spivak-intro-diff-geom-v1-3ed

‘A Comprehensive Introduction to. DIFFERENTIAL GEOMETRY VOLUME ONE Third Edition MICHAEL SPIVAK PUBLISH OR PERISH, INC. “es Houston, Texas 1999 ACKNOWLEDGEMENTS Iamgreatly indebted to Richard 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 TTprefacetothefirstedition,reprintedonthesucceeding pages,excused thisbook’s deficiencies ongrounds that can hardly bejustified now that these “notes” truly have become abook. Atonetime Ihadoptimistically planned tocompletely revise allthismaterial forthemomentous occasion, butIsoon realized thefutility ofsuch anunder- taking. AsIexamined these fivevolumes, written somany years ago, Icould scarcely believe that Ihad once had theenergy tolearn somuch material, or even recall how Ihad unearthed some ofit. SoIhavecontented myselfwiththecorrection oferrorsbrought tomyatten- tion bydiligent 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 outthat since thefirst volumes ofthis series made their appearance in1970, references inthetext to“recent” results should beplaced incontext. Preface totheFirst Edition HOW THESE NOTES CAME TO BE and how they did not come tobeabook For many years Ihave wanted towrite the Great American Differential Geometry book. Today adilemma confronts any one intent onpenetrat- ing the mysteries ofdifferential geometry. Onthe one hand, one can consult numerous classical treatments ofthe subject inanattempt to form some idea how the concepts within itdeveloped. Unfortunately, amodern mathematical education tends to make classical mathematical works inaccessible, particularly those indifferential geometry. Onthe other hand, one can now find texts asmodern inspirit, and asclean in exposition, asBourbaki's Algebra. But athorough study ofthese books usually leaves one unprepared toconsult classical works, and entirely ignorant ofthe relationship between elegant modern constructions and their classical counterparts. Most students eventually find that this ignorance ofthe roots ofthe subject has its price --noone denies that modern definitions are clear, elegant, and precise; it's just that it's impossible tocomprehend how any one ever thought ofthem. And even after one does master amodern treatment ofdifferential geometry, other modern treatments often appear simply tobeabout totally different subjects. Ofcourse, these remarks merely mean that nomatter how well some ofthe present day texts achieve their objective, Inevertheless feel that an introduction todifferential geometry ought tohave quite different aims. There are two main premises onwhich these notes are based. The first premise isthat itisabsurdly inefficient toeschew the modern language ofmanifolds, bundles, forms, etc., which was developed precisely in order torigorize the concepts ofclassical differential geometry. Rephrasing everything inmore elementary terms involves incredible x Preface totheFirst Edition contortions which are not only unnecessary, but misleading. The work ofGauss, for example, which uses infinitesimals throughout, ismost naturally rephrased interms ofdifferentials, even ifitispossible torewrite itinterms ofderivatives. For this reason, the entire first volume ofthese notes isdevoted tothe theory ofdifferentiable manifolds, the basic language ofmodern differential geometry. This language iscompared whenever possible with the classical language, so that classical works can then beread. The second premise for these notes isthat inorder for anintroduction todifferential geometry toexpose the geometric aspect ofthe subject, anhistorical approach isnecessary; there isnopoint inintroducing the curvature tensor without explaining how itwas invented and what it has todowith curvature. Ipersonally felt that Icould never acquire asatisfactory understanding ofdifferentiable geometry until Iread the original works. The second volume ofthese notes gives adetailed exposition ofthe fundamental papers ofGauss and Riemann. Gauss' work isnow available inEnglish (General Investigations ofCurved Surfaces; Raven Press). There are also two English translations ofRiemann's work, but Ihave provided a(very free) translation inthe second volume. Ofcourse, Idonot think that one should follow all the intricacies of the historical process, with its inevitable duplications and false leads What isintended, rather, isapresentation ofthe subject along the lines vhich its development might have followed; asBernard Morin said tome, there isnoreason, inmathematics any more than inbiology, why ontogeny must recapitulate phylogeny. When modern terminology finally isintroduced, itshould beasanoutgrowth ofthis (mythical) historical development. And all the major approaches have tobepresented, for they were all related toeach other, and all still play animportant role. Preface totheFirst Edition xt Atthis point Iamreminded ofapaper described inLittlewood's Mathematician's Miscellany. The paper began "The aim ofthis paper is toprove ..." and ittranspired only much later that this aim was not achieved (the author hadn't claimed that itwas). What Ihave outlined above isthe content ofabook the realization ofwhose basic plan and the incorporation ofvhose details would perhaps beimpossible; what Ihave written isasecond orthird draft ofapreliminary version ofthis book. Thave had torestrict myself towhat Icould write and learn about within the present academic year, and all revisions and corrections have had to bemade within this same period oftime. Although Imay some day beable todevote toits completion the time which such anundertaking deserves, atpresent Ihave noplans for this. Consequently, Iwould like tomake these notes available now, despite their deficiencies, and with all the compromises Ilearned tomake inthe early hours ofthe morning. These notes were written while Iwas teaching ayear course indif- ferential geometry atBrandeis University, during the academic year 1969-70. The course was taken bysix juniors and seniors, and audited by afew graduate students. Most