm ||||)and S""‘ denotes 3DCMp, then 1F1SW”! 3.0 D =‘/\jp*Q) D _1 byTheorem 26,andthefact —‘ thatsupport jp*w CD. 1i.1'c21tIs?'022 inf/reRealm rJA(ge/n'a2'r Yéjiulrigs‘ 449 Now, forqEB—{p},wecandeﬁne Ytq)=Xtq)/|X(q)i, ,- andX:BB—>TMissmoothly homotopic toXI3B—>TM. So (3) fX*:rrg*(j_,,*w) =fX*:rrg* dp B B =JX*rr;*p byStokes’ Theorem BB =/ l7*Ff2*PaB =/ (Ff2°l7)*P-BB From thedeﬁnition oftheindex ofavector ﬁeld, together with equation (2),it follows that (4) f(rt;oX*)p =index ofX atp. aB K-"K*\\:_/'0:¢ Equations (l),(3),(4)together imply 29.COROLLARY. IfXand Yaretwo vector ﬁelds with only ﬁnitely many zeros onacompact orientable manifold, then thesumoftheindices ofXequals thesum oftheindices ofY. Atthemoment, wedonoteven know that there isavector ﬁeld onMwith ﬁnitely many zeros, nordoweknow what thisconstant sum oftheindices is (although ourterminology certainly suggests agood guess). Toresolve these questions, weconsider once again atriangulation ofM.'Wecanthen ﬁnd a vector ﬁeld Xwithjustonezero ineach k—simplex ofthetriangulation. We begin bydrawing theintegral curves ofXalong the1—simplexes, with azero at cach 0-simplex andatonepoint ineach 1—siinplex. Wethen extend thispicture 450 Cfzapter 11 toinclude theintegral curves ofXonthe2—simplexes, producing azero atOnt I point incach ofthem. Wethen continue similarly until then—simplexes are filled. 30.THEOREM (POINCAREHOPF). The sum oftheindices ofthisvectoi" field (and hence ofanyvector ﬁeld) onMistheEuler characteristic x(M). Thus, for2;‘=JTITM—>Mwehave )((§) =;((M) ~p.. PROOF. Ateach O-simplex ofthe triangulation, the vector ﬁeld looks like with index 1. Now consider thevector ﬁeld inaneighborhood oftheplace where itiszero ona1-simplex. Thevector field looks likeavector ﬁeld onIR"=R1><]R"_‘ which points direcTly inwards onR‘><{0}anddirectly outwards on{0}><llR”_'. + )1 Ck___ >--ﬁ—— _ 1(a)n=2 I (b)it=3 Excuzrsioiz intheRealm qfAlgebmic Yépology 451 ForH=2,theindex isclearly -1. Tocompute theindex ingeneral, wenote that fxtakes the“north pole” N=(0,...,0, 1)toitself and noother point goes toN.ByTheorem 8-12 wejusthave tocompute sig"nN f,y.Now atNwe canpick projection onllR"_‘ ><{0}asthecoordinate system. Along theinverse image ofthexi-axis thevector ﬁeld looks exactly likeﬁgure (a)above, where we already know thedegree is--1,sofyitakes thesubspace ofS"-1Nconsisting of tangent vectors tothiscurve intothesame subspace, inanorientation reversing way. Along theinverse image ofthe x2-, ...,x"_l -axes thevector ﬁeld looks like sof,y,,, takes thecorresponding subspaces ofS"-1Ninto themselves inanori- entation presewing way. Thus signN fy=--1,which istherefore theindex of thevector ﬁeld. Ingeneral, near azero within ak-simplex, Xlooks likeavector ﬁeld on R"=Rk><llR"‘k which points directly inwards onR"><{0}anddirectly outwards on{0}><llR"_k. The same argument shows that theindex is(-1)"- Consequently, thesum oftheindices is Q0"Q1+Q2—"-=X(M). Q‘ Weendthischapter with onemore observation, which wewillneed inthe lastchapter ofVolume V!Let£5=JTIE—>Mbeasmooth oriented k-plane bundle over acompact connected oriented rz-manifold M,and let(,)bea Riemannian metric for!_;'.Then wecanform the“associated disc bundle” and “associated sphere bundle” D={e:(e,e)__ 7 s={@;(¢» = iiiiif Itiseasy toseethat Disacompact oriented (12+k)-manifold, with 3D=S; moreover, theDconstructed foranyother Riemannian metric isdiffeomorpliic tothisone. WeletnotS—>Mberr|S...Pb'1-\_i4iiiiW. 452 C/lajltef 11 3l.THEOREM. Aclass ozEHk(M) satisﬁes rr0*(a) =0ifand only ifozisa multiple ofx(E). PROOF. Consider thefollowing picture. The toprow istheexact sequence Hftn-s H"(0)M>H"(s) § rr * ixllH"<M) for(D,S)given byTheorem 13.The map s:M—>D—Sisthe0-section, while EIM—>Distliesame O-section. Note that everything commutes. J'r0*=1'*o(rr|D)* since no=(2'r|D) 01', Si._CT0L, since extending aform toD —* does notaffect itsvalue ons(M), andthat 0(JT|.D)* =identity ofH"(M), since (ir|D) oifissmoothly homotopic totheidentity. Now letozEHk(M) satisfy rr0*(a) =0.Then i*(1r|D)*a =0,so(rr|D)*a E image e.Since D-Sisdifleomorpliic toE,and every element ofH,i‘(D —S) isamultiple oftheTliom class Uof§,weconclude that (rr|D)*a=c--e(U) forsomece IR. Hence oz=.'§*(i'r|D)*a =:c-§*(e(U)) =c-s*U =v-x(€)- The proof oftheconverse issimilar. +1‘ ];'xr.'i1i:riri1i influ’Realm of/1lge/Jraic Ybjiologj 453 PROBLEMS 1.Find H"‘(Sl ><---><Sl)byinduction onthenumber I?offactors. [Answeri Cliff] Hk = 2.(a)Use theMayer-Vietoris sequence todetermine Hk(M -—-{p}) interms ofHk(M), foraconnected manifold M. (b)IfMand Naretwo connected It-manifolds, letM#N beobtained by joining Mand Nasshown below. Find thecohomology ofM#Ninterms of that ofMandN. kl‘ N @/ / \_ W (c)Find Xforthen-holed torus. [Answerz 2--211.] (a)FindH"(Mobius strip). )FindH"(lP’2). c)Find Hk(lP’”). (Use Problem l-l5(b); itisnecessary toconsider whether a neighborhood oflP”'_l inP"isorientable ornot.) [Answerz dimH"(lP’") =1 ifkeven and 5it,=Ootherwise.] (d)FindHf‘(Klein bottle). (e)Find thecohomology ofM#(Mobius strip) and M#(Klein bottle) ifM isthen-holed torus. 4.(a)The figure below isatriangulation ofarectangle. Ifweperform the indicated identiﬁcations ofedges wedonotobtain atriangulation ofthetorus. Why not? A> .11.A 454 C/2(2j)l€?' U (b)The ﬁgure below does give atriangulation ofthetorus when sides areiden- tiﬁed. Find (Y0,O11,Q2forthistriangulation; compare with Theorem 5and Problem l. 5.(a)Foranytriangiilation ofacompact 2-manifold M,show that 30:2=20:; <11=30’-Yo—x(M)) (10010 "-1) >Q?~"--—2 __i l (X035(v+~/49-24x(M) ). (b)Show thatfortriangulations ofS2andthetorus T2=Sl><Slwehave S2: 01034 01136 (x234 T219027 01132] Q2214. Find triangulations forwhich these inequalities areallequalities. 6.(a)Find Hf(S"><Rm)byinduction onrt,using theMayer—Vietoris sequence forcompact supports. (b)Use theexact sequence ofthepair (S"><Rm,{p}><Rm) tocompute the same vector spaces. (c)Compute Hk(S" ><S"'_l), using Theorem 13. 7.(a)The vector space H1(R2 —N)may bedescribed asthesetofallse- quences ofrealnumbers. Using theexact sequence ofthepair (R2,N), show that Hcl(R2 -—-N)may heconsidered asthesetofallreal sequences {an} such that an=0forallbutﬁnitely many n. (b)Describe themap PD: H1(R2 -—N)—>H,_l(R2 -—-N)* interms ofthese descriptions ofH1(R2—-N) andH61(R2—-N). andshow thatitisanisomorphism. (c)Clearly Hcl(R2 -—-N)hasacountable basis. Show that H1(R2 -—-N)does not. Hint: Ifv,-=={aij} eH1(R2 -N),choose (b1,b2) ER2linearly indepen- dent of(011,012); then choose (b3,b4,b5) ER3linearly independent ofboth (¢113,m“,¢115) and(f123,=’124,f125); etc. Excumon intheRealm qfAlgebraic Yipologr 455 8.Show that thesquares inthediagram intheproof ofLemma 21commute, except forthesquare H"_‘(U riV)i. H"(M) jet) jPB H;.’+‘(U n1/)*__> Hj(M)* which commutes uptothesign (-—1)l‘, (Itwillbenecessary torecall how various maps aredeﬁned, which isagood exercise; theonly slightly diﬁicult maps are theones involved intheabove diagram.) 9.(a)LetM=M1UMQUMQ U---beadisjoint union oforiented n-manifolds. Show that Hcl‘(M) '~==@,-H:‘(M,-), this“direct sum” consisting ofallsequences (0z1,0zg,0z3, _..)with oz;EH:‘(M,-) and allbutﬁnitely many ct;=0EHg‘(M,-). (b)Show that Hk(M) '4]_[,-Hk(M,-), this“direct product” consisting ofall sequences (a1,0z2,cz3,...) with oz;6H2‘(M,-). (c)Show that ifthePoincare duality theorem holds foreach M,-,then itholds forM. (d)Theﬁgure below shows adecomposition ofatriangulated 2-manifold into three open sets U0,U1,and U2. Use ananalogous decomposition inndimen- sions toprove thatPoincare duality holds foranytriangulated manifold. \ kw‘: * ".*_. '='i=.1-;._\' I.-.3‘.-i'.»,:-,-‘>:‘,;i_“-_ §é}%;f_.:_.-,v .4.-7‘.T"i"i?1T~1I< t!‘,‘;§,iiET-§,'£*I t .