ofthem were familiar with the material in Calculus onManifolds, which isessentially regarded asaprerequisite. More precisely, the complete prerequisites are advanced calculus using linear algebra and abasic knowledge ofmetric spaces. Anacquaintance with topological spaces iseven better, since itallows one toavoid the technical troubles which are sometimes relegated tothe Problems, but I tried hard tomake everything work without it. The material inthe present volume was covered inthe first term, except for Chapter 10, which occupied the first couple ofweeks ofthe second term, and Chapter 11, vhich was not covered inclass atall. Wefound it necessary totake rest cures ofnearly aweek after completing Chapters 2, 3,and7. The same material could easily beexpanded toafull year course a Preface totheFizst Edition jnmanifold theory with apace that few yould describe asexcessively leisurely. Iamgrateful tothe class for keeping upwith myaccelerated pace, for othervise the second half ofthese notes would not have been written. Iamalso extremely grateful toRichard Palais, vhose expert knowledge saved meinnumerable hours oflabor. Wrath, 1970 TABLE OF CONTENTS Although thechapters arenotdivided intosections, thelisting foreach chapter gives some indication which topics aretreated, andonwhat pages. CHAPTER 1.MANIFOLDS Elementary properties ofmanifolds... -. 2....-.-.---.-.1 Examples ofmanifolds... 2.0eB Problems ara eh a ee nO) CHAPTER 2.DIFFERENTIAL STRUCTURES COstructUrcs eae iaa a ee ee 27 C@ functions 2...eeeeeeeeeBI Partialderivatives 22.1eeeeeeeeeee8S Critical points ©2. ee ee eeeee Immersion theorems ..2.2-1 eeeeee eeee42 Partitions ofunity 2... 2. eeee ee ee ee. 50 Problems... 062 0 te ee ee te ee CHAPTER 3.THE TANGENT BUNDLE Thetangent spaceofR" 2.2... 2.eeeeeeeeee63Thetangent spaceofanimbedded manifold... 2... 2... 67 Vector bundles... 22 ee eee ee TI Thetangent bundle ofamanifold .....---.-----.. 75 Equivalence classes ofcurves, andderivations .-. 2... -... 77 Vector fields.22 eeeeeeee. 82 Orientation©... ee ee ee eeBF Addendum. Equivalence ofTangent Bundles .......... 89 liye goo oo ooo eo booed doo poo oo obo EN) xii xiv Contents CHAPTER 4.TENSORS Thedualbundle 2... 2.-eeeeeee eeeees(107 Thedifferential ofafunction ................. 109 Classical versus modern terminology ..............11 Multilinear functions... 2... 2-2. eee eeee WS Covariant and contravariant tensors .---..--....-.117 Mixed tensors, andcontraction ....-........... 121 Problems .2-2 2. 0 eee ee ee ee eee ee 17 CHAPTER 5.VECTOR FIELDS AND DIFFERENTIAL EQUATIONS Integral curves. 2. ee135 Existence anduniqueness theorems... 2.2... 2..139 Thelocalflow ...2... 0-2 eeeeeeeee es148 One-parameter groups ofdiffeomorphisms ...... 2... .148 Liederivatives... 2. 2ee ee ee ee. 150 Brackes ae i eeeS) Addendum 1.Differential Equations... ...2... ...164 Addendum 2.Parameter Curves inTwoDimensions. ......167 Problems .........-.-.--.---..--.... 1689 CHAPTER 6.INTEGRAL MANIFOLDS Prologue; classical integrability theorems ............ 179 Local Theory; Frobenius integrability theorem... ..2... .190 Global Theory... -2ee ee ee ee 194 CHAPTER 7.DIFFERENTIAL FORMS Alternating functions .2... 22-222. 2eeee 201 Thewedge product... 2.02.eeeeee ee.203 Tee podeoupeoodoonpooaoeboobooOry Differentialofaform.........-....-......210 Frobenius integrability theorem (second version)... ......215 Closed andexactforms ..2... 2... .-.-..-...218 ThePoincaré Lemma... -.22-1 7-222ee ee ee225 Problems 2... 2ee227 Contents au CHAPTER 8.INTEGRATION Classical lineandsurface integrals .2... 22... 7. -.239 Integrals oversingular k-cubes. 2... ee 246 Theboundary ofachain ............ we 248 Stokes’ Theorem... 222 ee ee 253 Integrals overmanifolds. .2... 2... eeee .256 Volume elements... 2.2-2 eee eeee ee 2258 Stokes’ Theorem... 2... 2eeeee eeeeee261 deRham cohomology .........--.--.2.24. .263 Problems .2... ee ee ee 283 CHAPTER 9.RIEMANNIAN METRICS Innerproducts. 6. ee ee 301 Riemannian metrics ©2... ee ee 308Length ofcurves 2... 00.2 eeeeeeeeBID Thecalculus ofvariations ..2... 2... 2.22... 316 The First Variation Formula andgeodesics ..... ~~... .323 Theexponential map... 0. ee 384Geodesic completeness»... ..-.-...0...2...341 Addendum. Tubular Neighborhoods... .......... .344 Problems .2-2... eee ee ee B48 CHAPTER 10. LIE GROUPS Liegroups..-..-2-2-2 eeeeeeeeeBT Leftinvariant vector fields... 2... 2.2eeee OTFLiealgebras 2. ee es. 396Subgroups andsubalgebras... 2. ee 79 Homomorphisms..2...00-2200. 2eees.380 One-parametersubgroups... 2-2-2... 1eeeess382 Theexponential map... 2.2... 2..-2..22...-384 Closedsubgroups ©...2eeeeeeeeeeees89] Leftinvariant forms 2... 2... 22 eeeee .94 Bi-invariant metrics. .2. eeee ee ee 400 Theequationsofstructure... -2.....--........ 402 Problemsea i eeiir200) xvi Contents CHAPTER 11. EXCURSION IN THE REALM OF ALGEBRAIC TOPOLOGY Complexesandexactsequences©...2.eeeeee419 TheMayer-Vietoris sequence... -2.2...424 Triangulations©...eeee426 TheEuler characteristic. .2... eee ee ee 428Mayer-Vietoris sequence forcompact supports... .......430Theexactsequence of'apair 2... 22.1eee es482Poincaré Duality... 2.0... -2-0 eeee.2439 TheThomclas ... 2... 2-2-2. 02s eee eee es442 Index ofavector field... .-..-....--2.......446 Poincaré-HopfTheorem... ..-.-.-.....-+.. 450 Eroblemns rai iia ee a ee 23) APPENDIXA ToChapter] 2... ee459 Problems....22... eee ee ee es 467 ToChaptr2 22-20... 0. ee ee eea7 Problems ..2... 0eeee eeeeeeee ee472 ToChapter6 ....-...-..-2--..-222.... 473 ToChapters7,9,10 .....--......-2222...474 NOTATION INDEX... ...-.-.-.-2-...--22....-477 A Comprehensive Introduction to DIFFERENTIAL GEOMETRY VOLUME ONE CHAPTER 1 MANIFOLDS TXnicestexample ofametricspaceisEuclidean n-spaceR",consisting ofalln-tuples x=(x!,...,x") witheachx!