-=='=E'=%:'1iZ€é?Eii:sa€it E€'=.i=.t:-'2-;'=I=.;i--k.=1... / I _—"'-""~-;:L‘.-;=;*—»- ‘=1-.-',;_.,;--.,-:’.=-3;..,-_ .%_;, 'F ~v :.--:.'---=-~-.'--- -' _.=..——,,,_. _1t/ I\ :.-.-E".-ztmzziiiséiiiil‘ '_1r:."'-‘?‘.=1'.;'-'::;' 1.;;L.‘’ ;,. ~—-.._ .-1-;e:-._s.:.¢;r;é>=a:¢;".¢- ".:i=?.’,-1==)‘=-.*.-='rii;-.-,.-- ==.'-3'--‘J-"R=.=-=-111%‘; .-:~- ‘~t- '-‘Ev-??:1+~¢.¢<>:i=-:==:.1;)‘ ‘K1--1 '-1%E.-‘FEE-itii-E5i%::'$;ﬁr -1'#‘*'»‘*\.~Ir.-i ‘t=';§;::;~.F;_;;;=t»-5gm»;r ’ --"¢:*;.+:=i:;1g:s::-;- .~\‘.<-.'I~ii1.1;-<,:1:;- '==.*:;=’¢’:-‘;itt,>%_::&~.:z-"~%%'3*.*si=.:-:<;it=- trait?-;;¢‘e.=t"1=§§:._.. "s.=§§€?_=-.E=‘.~é1-.i'-I‘-'-.‘.".. "*-T¢El£EiE" .€$3ii-5"Z5535"-‘?5‘£i:. ‘"335’-il'€‘55 --.- ..1.i -'4"=t¢.1";ir:;,<11;!§.1:....:, -¢,—;v_-.1 _-._-.if--' - '~~ -"-iflgrf-,4: 1,‘.1.-1--~~-.r. .1~.1.-;, .¢.. 1' :1r ~31‘;-1<-::;;=<-E1 ~i:Es'..":P_.'.W.=¢5 we :.~_-=I -_1::'.:l-2) "1'3'»=i<.':i:m;1*i:;':;.t1;'g ‘.;.."-- Q :-_.:'-_.=.-' "I; ;.t.:‘.'. .- .i.__' '1‘ P" ...;= .-=>;=,{-;;>1;§§.;;»a;:_,t,-...,...... ‘Y...~;a-:es:+t-.~‘.; ’~».‘\':-1'1‘? .'-'-"~'*~ E'-'i->':i.\'e£.1t.\“€-C(:12i1~-1%:-11? 1:1. I“‘.‘.:-.‘.:':'.’.-;-.'-: -iii‘-Y-1‘ »'.-='¢.':-i%‘t;*;‘.-;::!:=Q- E‘-1'-_ °'-‘-.1‘ ‘--.;v;-" wit:;g?f£=ig§;§:;:_eig~;;;,-1'.;z5§21;# ill,I 4J '1j eg-< “Ii =<' .. . .._-<.- .r....' .z_.-\,.‘_<.,~- ->i~.-,..-.-.--. Ii..- i '- '="::'*‘=*->'-"-f§==‘ =-i;'=:-.--5' ..<=;::;‘.='~ =1"B~ Q ' “ - €'~'-‘.1.-1:-;'~¢>>"=2‘.?=:'.:Y. '1?"-:2 21:32‘; ::'.' '~:'.» ~13’-115:1-:~t-:-gr-;=.t21 '-i‘*i‘i,i£1i:1.¥'::::}.~le.-';;i Qt ‘Ti=13‘:‘.%";‘:2>§~1?€5.=';-";:=I: =,=i:,it'*f§'Q5f}l?j1£iE?iiif>1‘-> .I'.‘-.z, __;:;';',-'.-1‘.:1.'~"" “' ‘. \.s\1";= U0isunion ofshaded U2isunion Ofshaded U1isunion ofunshacled% 456 Chapter 11 I0.Let§=rt:E—>Mand.§'=rt’:E’—>Mbeoriented k-plane bundles. over acompact oriented manifold M,and(f,f)abundle map from 5’toE‘ which isanisomorphism oneach ﬁbre, (a)IfUeHf(E) andU’eHf(E") aretheThomclasses, thenf*(u)=U’. (b)f*(;((§)) =;((§’). (Using thenotation ofProblem 3-23, wehave f*()((§)) == X(f*(§))-) 1I.(a)Let£5=rt:E—>Mbeanoriented k-plane bundle over anoriented manifold M,with Thom class U.Using Poincaré duality, prove theThom Isomorphism Theorem: The map H"(E) —>H§+l‘(E) given byozl—>ctuUis anisomorphism forallI. (b)Since wecanalsoconsider Uasbeing inHk(E), wecanform UuUE H;.2"(E). Using anticommutativity ofA,show thatthisis0forkodd. Conclude thatUrepresents 06Hk(E), sothatX(§l =0.Itfollows, inparticular, that )((§) =0when §=rt:TM —>MforMofodddimension, providing another proof that )((M) =Ointhiscase. I2.Ifavector ﬁeld Xhasanisolated singularity atpEM", show that the index of--X atpis(--1)" times theindex ofXat1).This provides another proofthat ;((M) =0foroddn. 13.(a)Letp1,...,p,-EM.Using Problem 8-26, show that there isasubset DCMdiffeomorphic totheclosed ball, such that all19,-Einterior D. (b)IfMiscompact, then there isavector ﬁeld XonMwith only onesingu- larity. (c)Itisafactthat aC°° niap f:S”“l —>S"_1 ofdegree 0issmoothly lio- motopic toaconstant map. Using this,show thatif;((M) =0,then there isa nowhere 0vector ﬁeld onM. (d)lfMisconnected and notcompact, then there isanowhere 0vector ﬁeld onM.(Begin with atriangulation toobtain avector ﬁeld with adiscrete setoi zeros. join these byaraygoing toinﬁnity, enclose thisrayinacone, andpush everything offtoinﬁnity.) (e)lfMisaconnected nianifold-with-boundary. with BM#I5,then there isa nowhere zerovector ﬁeldonM. Excuiztiaii inllllfRealm qfAlgebraic Yfijialogy 457 14.This Problem proves deRham’s Theorem. Basic knowledge ofsingular coliomology isrequired. Vilewilldenote thegroup ofsingular lc-chains ofX byS;,(X). Foramanifold M,weletSf°(M) denote theC°°singular k-chains, and letitSf°(M) —>S;,(M) betheinclusion. Itisnothard toshow that there isachain map r:S;,(M) —>Sf°(M) sothat 1'oi=identity ofS,‘;’°(M), while 1'O1'ischain homotopic totheidentity ofS;((M) [basically, Tisapproximation byaC°°chain]. This means that weobtain thecorrect singular cohomology ofMifweconsider thecomplex Hom(Sf°(M), R). (a)Ifcuisaclosed k-form onM,letR/1(0)) 6Hom(Sf°(M),R) be R/i(w)(c) =fa). Show that R/zisachain map from {Ck(M)} to{Hom(Sf°(M),R)}. (Hail: Stokes’ Theorem.) Itfollows that there isaninduced map R/ifrom thedeRham cohomology ofMtothesingular cohomology ofM. (b)Show that R/zisanisomorphism onasmoothly contractible manifold (Lem- mas 17,l8,and19willnotbenecessary forthis.) (c)Imitate theproof ofTheorem 21,using theMayer-Vietoris sequence for singular cohomology, toshow thatifR/iisanisomorphism forU,V,andUOV, then itisanisomorphism forUUV. (d)Conclude that R/zisanisomorphism ifMisofﬁnite type. (Using the method ofProblem 9,itfollows that R/iisanisomorphism foranytriangulated manifold.) (e)Check that thecupproduct deﬁned using Acorresponds tothecupproduct deﬁned insingular cohomology. APPENDIX A CHAPTER 1 Following thesuggestions inthischapter, wewillnow deﬁne amanifold tobe atopological space Msuch that ()MisHausdorfi“, (2)Foreach x6Mthere isaneighborhood Uofxandaninteger n30 such that Uishomeomorphic toR".>-| Condition (l)isnecessary, forthere iseven a1-dimensional “manifold” which is notHausdorff. ltconsists ofRU{=i=}where =i=¢R,with thefollowing topology: AsetUisopen ifandonly if ]\3|1--I()UORisopen, ()If=-r=EU.thcn (UOR)U{0}isaneighborhood ofO(inR). Thus theneighborhoods of=r=lookjustlikeneighborhoods of0.This space may alsobeobtained byidentifying allpoints except 0inonecopy ofRwith thc corresponding point inanother copy ofR.Although non-Hausdorff manifolds areimportant incertain cases, wewillnotconsider them. \/Vehave justseen thattheHausdorff property isnota“local property”, but local compactness is,soevery manifold islocally compact. Moreover, aHaus- dorfflocally compact space isregular, soevery manifold isregular. (Bytheway, thisargument does notwork for“infinite dimensional” manifolds, which arelo- cally likeBanach spaces; these need notberegular even ifthey areHausdorff) Ontheother hand, there aremanifolds which arenotnormal (Problem 6).Ev- erymanifold isalsoclearly locally connected, soevery component isopen, and thus amanifold itself. Before exhibiting non-metrizable manifolds, weﬁrstnote that almost all“nice” properties ofamanifold areequivalent. THEOREM. Thefollowing properties areequivalent foranymanifold M: )Each component ofMis0-compact. ::"".-B‘()Each component ofMissecond countable (hasacountable base forthe topology). Mismetrizable. 1"?S} (d)Misparacompact. (Inparticular. acompact manifold ismetrizable.) 459 460 Jlpperzalix A FIRST PROOF. (at=>(b)follows immediately from thesimple proposition that a0-compact locally second countable space issecond countable. (1))=>(c)follows from theUrysohn metrization theorem. (c]=>(cl)because any metric space isparacompact (Kelley. General 75/10/ogr. pg.l60). The second proof does notrelyonthisdiflicult theorem. (cl)=>(a)isaconsequence ofthefollowing. LEMMA. Aconnected. locally cotnpact, paracompact space iso-compact. Pica)? There isalocally ﬁnite cover ofthespace byopen setswith compact clo- sure. IfU0isoneofthese. then U0canintersect onlya ﬁnite number U1,...,U“, ofthe 0thers. Similarly U0UU1U---UU,“intersects only U,,,+1, ...,U,,,; and soon.The union 0 U0U...L,|L,m L,|..,L,|[j,,:L,|...__U0L,|,.,U[jn| U.,.L,|U”:L,|... isclearly open. Itisalso closed. forifxisintheclosure, then xmust bein theclosure ofaﬁnite union ofthese U,-.because .\-hasaneighborhood which intersects only ﬁnitely many. Thus xisintheunion. Since thespace isconnected, itequals thiscountable union ofcompact sets. This proves theLemma andtheTheorem. SECOJVD PROOF. (a)=>(b)=>(c)and (d)=>(a)asbefore. (c)=>(a)isTheorem 1-2. (a)=>(d).LetM=C1UCQ U---.whcre each C;iscompact. Clearly C1has anopen neighborhood U1with compact closure. Then U1UC2hasanopen neighborhood U3with compact closure. Continuing inthisway, weobtain open setsU1with U;compact and CU,-+1, whose union contains allC,-_andhence isM.Itiseasy toshow from thisthat Misparacompact. *3‘ Itturns outthatthcre areeven 1-manifolds which arenotparacompact. The construction ofthc-se examples requires theordinal numbers, which arebriefly cxplained here. (Ordinal numbers willnotbeneeded fora2-dimensional ex- ample locome later. ORDINAL NUMBERS Recall that anordering <onasetAisarelation such that (1)a<band/2<cimplies a<cforalla,b,c EA(transitivityt Ajijierrr/it A 46I (2)Foralla,bEA,oneandonly oneofthefollowing holds: (1.-=li (ii)a</1 (trichotomy). (iii)b<ct(also written a>b"I-60\_/ Anordered setisjust apair(A,<)where <isanordering onA.Twoordered sets(A,<)and(B.-<)areorder isomorphic ifthere isaone-one onto function f:A—>Bsuch that a<13implies f(a) -<f(b): themap fitselfis called an order isomorphism. and f-1iseasily seen tobeanorder isomorphism also. Anordering <onAisawell—ordering ifevery non-empty subset BCA hasaﬁrsl clement. that is.anelement bsuch thatb519'forallb’6B.Some well-ordered setsareillustrated below: inthisscheme wedonotlistanyofthe< relations which areconsequences oftheones already listed. til {0} 0<l (A={0,!}} 0<1<Z‘ (A={0,1,2}‘= 0<1<2<3 etc. 0<1<2<3<--- O<l<2<---<w (tvissomeset;-’:O,l,2,3....1 (w+1is,forthepresent. 0<1<2<<--<w<w+l justasetdistinctfrom those already mention edt 0<1<2<---<w<w+l<a>+2<.-- 0<l<2<---<w<w+1<a>+2<...<w-2 0<1<2<---<w<w+I<a>+2<.--<w-2<a>-2+l<--- 0<l<2<~--<w<w+l<w+2<---<w-2<w-2+l<---<w 0<1<2<---<w<-~-<w-2<~-~<w-3<...<--. 0<1<2<~--<a><~-<w-2<---<0)-3<---<-~<w2. 462 Aj)j)crzd2'x A Any subset ofawell-ordered setis.ofcourse, also awell-ordered setwith thr same ordering. Inparticular. asubset Bofawell-ordered setAiscalled an (initial) segment ifb6Bandct<bimply aEB.Itiseasy toseethatifBisa segment ofA.then either B=Aorelse there issome a6Asuch that B={a'eA:a'<a}; infact. ctistheﬁrst element ofA-—-B.Notice that each setonourlistisa segment ofthesucceeding ones. Itisnothard toseethat notwo setsonourlist areorder isomorphic. Forexample. 0<1<---<w and 0<l<---<w<w+1 arenotorder isomorphic because thesecond hasboth alastandanext tolast element. while thefirstdoes not. Butthere isamuch more general proposition which willsettle allcases atoncc: l.PROPOSITION. lfB72Aisasegment ofA.then_B isnotorder isomor- phic toA.Infact, theonly order isomorphism from Btoasegirreirl ofAistht identity. PROOF. If_/i:B—>B’CAisanorder isomorphism and B’isasegment ofA.then fortheﬁrstclement bofB(and hence ofA)weclearly must have f(b) =b.Then f(b’) must beb’.where b’isthesecond element. And soon. even forthe“wlh” element (the ﬁrst one after theﬁrst, second, third, etc.)l The way weprove thisrigorously isamazingly simple: Iff(b) =,ébforsome bEB. justconsider theﬁrstelement of{b6B1f(b) ¢b}:anoutright contradiction appears almost immediately. *1‘ Proposition lhasacompanion. which makes thestudy ofwell-ordered sets simply delightful. 2.PROPOSlTION. If(A,<)and (B,-<) arewell-ordered sets, then oneis order isomorphic toasegment oftheother. PROOF. Wematch theﬁrstelement ofAwith theﬁrstofB,thesecond with thesecond, ,the“a>‘h” with the“ﬁlth”, etc., until werunoutofoneset. Todothisrigorously, consider order isomorphisms from segments ofAonto segments ofB.Itiseasy toshow that anytwosuch order isomorphisms agree onthesmaller oftheir twodomains (just consider thesmallest element where Aj)j;endzLt' A 463 they don’t). Soallsuch order isomorphisms canbeputtogether togiveanother. which isclearly thelargest ofall.lfit isdefined onallofAwearedone. llit isnot,then itsrange must beallofB(orwecould easily extend it)andweare stilldone. *1‘ Suppose wcdeﬁne arelation <between well-ordered setsbystipulating that (A.<) -<(B,<) when (A.<) isorder isomorphic toaproper segment ol (B.<).Transitiyity of-<isobvious, andPropositions land2show thatweal— most have trichotomy. ‘E/&1inost”, because thecondition “(A, <)=(B,<)”musl bereplaced by“(A, <)order isomorphic to(B.<)". Toobviate thisdifficulty weneed only work with order isomorphism classes ofwell-ordered sets,instead ofwith thewell-ordered setsthemselves. These order isomorphism classes are called ordinal numbers. They arebeautilul:* 3.PROPOSITlON. -<isawell—ordering oftheordinal numbers. PROOF. Given anon—empty setAofordinal numbers, let(A,<)beawell- orclered setrepresenting oneofitselements <1.Toproduce asmallest element ofAwe(‘anobviously ignore elements 3oz.Every element <ozisrepresented byanordered setwhich isorder isomorphic tosome proper segment ofA: each ofthese isthesegment Consisting ofelements ofAlesstliat some aeA. Consider tlieleast ofthese a’s.Itdetermines asegment which represents some [56,>4~.This )3isthesmallest element ofA».*1‘ Notice that ifaisanordinal number, represented byawell-ordered set (A.<),then thewell-ordered setofallordinals 16<ozhasaparticularly simple representation: itisorder isomorphic totheset(A.<)!Roughly speaking: An ordinal number isorder isoniorphic tothesetofallordinals lessthan it. Ifozisanordinal number, wewilldenote by01+Ithesmallest ordinal after oz (ifozisrepresented bytliewell-ordered set(A,<).then oz+Iisrepresented byawell-ordered setwith just one more element, larger than allmembers ofA).Notice thatsome ordinals arenotoftheform a+1foranyct;these arecalled limit ordinals, while those oftheform oz+Iarecalled successor *Only onefeature mars thebeauty oftheordinal numbers aspresented here. Each ordinal number isahorribly large set;itwould bemuch nicer tochoose onespeciﬁc well- ordered selfrom eacli order isomorpliism class, anddeﬁne tliese speciﬁc setstobetlie ordinal numbers. There isaparticularly elegant waytodothis,duetoyonNeumann, which canbefound intheAppendix toKelley, Genera! 75/:0/051'. 464 Appendz'xA ordinals. Wewillalso denote some ordinals bythesymbols appearing before: 0,l,2.3....,w,w+1,... ,etc. Our listofwell-ordered setsonly begins tosuggest thecomplexity which well- ordered setscanachieve. With alittle thought, onecanseehow thesymbols (1)3,cu‘. would appear [symbols likew3+wz-3+cu-4+6would beused somewhere between (03and(04): after allthese onewould need cu ﬁfe) to_w ,.... andafter allthese thesymbol sopops up.After 2 3 w u:“" so,s0,....s0"’,...,st;"’ ,...,s0"’ UTIC COITICS l() 81,82,-.-38$.‘-.,8@u’,..-,sEn,»..,£iEFH,-. . andthisisonly thebeginning! Alithcwell-ordered setsmentioned sofararemzmlab/c. There areindeed an enormous number ofcountable well-ordered sets: 4-.PROl’OSlTlOl\". LetQbethecollection ofallcountable ordinals [ordinals represented byacountable well-ordered set). Then Qisuncountable. PROOF. ByProposition 3.(Q.<)isawell-ordered set.lfitwere countable. ii would represent acountable ordinal cz6Q.Bytheremark after Proposition 3. thiswould mean that Qisorclcr isomorphic tothecollection ofordinals -<o. i.c..toaproper segment ofitself. contradicting* Proposition l,+2‘ Wehave thus established theexistence ofanuncountable ordinal. Our Spr- cilic cxainplc. represented byQ,isclearly theﬁrst uncountable ordinal; any mcmber ofQiscountable, and consequently hasonly countably many pre- (lecessors. (ltishopeless totryto“reach” Qbycontinuing thelisting ofwell- ordered setsbegun above. foronewould have togouncountably far,anden- counter setswith anuncountable number ofdegrees ofcomplexity. Aleap of faith isrequired.) Although thecountable ordinals exhibit uncountably many degrees ofconi- plcxity. they areeach simple inoneway: *Byclclctin_:r thewords countable anduncountable inthisproofone obtains thc“Burali- Foni Paradox": theseiOrr!ofallordinal numbers iswell-orclered, soitrepresents an ordinal or501:1.and hence isorcler isomorphic toaninitial segment ofUni’. Fora resoluiion ofthisparadox, seel§clley’s Appendix. Aj)j)erzdzLt- A 465 5.PROPOSITION. IfozEQisalimit ordinal, then there isasequence ﬁt-< ﬁg<ﬁg-< -<01,such that every ﬁ-<:01satisfies ﬁ-<ﬁ,,forsome H(wesay that {ﬁn} is“coﬁnal” ina). PROOF. Since oziscountable, allitsmembers canbelisted (innot—necessarily increasing order) 1/(,1/2,}/3,. ...Letﬁt==yiandletﬁ,,.