€R,whereRisthesetofreal numbers. Whenever wespeak ofR”asametric space, weshall assume thatit has the “usual metric” ” d(x,» =|°o%- xi, f= unless another metric isexplicitly suggested. Forn=0wewillinterpret R®as thesingle point 0€R. Amanifold issupposed tobe“locally” likeoneofthese exemplary metric spaces R".Tobeprecise, amanifold isametric space Mwith thefollowing property: Ifx€M,then there issome neighborhood Uofxandsome integer n>Osuch thatUishomeomorphic toR”. The simplest example ofamanifold is,ofcourse, just R”itself; foreach x€R”wecantake UtobeallofR". Clearly, R"supplied with anequiva- lentmetric (one which makes ithomeomorphic toR”with theusual metric), isalso amanifold. Indeed, ahasty recollection ofthedefinition shows that anything homeomorphic toamanifold isalso amanifold—the specific met- ricwith which Misendowed plays almost norole, andweshall almost never mention it. [Ifyouknow anything about topological spaces, youcanreplace “metric space” by“topological space” inourdefinition; thisnewdefinition allows some pathological creatures which arenotmetrizable and which 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 ballinR";inthis case wecan take Utobetheentire open ball since anopen ball inR”is homeomorphic toR".This example immediately suggests thenext: anyopen 1 2 Chapter 1 subset VofR”isamanifold—for each x€Vwecanchoose Utobesome open ballwith x€UCV.Exercising amathematician’s penchant forgeneralization, ee U v7 NLnT weimmediately announce aproposition whose proof islefttothereader: An open subset ofamanifoldisalsoamanifold(called,quitenaturally,anopen submanifold oftheoriginal manifold). Theopen 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€V)which ishomeomorphic toR"byahomeomor- phism ¢:U>R",then $(V) CR®isanopen setcontaining $(x). Conse- ¢ CS ”wa quently, there isanopen ballWwith¢(x) «WC ¢(V). Thus x€¢7"(W) C VCU. Since $:V—R”iscontinuous, theset¢~"(W) isopen inV,andthus open inM;itis,ofcourse, homeomorphic toW,andthus toR”.This compli- cated little argument justshows thatwecanalways choose theneighborhood U inourdefinition tobeanopen neighborhood. Manifolds 3 Withalittlethought, itbegins toappear that,infact,Umustbeopen. But toprove this, weneed thefollowing theorem, stated here without proof.* 1.THEOREM. IfUCcR"isopen and f:U->R”isone-one andcontinu- ous,then f(U) CR”isopen. (Itfollows thatf(V) isopen foranyopen VCcU, sof~iscontinuous, andfisahomeomorphism.) Theorem 1iscalled“Invariance ofDomain”, foritimpliesthattheproperty of being a“domain” (aconnected open set)isinvariant under one-one continuous mapsintoR”.Theproofthattheneighborhood Uinourdefinition mustbe open isasimple deduction from Invariance ofDomain, lefttothereader asan easyexercise (itisalsoeasytoseethatifTheorem |werefalse,thentherewould beanexample where theUinourdefinition wasnotopen). Wenextturnourattention totheinteger »appearing inourdefinition. Notice that»maydepend onthepoint x.Forexample, ifMCR?is M={(x,yz):2=0}U{(x,y,z)ix=0andz=I} =M,UM, then wecanchoose 1=2forpoints inM;andn=1forpoints inMz.This KM _—_ M example, bytheway, isanunnecessarily complicated device forproducing one manifold fromtwo.Ingeneral, given M;andM2,withmetrics diandd2,we canfirstreplace each d;with anequivalent metric d;such that d;(x, y)<1for allx,y€Mj;forexample, wecandefine Adi 5.dj=Tea ordj=min(d;, 1). *Allproofs require some amount ofmachinery. The quickest routes arcprobably pro- vided byVick, Homology Theory andMassey, Singular Homology Theory. Anold-fashioned,butpleasantly geometric, treatment maybefoundinNewman, TopologyofPlaneSets. 4 Chapter 1 Then wecandefine ametric donM=MyUMpby d(z,y)=(ieifthereissome/suchthatx,y€M;1 otherwise (weassume thatM,andM)aredisjoint; ifnot, theycanbereplaced bynewsets which are). Inthenewspace M,both M;andM)areopen sets. IfM,andM2 aremanifolds, Misclearly amanifold also. This construction canbeapplied toanynumber ofspaces—even uncountably many; theresulting metric space is called thedisjoint union ofthemetric spaces Mj.Adisjoint union ofmanifolds isamanifold. Inparticular, since aspace with onepoint isamanifold, soisany discrete space M,defined bythemetric dana{ioe?1ifxsy. Although different n’smay berequired atdifferent points ofamanifoldM, itwould seem thatonly one»canwork atagiven point x€M.Fortheproof ofthisintuitively obvious assertion wehave recourse once again toInvariance ofDomain. Asafirststep,wenotethatR”isnothomeomorphic toR™when n#m,forifn>m,then there isaone-one continuous map from R”into anon-open subset ofR”.The further deduction, thatthe»ofourdefinition isunique ateach x€M,islefitothereader. This unique 7iscalled the dimension ofMatx.Amanifold has dimension norisn-dimensional orisan n-manifold ifithasdimension nateach point. Itisconvenient torefer tothe manifold MasM” when wewant toindicate that Mhas dimension n. Consider once more adiscrete space, which isa0-dimensional manifold. The only compact subsets ofsuch aspace arefinite subsets. Consequently, an uncountable discrete space isnoto-compact (itcannot bewritten asacountable union ofcompact subsets). The same phenomenon occurs with higher-dimen- sional manifolds, asweseebytaking adigoint union ofuncountably many manifolds homeomorphic toR”. Inthese examples, however, themanifold is notconnected. Wewillofien need toknow that thisistheonly way inwhich o-compactness canfailtohold. 