|.t betheﬁrst 1/inthe listwhich comes after ﬁ,,.‘Q’ 6.COROLLARY. IfozEQ.then ozisrepresented bysome well-ordered subset ofIR.However, nosubset ofIRisorder isomorphic toQ. PROOF. Suppose there were one, andhence asmallest, ozEQnotrepresented bysome subset ofIR.ltcannot happen that oz=1ﬁ+1,forthen ﬁwould be represented byasubset ofIR,thus alsobyasubset of(-—oo,0) andacould byrepresented byasubset ofIR.SobyProposition 5,there isasequence ﬁt-<ﬁg-<:ﬁg-< <01coﬁnal in01.Then ﬁ,-isrepresented byasubset oi (-00, f),andwecaneasily arrange thatthesubset representing ﬁ,-isasegment ofthesubset representing ﬁjfori<j.The union ofallthese setswould then represent a,acontradiction. Ifasubset ofIRwere order isomorphic toQ,then there would beuncountably many disjoint intervals inIR,namely those between thepoints representing Q and01+1forallcrEQ.This isimpossible. *1‘ The ﬁrst example ofanon-metrizable manifold isdeﬁned interms ofQ. Consider Q><[0,1),with theorder <deﬁned asfollows: (a,s)<(ﬁ,r) ifa-<ﬁorifa==ﬁands<r. This canbepictured asfollows: o >¢ ~">4 > o—>o—>oi> --->(0.0) (1.0) (2.01 (02.0) (w+1.0) (w+2.0) {nu-2,0) The setQ><[0,1)with tlieorder topology (asubbase consists ofsetsoftheform {.\'Ix<xo}and {.\'1.\'>_\'0}) iscalled theclosed long ray(with “origin” (0,0)). andL"'=Q><{O,1)—{(0,0)} isthe(open) long ray. The disjoint union oftwo copies oftheclosed long raywith their origins identiﬁed isthelong lineL.To distinguish L"'andL.thenames “half-lon gline" and“long line” may alsobe used. The Corollary toProposition 5implies easily that thelong rayandthe long lineare1-dimensional manifolds; aside from thelineandthecircle, there arenoother connected l—manifolds. 466 Appetzdz'xA Quite afewnew 2-manifolds cannow beconstructed: L+><Sl (half-long cylinder), L><S‘ (long cylinder). L+><R (half-long strip). L><lli (long strip), L><L (bigplane). L><L+ (bighalf-plane), L"'><L+ (bigquadrant). ldentifying allpoints ((0,0),6) intheproduct oftheclosed long rayand S‘ produces another 2—manifold. which might becalled the“big disc”. There isanother way ofproducing anon-metrizable 2—manifold which does notuseQatall.Webegin with theopen upper half-plane R1={(x,y)ER22 _]'>0}and another copy R2><{0}oftheplane; wewill denote this setbyR5. and denote thepoint (x,y,0) by(X,)=)o. Deﬁne amap ft}:(R§)_|_ —>Riby gf‘il,i’il', 7" -‘‘~ -1--)' 5I/IIf-1 Iowano==<>-.t=.r>- Consider thedisjoint union ofR1and Rg,with pE(R§,).|. andf0(p) 6R1 identified. This isaHausdorff manifold; thefollowing diagram shows two open setshomeomorphic toR2.The manifold itself is,infact, homeomorphic stt-"'- {1jI'il"-it) q__,IU sssssses)::::::ts113%;-(113%)-tIIIIIIIuI|II.___-v—¢ k’-'_-__.-—f"_XIII43"\I ‘i‘O91IJ i‘ ,_. '.;;_—-. toR2;wecould have thrown away R1tobegin with since itisidentiﬁed bya homeomorphism with (R%,)+ . Butconsider now, foreach a6R,another copy ofR2,sayR2><{ct},which wewilldenote byR5.Define jg:(R§)_,_ —>R1by fi=<<~».*">@> =<~+1'-\=rt i-'\11 Apjiemlix A 467 Inthedisjoint union ofR1andallR3,aERwewish toidentify each pE(R‘:;)+ with fa(p) 6R3, Wemay dispense with Ricompletely, and inthedisjoint union ofallRf,identify each (x,y),, and (x',,v’)), forwhich y=y’>0and .v_r+a =.r‘,v+b. The equivalence classes, ofcourse. areaspace homeomorphic toR1, sowewillconsider R12,asubset oftheresulting space. This space is still aI-lausdorﬁ‘ manifold, but itcannot besecond countable, forithas an uncountable discrete subset, namely theset{(0,0)a}. This manifold, thePriifer manifold, and related manifolds, have some very strange properties, developed intheproblems. PROBLEMS I.(a)Awell-ordered setcannot contain adecreasing inﬁnite sequence xt> X2>x3> . (b)Ifwe denote (oz+1)+1bya+2,(01+2)+1byct+3,etc., then anyoz equals )3+21 foraunique limit ordinal ﬁand integer n130.(Thus one can deﬁne evenand oddordinals.) 2.Letcbea“choice function”, i.e.,c(A) isdeﬁned foreach setA92El.and c(A) EAforallA.Given asetX,awell-ordering <onasubset YofXwill becalled “distinguished” ifforallyEY. 1'==c-(Y -—{y'EYI1"<y}). (a)Show that ofanytwodistinguished well—orderings, oneisanextension of theother. (b)Show that there isawell-ordering onX.(Zorn’s Lemma may bededuced from thisfactfairly easily.) (c)Given twosets,show thatoneofthem isequivalent to(can beputinone-one correspondence with) asubset oftheother. (d)Show that onanyinfinite setthere isawell-ordering which represents a limit ordinal. (e)From (d),and Problem l,show that ifXand Yaredisjoint equivalent infinite sets, then XUYisequivalent toY. 3.(a)L+and Larenotmetrizable. ' (b)Ifx15x;5X35 isasequence inL+. then {xn} converges tosome point. Consequently, anysequence hasaconvergent subsequence (but L"'is notcompactl). 468 Appendix A (c)If{xn} and{y,,} aresequences inL"'with x,,5yn5x,,+1 foralln,then both sequences converge tothesame point. (d)L+(and alsoL)arenormal. (Use (c)). (e)More generally, anyorder topology isnormal (completely different proof ). (f)Iff:L"'—>Riscontinuous, and2'>s,then oneofthe setsf"]((——oo,s]) andf“"({r,oo)) iscountable. (g)Iff:L"'—>Riscontinuous, then fiseventually constant. 4-la)Ll“isnotcontractible. Hint: Given H:L+>< [0,1]—>L+with H(x,O) = xforallx,show that forevery Iwehave {H(x,t)} =L+. (b)rr[(L"') =Ir)(L)=0.Similarly forL+><R,L><R,L><L,L><L+,L+>< L‘. (c)rr1(L"' ><S1)=rrI(L ><S‘)=Z.. 5.(a)L"'andLarenothomeomorphic. Hint: Imitating Problem l—l9, deﬁnt “paracompact ends". (b)L+><Rand L><Rarenothomeomorphic; L+><S‘andL><S]arenot homeomorphic. (c)Ofthe2-manifolds constructed from L+orLwith IT]=0andonepara- compact end,only L+><Rhasthehomotopy typeofL+. (d)TheStone-Cech compactifications ofL><L,L"'><L,L"'><L+,andthc bigdiscarealldistinct. (Using Problem 3(g), onecanexplicitly construct thesc Stone-Cech compactiﬁcations. 6.(a)Show thatthePrtiler manifold PisHausdorli. (b)Pdoes nothave acountable dense subset. (c)Let Ubeanopen setinR1which istheunion of“wedges” centered at (0.0) forevery irrational a.Show that Uincludes awhole rectangle ofthe form .-31;..'a.-' (a.b) ><(0.-9). Hint: Let/1,,={azthewedge centered atahaswidth 31/21}. Since R=QUU"An.some A,,isnotnowhere dense. Aj),()emlzLr /I 469 (d)LetC;,C2 CPbe Ci={(0,0),, :airrational} C2={(0,0)a Iarational}. Show that C1and C;areclosed, butthat they arenotcontained indisjoint open sets. (e)Deﬁne H:P><[0,1]—>Pby f_fS ' 1+3); ) ify>0 H((x>J’la>3l = 1+3)’ _S+Sy a (x 1-32,)’ 1-32) ify50. G/":“\ 7%+£3 Show that Hiswell-deﬁned and that H(p, 1)ER1U{(0,0),,} forallpEP. Conclude thatPiscontractible. (f)P—{(x,y),,I_v<0}isamanifold-with-boundary P’,whose boundary isa disjoint union ofuncountably many copies ofIR. (g)The disjoint union oftwo copies ofP’,with corresponding points onthe boundary identiﬁed, isamanifold which isnotmetrizable, butwhich hasa countable dense subset. Itsfundamental group isuncountablc. 7.Itisknown thatevery second countable contractible 2-manifold isS2orR2. Hence theresult ofconstructing thePrtifer manifold using only copies R5for rational amust behomeomorphic toR2.Describe ahomeomorphism ofthis manifold onto R2. B.LetMbeaconnected Hausdorff manifold which isnotapoint. (a)IfACMhascardinality c(the cardinality ofR),then theclosure Ahas cardinality t‘. (b)IfCCMisclosed andhascardinality t‘,then Chasanopen neighborhood with cardinality r. (c)LetpeM,There isafunction f:Q—>(setofsubsets ofM)such that f(oz) hascardinality cforalloz6Q,andsuch that f(0)={Pl f(a) isanopen neighborhood oftheclosure ofUﬂga f(,B). (Consider functions deﬁned oninitial segments ofQwith these same properties, andapply Zorn’s Lemma. Alternatively, onecanrequire _/(oz) tobetheresult of applying thechoice function tothesetofallopen neighborhoods oftheclosure 470 Ajlpezzdzar A ofU5“, f(ﬁ) with cardinality t‘.Then there isaunique fwith therequired properties. This isanexample ofdeﬁning afunction by“transﬁnite induction”,) (d)Afunction f:Q—>(setofsubsets of[0,1])with theproperties ofthefune- tioninpart (c)iseventually constant. (e)Mhascardinality t.(Given p’EM,consider anarefrom ptop’.) 9.(a)Aconnected 1-manifold whose topology istheorder topology forsome order, ishomeomorphic toeither therealline, thelong line, orthehalf-long line. (b)Every 1—manifold Mcontains amaximal open submanifold Nwhose topol- ogyistheorder topology forsome order. (c)IfMisconnected andN72M,then Mishomeomorphic toS]. Aj)pemlix A 471 CHAPTER 2 The long rayL"'canbegiven aC°° structure, and even aC"’structure. Toseethis weneed theresult ofProblem 9-24—any C°° [orCw] structure onamanifold Mhomeomorphic toRisdiﬁeomorphic toRwith theusual structure. This implies that itisalso diffeomorphic to(0,1),andconsequently that thestructure onMcanbeextended ifMisaproper subset ofL"'. An easy application ofZorn’s Lemma then shows that C°°andC"’structures exist onL+. Idonotknow whether allC°°structures onL+arediﬁeomorphic. Itis known that there areuncountably many inequivalent C”structures onL+, If pEL"'.andL4‘), denotes allpoints 5p,then L+-—L+), isclearly homeomor- phic toL+. If(9isaC“’structure forL+,then ityields aC“structure for L+-—L"'_,,, andhence forL+. These arealldistinct, inother words, there is noC“map fIL+-—L+p—>L+-—L+q q>/J with aC°°inverse. Infact, wemust have f(q) :>q,and then itiseasy tosee P QL 1 -. ..__. , ,f(e).-. ,__.‘I that wemust 3l$0 have ffffqll >f(q): ffffffqlll >ffff‘/ll, em The increasing sequence q,f(q),f(f(q)), ...hasalimit point xsEL"'-—L"'_,,. and f(x0) =9:0.Now fcannot betheidentity onallpoints >xo(forthen itwould betheidentity everywhere, since itisC"’).Soforsome qt>xowe have _/(qt) 72qt;wecan assume f(q1) >q;,since wecan consider f'_1 in thecontrary case. Reasoning asbefore, weobtain xi>xowith f(x1) =—-x1. Continuing inthisway, weobtain xo<xt<x;< with f(x,,) -=x,,.This sequence hasalimit inL+-—L+_,,, butthisimplies that f(x) =xforallx,a contradiction. AC"’structure exists onthePrufer manifold; thisfollows immediately from thefact that tliemaps used foridentifying points invariou8 (REM with points inR1,areallC"’.Idonotknow whether every 2-manifold hasaC°° structure. Using theC“’structure onL+,wecangetaC"’structure onL"'><L"'. How- ever, themethod used forobtaining aC"’structure onL+willnotyield acomplex attagyzzic struclure onL"'><L"';theproblem isthatacomplex analytic structure onR2may bcconformally equivalent tothedisc, andhence extendable, butit 472 AppendixA may alsobeequivalent tothecomplex plane, andnotextendable. Infact, i1 isaclassical theorem ofRado that every Riemannian surface (2-manifold with acomplex analytic structure) issecond countable. Ontheother hand, amod- iﬁcation ofthePriifer manifold yields anon-metrizable manifold ofcomplex dimension 2.References tothese matters aretobefound in Calabi andRosenlicht, Complex Analytic Manﬁirlds witltoui Countable Base, Proc. Amer. Math. Soc. 4(l953), pp.335-34-O. H.Rneser. Attafieisc/re Siruclzlm‘ unclAliza‘/zlbarkezif, Ann. Acad. Sic. Fennicae Se- riesA,I25l/5 (l958), pp.l—8. PROBLEMS 10.Prove that lorq>pthere isnonon-constant C"’map f:L"'-—L"',, —> L+-L*q. ll.Let(Y,p)beametric space and letf:X—>Ybeacontinuous locally one-one map. where XisHausdorﬁl connected, locally connected, and locally compact. (a)Every twopoints x,yeXarecontained inacompact connected CCX. (blLetd(x,_v) bethegreatest lower bound ofthediameters off(C)(inthe p-metric) foralleompact connected Ccontaining xand _1'.Show that disa metric onXwhich gives thesame topology forX. 12.Ofthevarious manifolds mentioned intheprevious section, trytodeter- mine which canbeimmersed inwhich. Apjlerzdix A 473 CHAPTER 6 Problem A-6(g) describes anon-paracompact 2-manifold inwhich twoopen half—planes areadense set.\'Vewillnow describe a3—dimensional version with atwist. LetA=={(x,y,z) ER3Iy#0},andforeach aEIRletR3beacopy ofR3. points inR3.being denoted by(x,y,z),,. Inthedisjoint union ofAandallIR2. aERweidentify (X,y,z)¢, fory>0with (a+yx,y,z +a) (x,y,z),, fory<0with (a+yx,y,z —a). The equivalence classes form a3—dimensional Hausdorff manifold M.Onthis manifold there isanobvious function “Z”, and thesets2=constant form a foliation ofMbya2-dimensional manifold N.The remarkable factabout this 2-dimensional manifold Nisthatit isconnected. For,thesetofpoints (x,y,c) E Awith y>0isidentiﬁed with thesetofpoints (x,y.c —a),,6R3.with y>0. Now thefolium containing {(x,y,c —a),,} contains thepoints (x,y,c —a),, with _v<0.and these areidentiﬁed with thesetofpoints (x,y,c -—2a) 6A with _v<0.Since wecanchoose a==c/2,weseethat allleaves ofthefoliation arethesame astheleafcontaining {(x,y,0) :y<0}CA. This example isduetoM.Rneser, Be£spie/ eincrdim6nsi0'nJer1z0/l8?ld€ﬂ ana[31tzl<~ char:Abbildung zwzschen zJibare'ib<";alz!ba2'en Adarzzizgﬁzltigkeitew. Archiv: Math. ll(l960), pp.280-281. 474 A;i;i.<-max/1 CHAPTERS 7,9,10 l.Vilehave seen thatanyparacompact C°°manifold hasaRiemannian metric. The converse also holds, since aRiemannian metric determines anordinary metric. 2.Problem A-ll implies that amanifold Nimmersed inaparacompact mani- foldMisparacompact, butamuch easier proof isnow available: Let(,)be aRiemannian metric onM;iff:N—>Misanimmersion, then Nhasthe Riemannian metric f*( ,). Wecannowdispense with theargument intheproof ofTheorem 6-6which wasused toshow thateach folium ofadistribution onametrizable manifold is alsometrizable, forthefolium isasubmanifold, andhence paracompact. 3.Since there isnoRiemannian metric onanon-paracompact manifold M. thetangent bundle TM cannot betrivial. Thus thetangent bundle ofthelong lineisnottrivial, noristhetangent bundle ofthe Prufer manifold, even though thePriifer manifold iscontractible. (On theother hand, abasic result about bundles says that abundle over aparacompact contractible space istrivial. Compare pg.V.272.) 4.The tangent bundle ofthelong lineLisclearly orientable, sothere can- no!beanowhere zero 1-form cuonL,forwand theorientation would cle- termine anowhere zero vector ﬁeld, contradicting thefactthatthetangen! bundle isnottrivial. Thus, Theorem 7—9fails forL.Notice also that ifM isnon-paracompact, then TMisdeﬁnitely notequivalent toT*M, since an equivalence would determine aRiemannian metric. Sothere areatleast two inequivalent non-trivial bundles over M. 5.Although theresults intheAddendum toChapter 9canbeextended to closed, notnecessarily compact. submanifolds, theycannot beextended tonon- paraconipact manifolds, ascanbeseen byconsidering the0—dimensional sub- manifold {(0.0)a} ofthe Priifer manifold. 6.ALiegroup isautomatically paracompact, since itstangent bundle istrivial. More generally, alocally compact connected topological group is0-compact (Problem 10-4}. Ajljierzdix A 475 7.ltisnotclear that anon-paracompact manifold cannot have ani]‘lCl€li]]ilf metric (anon-degenerate inner product oneach tangent space). This will bf proved inVolume II(Chapter 8,Addendum l). PROBLEM I3.Isthere anowhere zero 2-form onthevarious non-paracompact 2-mani- folds which have been described? CHAPTER 1 5-?