2.THEOREM. If¥Xisaconnected, locally compact metric space, then ¥is o-compact. PROOF. For each x€Xconsider those numbers r>0such that the closed ball {ye X:d(x,y) sr} Manifolds 5 isacompact set(there isatleast onesuch r>0,since ¥islocally compact). Thesetofallsuch+>0isaninterval. If,forsomex,thissetincludes allr>0, then ¥iso-compact, since co X=Ute xX:d(x,y)<n}. n=! Ifnot, then foreach x€Xdefine r(x) tobeone-halftheleastupperboundof allsuch r. The triangle inequality implies that {ye Xid(xs,y) Sr} C{ye Xsd(x,y)s1+d(m,2)}, sothat fy€Xsd,y)S17—d(xi,x2)} C(yeXsd,y)S17}, which implies that 1 (i)rer)2r2)—5d X2). Interchanging x;andx2gives 1 (2)Ira)—ra)L S5dx2), sothefunction r:¥—Riscontinuous. Thishasthefollowing important consequence. Suppose ACXiscompact. LetA’betheunion ofallclosed balls ofradius r(y) andcenter y,forally€A.Then A’isalsocompact. The proof isasfollows. Let 21,22, 23,-.. beasequence inA’. Foreach ithere isayj€Asuch that2;isintheballofradius r(y;) with center y;.Since Aiscompact, some subsequence ofthey;,which wemight aswell assume isthesequence itself, converges tosome point y€A.Now theclosed ballBofradius 3r(y) and 3r(y) ro(y)y 6 Chapter 1 center yiscompact. Since y;>yand since thefunction riscontinuous, eventually theclosed balls {ye X:d(y, yi) sru)} arecontained inB.Sothesequence 2;iseventually inthecompact setB,and consequently some subsequence converges. Moreover, thelimit point isactually intheclosed ballofradius r(y) andcenter y(Problem 10).Thus A’iscompact. Now letxo€Xandconsider thecompact sets Ay={xo} Ans1 =An’. TheirunionAisclearlyopen.Itisalsoclosed.Toseethis,suppose thatxis apoint intheclosure ofA.Then there issome y€Awithd(x,y)<3r(x). By(1), 1 PQ")2r(x)~5x¥) 12 2 >r(x)-ory(x)=37) >d(x,y). This shows that ify€An, then x€Ay’, sox€A. Since ¥isconnected, and A#9isopen andclosed, itmust bethat ¥=A, which iso-compact. After thishassle with point-set topology, wepresent thelong-promised exam- ples ofmanifolds. The only connected 1-manifolds arethelineRand thecircle, orI-dimensional sphere, S',defined by S'=&€R*:d(x,0) =1}. Manifolds 7 The function f:(0,2r) +S!defined byf(6) =(cos6,sin6) isahome- omorphism; itiseven continuous, though notone-one, on[0,27]. Wewill oftendenotethepoint(cos6,sin@) €S!simplyby6€[0,2z]. (Ofcourse,it isalways necessary tocheck that useofthisnotation isvalid.) The function g:(-m,2) —S',defined bythesame formula, isalsoahomeomorphism; together with fitshows thatS!isindeed amanifold. There isanother waytoprove this,better suited togeneralization. The pro- jection Pfrom thepoint (0,1) onto thelineRx{—-1} CRxR,illustrated in 1) x, Pr) 2 Oy F Le theabove diagram, isahomeomorphism ofS!—{(0,1)}ontoRx{—1}: thisis proved most simply bycalculating P:S!—{(0,1)} >Rx{-1} explicitly. The point(0,1)maybetakencareofsimilarly, byprojecting onto Rx{1},oritsuf- ficestonotethatS?is“homogeneous” —thereisahomeomorphism takingany point intoanyother (namely, anappropriate rotation ofR?). Considerations similar tothese now show that then-sphere S"={xER"! :d(x,0) =D} isann-manifold. The2-sphere $?,commonly known as“thesphere”, isour firstexample ofacompact 2-manifold orsurface. From these fewmanifolds wecan already construct many others bynoting thatifMjaremanifolds ofdimension nj(=1,2), then MyxMpisan(my-+12)- manifold. Inparticular S'x--+x S! —— ntimes iscalled then-torus, while S!xS!iscommonly called “the torus”. Itisob- viously homeomorphic toasubset ofR4,anditisalsohomeomorphic toa certainsubsetofR3whichiswhatmostpeoplehaveinmindwhentheyspeakof 8 Chapter 1 “the torus”: This subset may beobtained byrevolving thecircle {0,y,z) €Rs(y— 1)?+27 =1/4} around thez-axis. The same construction may beapplied toany I-manifold CAD contained in{(0, =)€R?:>0}.Theresulting surface, called asurface of revolution, lascomponents homeomorphic either tothetorus ortothecylinder S!xR,thelatter ofwhich isalsohomeomorphic totheannulus, theregion of theplane contained between twoconcentric circles. The nest simplest compact 2-manifold isthe2-holed torus. Toprovide amore ==) explicit description ofthe2-holed torus,itiseasiest tobeginwitha“handle”, a space homeomorphic toatorus with ahole cutout; more precisely, wethrow Manifolds 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 union of1handles and asphere with 1holes, andthen identifying points on theboundary ofthei*®handle withcorresponding points onthei**boundary piece ofthesphere. There isone2-manifold ofwhich most budding mathematicians make the acquaintance when they stillknow more about paper andpaste than about 10 Chapter 1 metric spaces—the famous Mébius strip, which you “make” bygiving astrip ofpaper ahalftwist before pasting itsends together. This canbedescribed aS analytically astheimage inR?ofthefunction f:[0,2] x(—1,1)+R?defined by J(6,1)=(2058 +1c0s$cos,2sin9+1cos$sind, rsin$). unitvector sin$ eaeae) cos$sing 4/S3-2 cos$ eS —-SS8cos 2cos8 If'we define fon[0,217] x[-1, 1]instead, weobtain theMabius strip with aboundary; asinvestigation ofthepaper model willshow, thisboundary is homeomorphic toacircle, nottotwodisjoint circles. Withourrecently intro- duced terminology, theMabius strip canalso bedescribed as[0,1] x(1,1) with (0,1)and (1,—1) “identified”. i M 0 Ul 7 y + a: oe Wehave notyethadtomake precise thisnotion of“identification”, butour next example willforce theissue. Wewish toidentify each point x€S?with Manifolds i itsantipodal point —x€S?.Thespace which results, theprojective plane, P?, isalotharder tovisualize than previous examples; indeed, there isnosubset ofR?which represents itadequately. aiS Theprecise definition ofP?usesthesame trick thatmathematicians always usewhen they want twothings which arenotequal tobeequal. The points ofP?aredefined tobethesets{p,—p} forp€S?.Wewilldenote thisset by[p]€P?,sothat[—p] =[p]. Wethushave amap f:S?>P?given byf(p) =Lp],forwhich f(p) =f(g) implies p=+g. Wewillpostpone forawhile theproblem ofdefining themetric giving thedistance between two points [p]and[9],butwecaneasily saywhat theopen setswillturn outtobe (andthisisallyouneed toknow inorder tocheck thatP?isasurface}. Asubset UCP?willbeopenifandonlyif{-'(U) CS?isopen.Thisjustmeans that theopen setsofP?areoftheform f(V) where VCS?isanopen setwith theadditional important property thatifitcontains pitalsocontains —p. Sze 12 Chapter 1 Inexactly thesame way, wecould have defined thepoints oftheMébius strip Mtobe allpoints (s,1) €(0,1) x(-1,1) together with allsets{(0,1),(1,-1)}, denoted by[(0,4)]or[(1, -9]. There isamap f:[0,1] x(-1,1) >Mgiven by (st) if's£0,1 £9)={ . [G0] ifs=0or1, andUCMisopenifandonlyiff-'(U) C[0,1]x(-1,1) isopen,sothat theopensetsofMareoftheformf(V)whereVisopenandcontains (s,—1) whenever itcontains (s,1)fors=0or1. TogetanideaofwhatP?lookslike,wecanmakethingseasierforourselves byfirstthrowing away allpoints ofS?below the(x,»)-plane, since theyare identified with points above the(x,y)-plane anyway. This leaves theupper hemisphere (including thebounding circle), which ishomeomorphic tothedisc D?={x€R?:d(x,0) <1}, andwemust identify each p€S!with —p€S'. Squaring things offa bit,thisisthesame asidentifying points onthesides ofasquare according tothescheme shown below (points onsides with thesame label areidentified insuch away thattheheads ofthearrows areidentified with each other). The dotted lines inthispicture arethekeytounderstanding P?.Ifwedistort the A LoA Manifolds 13 region between themabitweseethatthefrontpartofBfollowed bythe back part ofA,attheupper left,istobeidentified with thesame thing atthe lower right, inreverse direction; inother words, weobtain aMébius strip with A ‘A xB ox, x,A _KeNA 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 A 4 |. AUR“Ee, ix47) fla \SN a) So ee oma Lm» | Ow ae aev ee] oe NEE * AHse thedisjoint union ofadiscandaMobius stripwithaboundary, byidentifying points ontheboundary andpoints ontheboundary ofthedisc, both ofwhich arecircles. Thus tomake amodel ofP?wejusthave tosewacircular piece of clothandaclothMobius striptogether alongtheiredges.Unfortunately, alitle experimentation willconvince youthatthiscannot bedone (without having the twopieces ofcloth pass through each other). Thesubset ofIR?obtained astheunion oftheMobius strip andadisc, al- though nothomeomorphic toP?,canstillbedescribed mathematically interms ofP?.There isclearly acontinuous function f:P?>R3whose image isthis subset; moreover, although fisnotone-one, itislocally one-one, that is,every point p€P?hasaneighborhood Uonwhichfisone-one. Suchafunction f 14 Chapter 1 iscalled atopological immersion (thesingle word “immersion” hasamore spe- cialized meaning, explained inChapter 2).WecanthussaythatP?canbe topologically immersed inR?,although nottopologically imbedded (there isno homeomorphism ffrom P?toasubset ofR3).InR*,however, with anextra dimension toplay around with, thedisc canbeadded soasnottointersect the Mébius strip. Another topological immersion ofP?inR3canbeobtained byfirstimmers- ingtheMobius strip sothatitsboundary 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 thefactthat theMébius strip istheprojective plane with a discremoved.) This inner circle canbereplaced byaquadrilateral. When the AzAlB2 pee Seo fe eat AzAtaf Ba B RO ‘Az||Az => (2 |— > Ar],a2 resulting figure isdrawn upinto3-space andtheappropriate identifications are made weobtain the“cross-cap”. The cross-cap together with thedisc atthe bottom isatopologically immersed P2. €2 <>D/C Le_> > Manifolds 15 ‘The onegapinthepreceding discussion isthedefinition ofametricforP?. ‘Themissing metric canbesupplied byanappeal toProblem 3-1,which will later beused quite often, and which thereader should peruse sometime before reading Chapter 3.Roughly speaking, itshows thatthings likeP?,which ought tobemanifolds, are.(Those who know about topological spaces willrecognize itasadisguised case oftheUrysohn Metrization Theorem.) Forthepresent, however, wewillobtain ourmetric byatrick that simultaneously provides an imbedding ofP?inR*,Consider thefunction f:S?>R¢defined by I(x, y,2) =(yz,x2,xy,x?+2y?+32), Clearly f(p) =f(—p). Wemaintain that f(p) =f(g) implies that p=+q. ‘Toprovethis,suppose thatf(x,y,z)=f(a,b,c). Wehave,firstofall yz=be () xz=ac xy=ab. lfa,b,¢ #0, this leads to bx oe fe) a Q) ex ze—. Now (x+y +2) =x? +yr+2? +Uxy xz+yz} =14+2xy+xz +yz), sowealso have tytzP=(a+bto), hence (3) atb+cost(xt+y+z). Using (2),thisgives 4 5 atbtentx(1 +245)=ax(42**),a a a 16 Chapter 1 sox=+a. Similarly, weobtain »=£b, z=£6,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. Ifx#0, then (1)would immediately give y=z=0,sothat (x, ¥,2)=(£1,0,0). Buty= z=0implies (by(1)again) that be=0,sob=0orc=Oand (a,b,c) =(0,41,0)or(0,0,1). These equations clearly contradict x?42y?+32? =a?+2b?+307. Thus x=0also, and wehave (4)yz=be ;peal(5)-2y?+32?=2b?+3c? ®)P+=i. But(6)implies that 2)?4327 =2y?+.3(1 —y*) =3-y', andsimilarly forbandc,so(5)gives 3-y? =3-2? (7) yas. Now (4)gives (3) z=te (this holds even ify= =0, since then z,¢=+1). Clearly, (4)also shows that thesame sign holds in(7)and (8).which completes theproof. Since f(p) =f(g)precisely when p=+g,wecandefine f:P?>R*by FP) =fe). This map isone-one andwecanuscittodefine themetric inP?; d((p}, fal)=(FLD, Fad) =a(S), F)- Manifolds 17 Then onecancheck that theopen setsareindeed theones described above. Bytheway,themap g:P?>R3defined bythefirst 3components of/, 8(Lx,¥,2])=(vz,x2,xy) isatopological immersion ofP?inR?.Theimage inR?isSteiner’s “Roman surface”. YY 2DiCSIS With thenewsurface P?atourdisposal, wecancreate other surfaces in thesame way asthex-holed torus. Forexample, toahandle wecanattach aprojective spacewithaholecutout,or,whatamounts tothesamething,a Mobius strip. The closest wecancome topicturing thisisbydrawing across- capsticking onatorus. Wecanalsojoin together apair ofprojective planes with holes cutout,which amounts tosewing twoMébius strips together along their boundary. Although thiscanbepictured astwocross-caps joined together, ithasanicer, andfamous, representation. Consider thesurface obtained from thesquare with identifications indicated below; itmay alsobeobtained from thecylinder [0,1] xS?byidentifying (0,x)€[0,1]xS?with(1,x’),wherex’ isthereflection ofxthrough afixed diameter ofthecircle. Notice that the 18 Chapter 1 identifications onthesquare force P},P2,P3,andPstobeidentified, sothat theset{P1, Po,Ps,Ps} isasingle point ofour new space. The dotted lines Py Ps A B B (0,x) (1,x’) P. A Pi 2 ‘ {Pi, Pa} {P3, Pa} below,dividing thesidesintothirds,formasinglecircle,whichseparates the surface intotwoparts, oneofwhich isshaded. A2 Ad As,Ar,Ai oeI\BRSe) UBy als. ne|id |Byams, By By Rearrangement ofthetwo parts shows that this surface isprecisely two Mobius strips with corresponding points ontheir boundary identified. The description interms of[0,1] xS'immediately suggests animmersion ofthe surface. Turning oneendofthecylinder around andpushing itthrough itself orients thelefi-hand boundary sothat (0,x)isdirectly opposite (1,x’),towhich itcanthenbejoined, forming the“Klein bottle”. VA Manifolds 19 Examples ofhigher-dimensional manifolds willnotbetreated innearly such detail, but, inaddition tothefamily ofn-manifolds S$”,wewill mention the related family of“projective spaces”, Projective n-space P”isdefined asthe collection ofallsets{p,—p}forp€S”.Thedescription ofthe open setsinP” isprecisely analogous tothedescription forP?.Although these spaces seem to form afamily asregular asthefamily S”,wewillseelater that thespaces P” foreven ndiffer inavery important wayfrom thesame spaces forodd1. 