(X)Hﬂ MN P2 pr: Rn S! Sn BM CHAPTER 2 Al Cf C0 CDC Cw D='f(ﬁ') GL(n, IR) O(n) Rt») (W,‘ul SL(n, IR) SO(n) tr,U) 8/,)Hr;Bx‘lip’Bx‘JP 3 Bx‘P 3B F56’ CHAPTER 3 Df £”(X) 5*“ fa-= 1*.»23 l9 4 ll 19 l 6 7 19 6l 34 34 28 34 35 6l 6] 62 29 6l 62 28 35 39 36 65 72 83 65 65 f*(€) 101NOTATION INDEX MP (M,iii» lR"p TM T(M, 1') T11?” Ti- A}, J? .\'-*' ix,“in v,. I-‘(fl [v1,...,vn] El/1 -‘EGBti -5!X$2 dc FE dc‘ ET It: iBx‘ CHAPTER 4 if d.\" End( V) f‘L1 Hom(V, W) T*M Tos WV) T"‘(E> 7it(V) 'I;'<v> ms) ﬁ‘,"(v)76 68 64 75 68 64 103 82 83 81 76 64 80 84 72 lOl 102 81 80 83 09 10 2l 07ll6 ll9 l3 31 O9 l6 l6 17 20 2l 2l 22 478 '1?‘($1 V’ .1- 1", rill §!.'.:1;{"511....1,, Pi|...!;, Y. w(X) 11-‘ CHAPTER 5 I/ll r"’ exp/I Ly/I Lx] L)()' Lyw 0(t2) [X,Y] 0rx(t‘) =0r(t‘.x) (“'1/Ylq 4': CHAPTER 7 Ali Ali curlX divX (H1 dw glad /w 1(5) :,.w Lyw 5:.Nbta£z'0n Index I.lT .j}kln;w](V) 10: Mt30: Jttmwifvl ;2s T°<">:34 v_iw;2s 2*,"""1tv>0. ps4 8'.08 0-(v1.....v;() .\0q 0-(v1,....v;() Oi.( mm; ‘ QW)s2°(1/1 /\ 171 6l :7} CHAPTER 8 174 B4(M) L50 Bf(M) (‘(543) L50 cR_,, Q7? deg] 153 |d,\-1/\---/\ d.\-"I L4-3 dél L35 d(~J,, :44 d®Ia.lJ.r) f’ f‘ 202 H"‘<M1 205 ff‘(M1 238 ' 238 /tiilvrl 219,235 ~‘f(f,sJ 210,213,215,234- M. Q3’; J17,- 224 ||P|i 215 P" 22'; Siam,I 234 w(r>) 202 Z"‘(M)23l 23l 23l 23l 20l 227 206 202 227 2l5 201 201 203 263 268 250 284- 275 258 252 290 297 292 274- 263 268 24-6 249 296 283 283 239 264 275 293 263 Z:f‘tM>A. —m O OJ lw] Br 8,-r 3:1 rf-it f(IJ J: fOJ JM 1-‘Ff"(0) .j0,1]° u >6 A CHAPTER 9 cosh cosh‘, d(11.17) ds (IV Euc(V) EH00?) Eu») exp f*(,l ts") [U11I L2M,,lfJfdx-l-gd_1‘.N0iat2'02i Index 268 5 285 sinh 29] tanh 264, 29] W-L 264 Ji"(v) 264 ii 263 c7(u) 248, 285 5.} 285 03 260 Fg. 24-6 Ft"-I _.....__E_____=__;__________.=,..g__.=__.g—._,1.,.....-‘V"-/"'\I%"\-._¢4"'\-<-f"\-/@4‘ ""'Ii>-'-*1'~239 243, 24-5, 246, 24-8. 257, 259, 288 294 246 299 299 266 CHAPTER l0 |A|322,ase A.-, 314 adX 314 Avftq)311 ici309 at.» 309 E01) 324 Em/(L1) €‘Xl) 302 exp(A) Gi_.(n ,IR) G 312 ,-= 344 gl(n. ilk)3l3 356 356 349 335 335 318 319 314 353 328 301 3l5 308 301 349 305 301 36.» 303 303 3l5 384 372 409 410 409 396 402 373 410 385 385 372 407 407 376404 480 la La 1-0 ..C(G) 003) 0(n) P P—~1 Ra SO(n) 2" ,7 4')» 1/1 plw/\Y?) w(natural g-valued 1-form, L~l tnAll W‘:._k:._Mtgﬁgﬁg,ﬁsE"">"'Q-:5_f f(a)do401 374 379 376 388 376 411 411 374 373 376 379 380 395 403, 4l0 403 376 403 400 400 4-00 403.Miia£z'on Index CHAPTER ll C*(M) ./A‘ s"tNlM14N PD R1.{F A. <5 P x(M) x(€)U APPENDIX A Ora’ cr+I £0 Q LU < -(419 446 432 453 439 457 442 426 4-23 435 428 445 439 464 463 464 464 461 461 4-63 Abelian Liealgebra, 376, 382, 395 Adams,J.F., 100 Adjoint T‘ofalinear transformation T,l03 Ado, l.D., 380 A]exander’s Horned Sphere, 55 Algebra. Fundamental Theorem of. 285,293 Algebraic inequalities, principle of irrelevance of,233 Alternating covariant tensor ﬁeld, 207 multilinear function, 20] Alternation, 202 Analytic manifold, 34 Annihilator, 228 Annulus, 8 Antipodal map, 278 point, ll Arclength, 312 function, 313, 332 Arcwise connected, 20 subgroup ofaLiegroup, 409 Area, generalized, 246 Associated discbundle, 45] sphere bundle, 45] Atlas, 28 maximal, 29 Auslander, L.,106 Banach space. 145 Base spacc, 7] Basis dual, l0? forM,,*,208 forS'Z"(p), 208 Bel0ngs toadistribution, [91 Besicovitch, A.5.,179 Big disc, 466 half-plane. 466 plane, 466lNDE.\' quadrant, 466 Bi—invariant metric, 40] Boundary, l9,248, 252 Bounded manifold, 19 Boy’s Surface, 60 Bracket, l54 ingl(n,lR), 378 in0(n,llR), 379 Bundle cotangent, 109 dual, I08 ﬁbre, 309 induced, 10] map, 73 n—plane, 7] normal, 344 ofcontravariant tensors, 120 ofcovariant tensors, ll? tangent, 77 []"i\rl£1l, 72, 210 vector, 71 Burali-Forti Paradox, 464 Calabi, E.,472 Calculus ofvariations, 3l6 Cartan, Elie,39,348,300Cartan’s Lemma, 230 Cauchy-Riemann equations, 200 Cayley numbers, 100 Chain, 248, 285 Chain Rule, 35,38 Change, inﬁnitely small, lll Chart, 28 Choice, 283 Choice function, 46? Circle, 6 Closed form, 218, 252 geodesic, 367 half—space, l9 long ray, 465 manifold, 19 subgroup ofa Liegroup, 391 submanifold, 49 482 Closed (continued, uptoﬁrst order, l6fl Coﬁnal, 465 Cohomology, 419 deRham, 263 group ofMwith realcoeﬁicients. 263 ofa complex, 42[ Commutative diagram, 65,420 Commutative Liealgebra. 376 Complete, geodesically. 341 Complex, 42] analytic structure. 47] numbers ofnorm l.373 Conjugate, 358 Constants ofstructure, 390 Continuous homomorphism, 387 Contractible, 220, 225. 236 Contraction, l2], 139. 227 Lemma, 139 Contravariant functor, l30 tensor ﬁeld, l2{) vector ﬁeld, 113 Convex geodesically, 363 polyhedron, 429 Coordinate lines, 159 Coordinate system, 28.158 Coordinates, 28 Cotangent bundle, 109 Covariant functor, 130 tensor ﬁeld, ll? vector ﬁeld, I13 Covet locally ﬁnite, 50 point-ﬁnite, 60 reﬁnement of,50 Cramer’s Rule, 372 Critical point, 40 inthecalculus ofvariations, 320 Critical value, 40 Cross section, 227 Cross-cap, l4 Cross-product, 299Index Cup product, 299, 439 Curl, 238 Cylinder, 8 C’manifold, 34 C0manifold, 34CO( distribution, 1/9 form, 207 function, 32 manifold, 29 manifold-with-boundary, 32 Riemannian metric, 308 structure onTM, 82 C°°—related, 28 Darboux integrable, 283 integral, 283 Darboux’s Theorem, 284 Debauch ofindices, 39,123 Decomposable, 228 Deﬁnition, invariant, 214 Deformation retraction, 279 Degenerate, 286 Degree, 275 mod 2,295 Density even scalar, I33, 209 oddscalar, l33, 259 relative scalar, 23l scalar, 133 Derivation, 39,78 ofaring, 83 Derived set,25 Descartes—Euler Theorem, 429 Determinant, 232 Difleomorphic, 30 Dillieomorpliism, 30 one—parameter gmup of,148 Diﬁerentiable, 27,28,31,32 atapoint, 31 manifold, 29 structure, 30 onthelong line, 47] Cube, singular, 246 011P”. 32 Differentiable (continued (structure continued. on1R",29 onS”,30 Differential, 210 equation, 136, 164 depending onparameters, 169 linear, 165 forms, 201 ofa function, 109 Dimension, 4 Direct sum, 421 Disc bundle, associated, 451 Discriminant, 233 Disjoint union, 4,20 Distribution, 179,181 ideal of,215 ontorus, 180 Divergence, 238 Theorem, 352 Domain, 3 DuBois Reymond’s Lemma, 355 Dual basis, 107 space, 107 vector bundle, 108 Einstein summation convention, 39 Elements ofnorm 1,308 Elliptical non-Euclidean geometry, 367 Embedding, 49 End, 23 paracompact, 468 Endomorplaism, 121 Energy, 324 Envelope, 358 Equations depending onparameters. 169 Equations ofstructure. 404 Equivalence (ofvector bundles), 72 weak, 96 Euclidean metric, 305, 315 motion. 374 n—space, 1483 Euler, 429 characteristic, 428 class, 445 Euler’s Equation, 320 Even ordinal, 467 relative scalar, 231 relative tensor, 134, 231 scalar density, 133, 209 Exact form, 218 sequence, 419, 422 ofapair, 433 ofvector bundles, 103 Exponential map, 334, 385 Exponential ofmatrices, 384 Extension, 432 Extremal, 320 Faith, leap of,464 Fibre, 64,68,71 Finite characteristic, 205 type, 438 First element, 461 First variation, 319, 327 Five Lemma, 440 Fixed point, 139 Foliation, 194 Folium, 194 Force ﬁeld, 240 Form, 207 diﬁerential, 201 leftinvariant, 374 right invariant, 400 f-related, 190 Frobenius lntegrability Theorem, 192 215 Fubini’s theorem, 254 Functor, 130 Functorites, 89 Fundamental Theorem ofAlgebra. 285,293 Fundamental Theorem ofCalculus, 254 484 Index Gauss’s Lemma, 337 General linear group, 61,372 Generalized area, 246 Geodesic, 333 closed, 36? reversing map, 401 Geodesically complete, 341 Geodesically convex. 363 Geodesy, 333 Germs ofk-forms, 432 Global theory ofintegral manifolds. 194 Gradient, 237 Gram-Schinidt ortlionortnalization process, 304 Gml-:. 84 Group Lie. 371 matrix, 372 opposite, 407 orthogonal, 372 topological, 371 Gtlillemin, V.V\l.,106 Hahn-Banach theorem, 145 Hair. 69 Half—10n_t! cylinder. 