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 ot:these “boundaries” donothave neighborhoods homeomorphic toR”,but they dohave neighborhoods homeomorphic toanimportant subset ofR".The (closed) half-space H”isdefined by H”={(x!,...,x") €RB":x”>0}. Amanifold-with-boundary isametric space Mwith thefollowing property: Ifx€M,then there issome neighborhood Uofxandsome integer n2Osuch that Uishomeomorphic toeither R"orH". Apoint inamanifold-with-boundary cannot have aneighborhood homeo- morphic toboth R”andH"”(Invariance ofDomain again); wecantherefore distinguish those points x€Mhaving aneighborhood homeomorphic toH”. Thesetofallsuchxiscalledtheboundary ofMandisdenoted by0M.IfMis actually amanifold, then [email protected] that ifMisasubset ofR”,then 2M isnotnecessarily thesame astheboundary ofMintheoldsense (defined for anysubset ofR”};indeed, ifMisamanifold-with-boundary ofdimension <7, then allpoints ofMwillbeboundary points ofM. Ifmanifolds-with-boundary arestudied asfrequently asmanifolds, itbecomes bothersome tousethislong designation. Often, theword “manifold” isused for“manifold-with-boundary”. Amanifold inoursense isthen called “non- hounded”; anon-bounded compact manifold iscalled a“closed manifold”. We willstick totheother terminology, butwillsometimes use“bounded manifold” instead of“manifold-with-boundary”. 20 Chapter 1 PROBLEMS 1,Show thatifdisametric onX,then both d=d/(1 +d) andd = min(1,d) arealsometrics andthattheyareequivalent tod(ie.,theidentity map 1:(X,d) +(X,d) isahomeomorphism). 2.If(Xj,d;)aremetric spaces, fori¢1,withmetricsd;<1,and¥;0.X;=@ fori#j,then(X,d)isametric space,where ¥=U,Xj,andd(x,y) = di(x, 7)ifx, €X;forsome i,while d(x, y)=1otherwise. Each X;isan open subset ofX,and ¥ishomeomorphic toXifandonly ifY=U;¥i where the¥;aredisjoint open setsand ¥;ishomeomorphic toX;foreach i. The space (¥,d)(oranyspacehomeomorphic toit)iscalledthedisjointunion ofthespaces X;. 3.(a)Every manifold islocally compact. (b)Every manifold islocally pathwise connected, andaconnected manifold is pathwise connected. (c)Aconnected manifold isarewise connected. (Apath isacontinuous image of[0,1], butanarcisaone-one continuous image. Adifficult theorem states that every path contains anarcbetween itsendpoints, butadirect proof of arcwise-connectedness canbegiven formanifolds.) 4,Aspace Xiscalled locally connected ifforeach x€Xitisthecase that every neighborhood ofxcontains aconnected neighborhood. (a)Connectedness does notimply local connectedness. (b)Anopensubsetofalocallyconnected spaceislocallyconnected. (©)Xislocally connected ifandonly ifcomponents ofopen setsareopen, so everyneighborhood ofapointinalocallyconnected spacecontains anopen connected neighborhood. (d)Alocallyconnected spaceishomcomorphic tothedisjointunionofitscom- ponents. (c)Every manifold islocally connected, andconsequently homeomorphic to thedisjoint union ofitscomponents, which areopen submanifolds. 5.(a)Theneighborhood Uinourdefinition ofamanifold isalwaysopen. (b)The integer ninourdefinition isunique foreach x. 6.(a)Asubset ofann-manifold isann-manifold ifandonlyifitisopen. (b)1fMfisconnected, then thedimension ofMatxisthesame forallx€M. 7.(a)IfUC Ris aninterval andf:U-Riscontinuous andone-one, thenfiseitherincreasing ordecreasing, Manifolds 21 (b)‘The image f(U) isopen. (©)The map fisahomeomorphism. 8.Korthisproblem, assume (1)(The Generalized Jordan Curve Theorem) IfACR" ishomeomorphic toS"“!,thenR"~Ahas2components, andAistheboundary ofeach. (2}IfBCcR®ishomeomorphic toD"={x€R”:d(x,0) <1},then R"~Bisconnected. (a)One component ofR"~ A(the“outside ofA”)isunbounded, andtheother (ihe“inside ofA”)isbounded. (b)IfUC R"isopen, ACUishomeomorphic toS"~? and f:U R”isone-one and continuous (sothat fisahomeomorphism on4A),then (inside ofA)=inside off(A). (First prove C.) (c)Prove Invariance ofDomain, 9.(a)Give anelementary proof thatR!isnothomeomorphic toR”for1>1. (b)Prove directly from theGeneralized Jordan Curve Theorem that R"isnot homeomorphic toR"form#n. 10.Intheproof ofTheorem 2,show thatthelimit ofaconvergent subsequence ofthez;isactually intheclosed ballofradius r(y) and center y. 11.Every connected manifold (which isametric space) hasacountable base foritstopology, and acountable dense subset. P 12.(a)Compute thecomposition f=S!~{(0,1)}} —> R!x{-1} >RI explicitly forthemap Ponpage 7,andshow thatitisahomeomorphism. (b)Dothesame forf:S"-! ~{(0,...,0, 1}>BR"). 13.(a)The textdescribes theopen subsets ofP?assetsoftheform f(V), where VCS?isopen andcontains ~pwhenever itcontains p.Show thatthis lastcondition isactually unnecessary. (b)Theanalogous condition ésnecessary fortheMobius strip, which isdiscussed immediately afterwards. Explain how thetwocases differ. 14.(a)Check thatthemetric defined forP?gives theopen setsdescribed in the text. (b)Check that P?isasurface. 22 Chapter 1 15,(a)Show thatP!ishomeomorphic toS!. (b)Since wecanconsider S”~' CS",andsince antipodal points inS"~! are stillantipodal when considered aspoints inS",wecanconsider P"-! CP”in anobvious way. Show thatP"—P"~" ishomeomorphic tointerior D”={x€ R”: d(x,0) <1}. 16.Aclassical theorem oftopology states that every compact surface other thanS?isobtained bygluingtogether acertainnumber oftori and projective spaces, and that allcompact surfaces-with-boundary areobtained from these bycutting outafinitenumber ofdiscs.Towhich ofthese “standard” surfaces arethefollowing homeomorphic? ‘ia §} |<> a |aholeinay.. SF ahole Ww (vy.aholein(umy”Eaholein(i Saholeme, 17.LetCCRCR?betheCantor set.Show thatR?—Cishomeomorphic tothesurface shown atthetopofthenext page. Manifolds 23 pony this circle isnotinthesurface these circles are /wwJinthesurface 18.Alocally compact (but non-compact) space ¥“has oneend” ifforevery compact CCXthereisacompact KsuchthatCCcKC¥andX~Kis connected. {a)R”hasone end ifn>1,butnotif1=1. (b)R”~{0}does not“have oneend” soR”~{0}isnothomeomorphic toR™. 19.This problem isasequel totheprevious one; itwillbeused inProblem 24. AnendofXisafunction ¢which assigns toeach compact subset CCX¥a non-empty component ¢(C) ofX~C,insuch away that C)CCyimplies (C2) C(GQ). a)IfCCRiscompact, then R~C hasexactly 2unbounded components, the “eft” component containing allnumbers <some N,the“right” onecontaining allnumbers >some N.If¢isanendofR,show thate(C) iseither always the “defi” component ofR—C,oralways the“right” one. Thus Rhas2ends. (b)ShowthatR”hasonlyoneend¢for2>1.Moregenerally, ¥hasexactly oneend¢ifandonly if¥“has oneend” inthesense ofProblem 18. (c)This part requires some knowledge oftopological spaces. Let€(¥) bethe setofallends ofaconnected, locally connected, locally compact Hausdorff spaceX.Defineatopology onXUE(X) bychoosing asneighborhoods Nc(eo) ofanend €9thesets Ne(€o)=€0(C)Ufends€:€(C)=e0(C)}, forallcompact C.Showthat¥U€(X)isacompact Hausdorff space. What isRU€(R), and R”UE(R") forn>1? 20.Consider thefollowing three surfaces. (A)Theinfinite-holed torus: (SS Se 24 Chapter 1 (B)Thedoublyinfinite-holed torus:***SSS B00 (C)The infinite jailcellwindow: —I i i HSee 86 SBieieie a —i — | VL ne— Weer Me WS (a)Surfaces (A)and (C)have oneend, while surface (B)does not. (b)Surfaces (A)and(C)arehomeomorphic! Hint: The region cutoutbythe ines inthepicture below isacylinder, which occurs attheleftof(A).Now draw intwomore Jines enclosing more holes, andconsider theregion between the twopairs. SK AOI JOQIOCta al}KOC lo or lo a Manifolds 25 21.(a)Thethree open subsets ofR?shown below arehomeomorphic. infinite swiss cheese (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 ofthe Urysohn Metrization Theorem, that foranyconnected manifold M,there isahomeo- morphism ffrom Mtoasubset ofthecountable product RxRx---. (a)IfMisaconnected non-compact manifold, then there isacontinuous function f:M—Rsuch that f“goes to00at00, ie,if{xn} isasequence which iseventually inthecomplement ofevery compact set,then f(xn) >00. (Compare with Problem 2-30.) (b)Given ahomeomorphism f:M>RxRx---anda g:M>Rwhich goesto00atco,definef:M>Rx(RxRx---)byf(x)=(g(x),f(@)). Show that f(M) isclosed. (c)There areatmost ¢non-homeomorphic connected manifolds (where ¢=280 isthecardinality ofR). 24.(a)Itispossible forR?—AandR?—B tobehomeomorphic eventhough A andBarenon-homeomorphic closed subsets. (b)IfACR?isclosed andtotally disconnected (theonly components ofAare points), then€(R? —A)ishomeomorphic toA.Hence R?— AandR?—Bare non-homeomorphic ifAand Barenon-homeomorphic closed totally discon- nected sets. (c)Thederived setA’ofAisthesetofallnon-isolated points. Wedefine A® inductively byA=A’andA@+) =(A)’. Foreachnthere isasubset An ofRsuchthatA,“consists ofonepoint. 26 Chapter 1 *(d)There are¢non-homeomorphic closed totally disconnected subsets ofR?. Hint: LetCbetheCantor set,and ¢)<cz<¢3<+--+asequence ofpoints inC.Foreachsequence m)<nz<---,onecanaddasetA,,suchthatits njderived setis{c}. (e)There are¢non-homeomorphic connected open subsets ofR?. 25.(a)Amanifold-with-boundary could bedefined asametric space Mwith theproperty that foreach x€Mthere isaneighborhood Uofxand an integer 1>0such that Uishomeomorphic toanopen subset ofH”. (b)IfMisamanifold-with-boundary, then 4Misaclosed subset ofMand 0M and M—@M aremanifolds. ()IfCi,i€7arethecomponents of9M, and I’CJ,then M—Ujey Giis amanifold-with-boundary. 26.IfMCR”isaclosed setandann-dimensional manifold-with-boundary, then thetopological boundary ofM,asasubset ofR",is9M. This isnot necessarily true ifMisnotaclosed subset. 27.(a)Every point (@,,¢) onSteiner’s surface satisfies b2c?+ac?+a2b? = abe. (b}If(,5,c) satisfies thisequation and04D=Vb?c? +a2c? 4a7b?, then (a,b,c) isonSteiner’s surface. Hint: Letx=be/D, etc. ()The set{(@,b,c) €RB:bc? +ac? +.a7b? =abc} istheunion oftheSteinersurface andofthe portions (—00,—1/2) and (1/2, 00)ofeach axis. CHAPTER 2 DIFFERENTIABLE STRUCTURES WwW:arenowreadytoapplyanalysistothestudyofmanifolds.Theneces- sary tools of“advanced calculus”, which thereader should bring along freshly sharpened, arecontained inChapters 2and 3ofCalculus onManifolds. Wewillusefreely thenotation andresults ofthese chapters, including some problems, notably 2-9, 2-15, 2-25, 2-26, 2-29, 3-32, and 3-35; however, wewill denote theidentity map from R”toR”by/,rather than byx(which willbe used often enough inother contexts), sothat//(x) =x!. Onageneral manifold Mthenotion ofacontinuous function f:M>R makes sense,butthenotion ofadifferentiable function f:M—Rdoesnot. This isthecase despite thefactthat Mislocally likeR”,where differentia- bility offunctions canbedefined. IfUCMisanopen setand wechoose ahomeomorphism ¢:U>R’",itwould seem reasonable todefine ftobe differentiable onUiffo6~!: R"=Risdifferentiable. Unfortunately, if yw:V=R"isanother homeomorphism, and UNV #9,then itisnot necessarily truethatfoy~!:R">Risalsodifferentiable. Indeed, since Soy =fog o(pow'), wecanexpect fey! tobedifferentiable forallfwhich make fo~" differ- entiable onlyif¢oy~!: R”>R"isdifferentiable. This iscertainly notalways vu ¢ R" R gow! —__ row, 27 28 Chapter 2 thecase; forexample, oneneed merely choose ¢tobehow, where h:R”>R” isahomeomorphism thatisnotdifferentiable. Ifweinsist ondefining differentiable functions onanymanifold, there isno way outofthisimpasse. Itisnecessary toadorn ourmanifolds with alittle additional structure, theprecise nature ofwhich issuggested bytheprevious discussion. Among allpossible homeomorphisms from UCMonto R",wewish toselect acertain collection with theproperty thatoy! isdifferentiable whenever $, areinthecollection. This isprecisely what weshall do,butafewrefinements willbeintroduced along theway. First ofall,wewillbeinterested almost exclusively infunctions f:R">R" which areC®(that is,each component function f!possesses continuous partial derivatives ofallorders); sometimes wewillusethewords “differentiable” or “smooth” tomean C®. Moreover, instead ofconsidering homeomorphisms from open subsets U ofMonto R’",itwillsuffice toconsider homeomorphisms x:U—x(U) CR” onto open subsets ofR". The useoftheletters x,’,etc., forthese homeomorphisms, henceforth ad- hered toalmost religiously, ismeant toencourage thecasual confusion ofapoint p€Mwith x(p) €R",which has“coordinates” x!