466 line, 46.’) strip, 466 Half-space, 19 Handle, 8 Hardy, G.H., 179 Has oneend, 23 Heinlein, Robert A.,84 Hausdorﬁf, 459 Homogeneous, 7 Homomotphism continuous, 387 ofLiealgebras, 380 Homotopic, 104, 277 Homotopy, 104, 277 Hopf, H.,342, 450Hyperbolit cosine, 356 sine, 356 tangent, 356 Ideal ofaLiealgebra, 410 Identiﬁcation, 10 Imbedding, 49 topological, 14 Immersed submanifold, 47 Immersion, 46 topological, 14,46 Implicit function theorem, 60 Indeﬁnite metric, 350 Independent infinitesimals, 314 Index ofinner product, 349 Index ofvector ﬁeld onamanifold, 447 on1R",446 Indices debauch of,39,123 raising andlowering, 351 Induced bundle, 101 orientation, 260 Inequalities, principle ofirrelevanc algebraic, 233 Inertia, Sylvester’s Law of,349 Inﬁnite volume, 312 Inﬁnitely small change, lll Inﬁnitely small displacements, 314 Inﬁnitesimal generator, 148 Inﬁnitesimals, independent, 314 Initial conditions, 136 ofintegral ctnve, 136 Initial segment, 462 Inner product, 227, 301 preserving, 304-, 372 usual, 301 lnside, 21 Integrability conditions, 189 Integrable distribution, 192 Integrable function Darboux, 283 Riemann 283 Hopf—Rinow—de Rham Theorem, 342 ,C Integral curve, 136 Darboux, 283 line, 239, 243 manifold, 179, 181 maximal, 194 ofa differential equation, 136 Riemann, 283 surface, 245 Integration, 136, 226, 239 Invariance ofDomain, 3 lnvariant, 128, 232 deﬁnition, 214 Irrelevance ofalgebraic inequalities, principle of,233 Isometry, 340 Isomorphic Liegroups. locally, 382 lsomorphism, natural. 108 lsotopic, 294 Jacobi identity, 155, 376 forthebracket inanyring, 378 _]acobian matrix, 40 Jordan Curve Theorem, 21,435 Kelley, _]_,460, 463, 464 Kink, 366 Klein bottle, 18,435 Kneser, H.,472 Kneser, M., 473 Lang, 5.,145 Laplace’s expansion, 230 Laplacian, 58 Law Oflnertia, Sy1vesler's, 349 Leaf, 194 Leap offaith, 464 Left invariant form, 394 11form, 400Indct 485 vector ﬁeld, 374 Left translation, 374 Length, 243, 305, 312 ofaCLITVC, 59 Liealgebra, 376 abelian, 376, 382, 395 commutative, 376 homomorphism of,380 ideal of,410 opposite, 407 Liederivative, 150 Liegroup, 371 arcwise connected subgroup of,409 closed stibgroup of,391 local, 415 normal subgroup of,410 topologically isomorphic, 388 Liesubgroup, 373 Lie’s fundamental theorem ﬁrst, 414 second, 415 third. 416 Limit ordinal, 463 set.60 Line integral, 239, 243 Linear diiierential equations, 165 systems of,171 Linear transformation adjoint of,103 contraction of,121 positive deﬁnite, 104 positive semi-deﬁnite, 104 Linl-ting number, 296 Lipschitz condition, 138 Littlew0od,E., 179 Lives atpoints, 119 Lobachevskian non-Euclidean geome- try,368 Local flow, 14-4 Liegroup, 415 one-parameter group oflocal diHeo- morphism, 148 spanned locally, 179 triviality, 71 Local theory ofintegral manifolds, 190 486 Index Locallt Mayer-Vietoris Sequence, 424 compact, 20 forcompact supports, 43] connected, 20 Measure zero, 40,4] ﬁnite cover, 50 Mesh, 239 isomorphic Liegroups, 382 Metric Lipschitz. 139 bi-invariant, 401 one-one, I3 Euclidean, 305, 3l5 pathwise connected, 20 indeﬁnite, 350 Long Riemannian, 308, 3ll cylinder, 466 usual, 312 line, 465 spaces, disjoint union of,4,20 ray, 4-63 Milnor,_].'W., 4-? closed. 465 Mod 2degree, 295 open, 46:3 Mobius strip, 10 Lower sum, 283 generalized, lO{l Multi-index, 208 Multilinear function. ]l5 Munkres,J.R., 34.106 MacKenzie. R.E., lO6 Magic, 214 Manifold, 1,459 analytic. 34 n-dimensional, 4 atlas for.28 n-forms, leftinvariant, 4-(JO boundary of.19 n-holed torus, 9 bounded. 19 n-manifold, 4 closed. I91 n-plane bundle, 71 C’, 34 n-sphere, 7 C0,34 n-torus. 7 C°°, 291 Natural Q-valued l—form, 403 cliﬂerentialnle, 29 Natural isomorpliisin, 108 climension of.4 Neighborhood, tubular, 345 imbedding inRN. 51? Newman, M.H.A.. 3 integral. 179, l8I Nice cover, 4-38 maximal, 194 Non-bounded, l9 non-nictrizable, 465, 466 N011-degenerate, 30] orieiitzltion of.86 Non—Euclidean geonietry smooth. ‘Z9 elliptical, 367 Manifold-wit]i—boundary. I9 Lobachevskian. 368 C°°, 31‘ Non-metrizable manifold, 465, 4-66 Map N01'l'-O1'i€1‘ll.&i')|t‘ between COI11p]€‘}CC5_ 4-‘2l bundle, 86 bundlc. '73 manifold, 86 ranl-: of.4ll Norm, 303 Massey. \’\’.S.. 3 ])1‘eser\'ing, 304_ 37') I\’lal1'i>£ gr0l1pS. 3'72 Normal Maximal intcgral manifold, I94 bundle, 344 Normal (continued) space, 459 subgroup ofaLiegroup, 4-10 outward unit, 351 Nowhere zerosection, 209 Odd ordinal, 467 relative tensor, 134, 288 scalar density, 133, 259 One-dimensional distribution, 179 One-dimensional sphere, 6 One—parameter group ofdiﬂeomorphisms, 148 oflocal diffeomorphisms, local. 148 One-parameter subgroup, 384 Open long ray,465 map, 60 submanifold, 2 Opposite group, 407 Liealgebra, 407 Ordet isomorphic, 461 isomorphism, 461 topology, 465 Ordered set,461 Ordering, 460 Ordinal numbers, 463 Orientable bundle, 86 manifold, 86 Orientation ofa bundle, 85 ofa manifold, 86 ofavector space, 84 preserving, 84,85,88,105, 248 reversing, 84,88,248 Orthogonal group, 61,372 Orthonormal, 304, 348 Ortlaonormalization process, Grain- Scbmidt, 304 Osgood’s Theorem, 284Index 487 Outside, 21 Outward pointing, 260 Outward unit normal, 351 Palais, R.5.,100, 225 Paracompact, 210, 459 end, 468 Parameter cuwes, special, 167 Parameterized byarclength, 313 Partial derivatives, 35 Partition, 239, 245 ofunity, 52 Pathwise connected, 20 Piecewise smooth, 312 Pig,Yfillow, 434 Poincare, H.,450 Poincare dual, 439 Poincare Duality Theorem, 441 Poincare-Hopi Theorem, 450 Poincare Lemma, 225 Poincare upper half?-plane, 367 Point inward, 98 outward, 98,260 Point-derivation, 39 Point-ﬁnite cover, 60 Polar coordinates, 36 integration in,266 Polarization, 304 Pollack, A.,l06 Positive deﬁnite, 104, 301 Positive element ofnorm 1,308 Positive senai-deﬁnite, l04 Product ofvector bundles, 102 tensor, ll6 Projection, 7,30,32 PI‘O_jCCtive plane, ll,435 space, l9,88 Pnoper map, 60,275 Prufer manifold, 46? Pseudometric. 95 488 Quaternions, 100 ofnorm l,373 Radial function, 4-35 Rado, T.,472 Ranl-z ofa form, 229 ofa map, 4-0,98 Rectiﬁable, 59 Reﬁnement ofacover, 50 Regular point, 40 space, 459 value, 4-0 Related vector ﬁelds. 190 Relative scalar, 134, 23l tensor, 134, 231, 288 Reparameterization, 244, 248 Retraclion, 264 deformation, 279 Revolution, surface of,8,321 deRham, G.,342 deRham cohomology vector spaces. 263 With compact supports, 268 dcRham’s Theorem, 263, 457 Riemann integrable, 283 integral, 283 sum, 283 Riemannian metric, 308, 3ll tisual, 312 Right invariant n-form, 400 Right translation, 374 Rinow, W.,342 Roman surface, l7.26 Rosenlicht, M., 472 Rotation group. 621?Zd€,1 Sard’s Theorem, 42,294 Scalar, relative, l34, 23l Scalar density, l33 Schwarz, H.,354 Schwarz inequality, 303, 362 Second countable, 459 Section ofavector bundle, 73 zero, 96 Segment, initial, 462 Self-adjoint linear transformation, I04 Semi-deﬁnite, positive, 104 Separate points andclosed sets, 95 Sequence exact, 419, 422 ofvector bundles, 103 Mayer-Vietoris, 424 forcompact supports ,4-31 ofa pair, 433 Shrinking Lemma, 51 Shrinking Lemma, 60 Shuttle permutation, 227 Simplex ofatriangulation, 42? singular, 285 Simply-connected, 287 Liegroup, 382 Singular cube, 246 simplex, 285 Skew-symmetric, 201, 378 Slice, I94 Slice maps, 54 Smooth, 28 homotopy, 277 manifold, 29 piecewise, 312 Smoothly contractible, 220 homotopic, 2'77 isotopic, 294 Solid angle, 290 Space ﬁlling cuwe, 58 Spanned locally, 179 Special linear group, 6] Special orthogonal group, 6'2 Sphere, 7 Sphere bundle, associated, 451 Standard n-simplex, 426 Singular cube, 246 Star-shaped, 221 Steiner’s surface, l7,26 Sternberg, S.,42,106 Stokes‘ Theorem, 253, 261, 285, 352 Stone-Cech compactiﬁcation, 468 Structure constants, 396 Subalgebra ofaLiealgebra, 379 Subbundle, 198 Subcover, 50 Subgroup Lie, 373 one-parameter, 384 Subnianifold, 49 C°°, 49 closed, 49 immersed, 47 open, 2 Successor ordinal, 464 Sum ofvector bundles, 'Whitney, 101 Support, 33,147 Surface, 7 area, 354 integral, 239 ofrevolution, 8,321 Sylvester’s Law ofinertia, 349 Symmetric bilinear form, 301 System oflinear diﬂerential equations. 171 cr-compact, 4,459 Tangent bundle, 77 Tangent space of1R",64 Tangent vector inward pointing, 98 ofa manifold, 76 ofR",64 outward pointing, 98,260 toactnve, 63,66Index 489 Tensor contravariant, 120 covariant, ll3 even relative, 134, 231 oddrelative, 134, 288 Tensor ﬁeld classical deﬁnition of,123 contravariant, 120 covariant, 1l3 mi:-ted, 121,122 Tensor product, ll6 Thom class, 442 Thom lsomorphism Theorem, 456 Topological group, 371 imbedding, I4 immersion, l4,46 Topologically isomorphic Liegroups, 388 Torus, /,8 n-holed, 7,9 Total space, 71 Totally disconnected, 25 Transitivity, 460 Translation left, 374 right, 374 Triangle inequality, 303 Triangulation, 426 simplex of,427 Trichotomy, 46] Trivial vector bundle, 72 Tubular neighborhood, 345 Two-holed torus, 8 Vick,_].w.,3 Wedge product, 203 Whitney, H.,106 W'hitney sum, 101 5 Tltese books were typeset using Donald E.Knutlfs TEX typesetting system, together with Berthold I-iorn’s DVIPSONE PostScript driver. The ﬁgures were produced with Adobe Illustrator, andnew ormodiﬁed fonts were created using Fontographei. The textfontisllpoint Monotype Bas1<ervi.1le—thou ghtheem-dash hasbeen modiﬁed—together with itsitalic. The elegant swashes oftheitalic 31andf cause problems inwords liketopology and apology, soaspecial gyligature was added; special ggandgfligatures were alsorequired. Although aBaskenrille bold faceisunhistorical, bold type wasuseful inspecial circumstances—mainly forindicating deﬁned terms. The bold facesupplied by Monotype, even the“semi-bold”, isobtru Sively extended, soanon-extended version wascreated. The somewhat bold appearance ofchapter headings, in16point type, results from thelinear scaling, aswellasthefactthattheupper caseBaskerville letters areofsomewhat heavier weight than thelower case. Ontheother hand, the tallinitial letters beginning each chapter were designed specially, since simple scaling would have made them unpleasantly heavy. Athicker setofnumeralswasconstructed forusewith theupper case lettering inchapter headings andstatements oftheorems, andspecial parentheses and other punctuation symbols were also required. Numerous other modifications ofthissort, including additional kerns andalterations ofsetwidths, were made forvarious purposes. Themathematics fonts areavariation oftheMo-r1vT2me fonts, nowbased on theMonotype Times New Roman family, together with theMonotype Times NRSeven and Times Small Text families—presumably these three families arebased ontheoriginal designs forTimes New Roman, which wascreated in three essential sizes: 9,7and5-;point. Theitalic fonts ofthese three families were used asthebasis forcreating the three separate “math italic” fonts—for useatordinary size, insuperscripts, and insecond order superscripts. The proportions andweights forthese were then used forthethree sizes ofthesymbol font and theother mathematics fonts, including bold symbols, script letters. andadditional special symbols, aswell as fortheextension font anditsbold version. The bold letters formathematics come directly from thebold fonts ofthe Times families, while blackboard bold letters were made byhollowing outthese bold letters. The Adobe Mathematical Pi2fontwasused fortheordinary sized German Fraktur letters, with suitably modiﬁed versions used forsuperscripts. The covers, painted bytheauthor inhisspare moments, areloosely based onSamuel Taylor Coleridge’s poem T720Rime oft/ze Arzcient Jldotinet. CORRECTIONS FOR VOLUME I pg.3,line3-:change d,-(x,y) <1to0',-(x, y)51. pg.14;relabel thelower leftpartofthecentral ﬁgure as /12 -42 -' ~4 \0 \ PP at;\ 1\ »\ - At i/it pg.19:replace thenext—to—last paragraph with thefollowing: . Thesetofpoints inamanifold-with-boundary thatdonothave aneighborhood homeomorphic toR"(butonlyonehomeomorphic toll-ll“)iscalled theboundary ofMandisdenoted byBM. Equivalently, x6BMifandonly ifthere isaneighborhood Vofxand ahomeomorphism ¢:V——>ll-ll"such that¢(x) =0.IfMisactually amanifold, then 8M=I3,and8Mitself isalways amanifold (without boundary)'. Q1. U1pg.43:Replace thelastlineanddisplayed equation with thefollowing: Since rank f=kinaneighborhood ofp,thelower rectangle inthematrix Q= Dk+1lf/k+1- --Dk+1WmBx! D",,r,k+l Dnvrlmpg.22:Replace thetopleftﬁgure with l pg.60,Problem 30:Change part(f)andaddpart(g): (f)IfMisaconnected manifold, there isaproper map f:M-—>R;thefunction fcanbemade C°°ifMisaC°°manifold. (g)Thesame istrueifMhasatmost countably many components. pg.61,Problem 32:Forclarity, restate part(c)asfollows: (c)This isfalseiff:M1-—>Risreplaced with f2M1—->Nforadisconnected manifold N. pg.70:Replace thelasttwolines ofpage 70andtheﬁrsttwolines ofpage 71withthefollowing: theorem oftopology). Ifthere were awaytomap T(M, 1'),ﬁbre byﬁbre, homeomorphically onto MxR2,then each upwould correspond to(p,v(p)) forsome v(p)6R2,andwecould continuously pickw(p) 6R2,corresponding toadashed vector, byusing the criterion thatw(p)should make apositive angle with v(p). pg.78:thethird display should read: 0=3(0)=£(fh) =f(P)5(l1) +lt(P)£(f) =0+5(f)- pg.103,Problem 29(d). Addthehypothesis thatMisorientable. pg.117: After thenexttolastdisplay, A(X1, ...,Xk)(p) =A(p)(X1(p), ...,X,t,(p)), add: IfAisC°°,then /TisC°°, inthesense that)T(X1,...,Xk) isaC°°function forallC°°vector ﬁelds X|,...,Xk. pg.118:Addthefollowing tothestatement ofthetheorem: IfAisC°°,then Aisalso. pg.119:Addthefollowing attheendoftheproof: Smoothness ofAfollows from thefactthatthefunction A,-,,__,-R is.A,(3/3x;,,...,3/Bx,-,,). pg.131,Problem 9:LetFbeacovariant functor from V, . pg.133. Though there isconsiderable variation interminology, what arehere called “odd scalar densities” should probably simply be called "scalar densities”; what arecalled “even scalar densities” might bestbecalled “signed scalar densities”. Inpart(c)ofProblem 10,weshould beconsidering thehofpart(a),notthehofpart(b)!Thus conclude thatthebundle ofsigned scalar densities (notthescalar densities) isnottrivial ifMisnotorientable. pg.134. Extending thechanged terminology from pg.133,weshould probably speak ofthebundle of“signed tensor densities oftype andweight w”(though sometimes theterm relative tensor isused instead, restricting densities tothose ofweight 1),when the transformation ruleinvolves (det/1)“, omitting themodiﬁer “signed” when itinvolves Idet/11"’. pg.143. Thehypothesis ofTheorem 3should bechanged sothatitreads: Letx6Uandletaha; betwomaps onsome open interval Isuch thata1(1),a;(I) CU, oo’(I)=f(t1t(I)) t'=1,2 and at1(tO) =o:2(t9) forsome toeI. Andtheﬁrstsentence oftheproof should bedeleted. pg.177. Problem 17,part(d)should begin: (d)Letf:M-—>N,andsuppose thatﬂu,=0. ForX,,,Y,, eM,, and . pg.198. InProblem 5,wemust alsoassume thateach A;EBA1isintegrable. pg.226. Inthecomutative diagram, thelower right entry should be“I-forms onN”. pg.233. Thereference “pg.V375” refers topg.375ofVolume V. pg.237. InProblem 26,replace parts (b)and(c)with: (b)Determine theid‘component ofv1x xv,,_1 interms ofthe(n—1)x(n—1)submatrices ofthematrix (U1)vn vxw=(v2w3 —v3w2,v3w1— vlwi, vlwz —vzwl).Inparticular, forR3,show that pg.292. InProblem 20,thecondition U;F1U;gé8should beU;r‘tU,~.,.1 56El. pg.408. Problem I6(b)should read: “For anyLiegroup G,show that ”. pp.408-410. Forconsistency with standard usage, Au!should bereplaced with Aut, andthen replace Endwith End. Inpart(g)of Problem I9,addthehypothesis thatHisaconnected Liesubgroup. pg.411. Thedisplay inProblem 21,part(c)should read: t-t>’""tw AtnAA11+<-1>"‘m A[AAw11+t-n'"'tt AtoA1111=0.