(p),...,x"(p). The only time thisnotation willbeconfusing (and itwillbe)iswhen wearereferring to themanifold R”,whcre itishard nottolapse back intothepractice ofdenoting points byxandy.Wewilloften mention thepair (x,U), instead ofxalone, justtoprovide aconvenient name forthedomain ofx. IfUandVareopensubsets ofM,twohomeomorphisms x:U>x(U)C R”andy:V>(V) CR"arecalled C™-related ifthemaps pox: x(UNV) >pUNV) xoytls (UAV) >x(U NV) areC®. This make sense, since x(UNV) andy(UNV) areopen subsets ofR". Also, itmakes sense, and isautomatically true, if UNV =9. 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 p€U. Wecaneven imagine amesh ofcoordinate lines onU,byconsidering the Differentiable Structures 29 inverse images underxoflinesinR"parallel tooneoftheaxes. Thesimplest example of'amanifold together withanatlas consists ofR"with anatlas Aofonly onemap, theidentity J:R”>R".Wecaneasily make the atlas bigger; ifUand Varehomeomorphic open subsets ofR",wecanadjoin anyhomeomorphism x:U->Vwith theproperty thatxandx7}areC®. Indeed, wecan adjoin asmany such x’saswelike—it iseasy tocheck that they areallC%-related toeach other. The advantage ofthis bigger atlas U isthatthesingle word “chart”, when applied tothisatlas, denotes something which must bedescribed incumbersome language ifone canrefer only toA. Aside from this, Udiffers only superficially from A;one caneasily construct U from A(and onewould befoolish nottodosoonce andforall).What hasjust been said fortheatlas {7}applies toanyatlas: 1.LEMMA. IfAisanatlas ofC™-related charts onM,then Aiscontained inaunique maximal atlas A’forM. PROOF. LetA’bethesetofallcharts ywhich areC°°-related toallcharts x€A.Itiseasy tocheck thatallcharts inA’areC®-related, soA’isanatlas, and itisclearly theunique maximal atlas containing A.¢ Wenowdefine aC®manifold (ordifferentiable manifold, orsmooth manifold) tobeapair (M,A),where Aisamaximal atlas forM.Thus, about thesimplest example ofaC®manifoldis(R",U),whereU(the“usualC°-structure for R””) isthemaximal atlas containing {7}. Another example is(R,'V) where V contains thehomeomorphism x+x°,whosc inverse isnotC®,together with allcharts C®-related toit.Although (R,U)and (R,V)arenotthesame, there isaone-one ontofunction f:R—Rsuchthat x€U ifand onlyifxofeV, namely, theobvious map f(x) =x.Thus (R,U) and(R,'V) arethesortofstructures onewould want tocall“isomorphic”. The term actually used is 30 Chapter 2 “diffeomorphic”: twoC®manifolds (M,A)and(N,B) arediffeomorphic if there isaone-one onto function f: M—>Nsuch that +.x€BifandonlyifxofeA.Ls KR Themapfiscalledadiffeomorphism, andf~"isclearly adiffeomorphism also. Ifwehadnotrequired ouratlases tobemaximal, thedefinition ofdiffeo- morphism would have hadtobemore complicated. Normally, ofcourse,wewillsuppress mention oftheatlasforadifferentiable manifold, andspeak elliptically of“the differentiable manifold M”; theatlas forMissometimes referred toasthedifferentiable structure forM.Itwill always beunderstood thatR”referstothepair(R”,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 Rwith anyatlas isdiffeomorphic to(R,U). Aproof ofthecorresponding assertion forR?ismuch harder, theproof forR3 would certainly betoodifficult forinclusion here, andtheproof oftheessential uniqueness ofC® structures onR"forn>5requires very difficult techniques from topology. Inthecase ofspheres, theprojections P;and P2from thepoints (0,...,0,1) and(0,...,0,—1) ofS"~! areeasily seen tobeC®-related. They therefore determine anatlas—the “usual C®structure forS”~!”. This alas may alsobe described interms ofthe21homeomorphisms fiSO ER" :x'>0}3RT gi:S™' {xe RY:x!<0} RT defined byfi(x) =gi(x) =(x1,...,x/7!,x/41,...,x"), whichareC%-related toP;andP;.There are,uptodiffeomorphism, unique differentiable structures onS”forn<6.Butthere are28diffeomorphism classes ofdifferentiable structures onS7,andover 16million onS3!. However, weshall notcome close toproving these assertions, which arepart ofthefield called “differential topology”, rather then differential geometry. (Perhaps most astonishing ofall isthequite recent discovery thatR*hasadifferentiable structure thatisnot diffeomorphic totheusual differentiable structure!) Other examples ofdifferentiable manifolds willbegiven soon, butwecan already describe adifferentiable structure A’onanyopen submanifold Nof Differentiable Structures 31 adifferentiable manifold (M,A);theatlasA’consists ofall(x,U) inAwith UCN. Just asdiffeomorphisms areanalogues forC° manifolds ofhomeomor- phisms, thereareanalogues ofcontinuous maps. Afunction f:M>Nis called differentiable ifforevery coordinate system (x,U) forMand (y,V) forN,themap yofox~!: R”—R”isdifferentiable. More particularly, f G®; » x, Re R iscalled differentiable atp€Mifyofox~? isdifferentiable atx(p) for coordinate systems (x,U) and (y’,V) with p€Uand f(p) €V.Ifthisis true foronepairofcoordinate systems, itiseasily seen tobetrue foranyother pair. Wecanthus define differentiability offonanyopen subset M’CM; asonewould suspect, thiscoincides with differentiability oftherestricted map S\M’: M’—N.Clearly, adifferentiable map iscontinuous. Adifferentiable function f:M—Rrefers, ofcourse, totheusualdifferen- tiable structure onR,andhence /fisdifferentiable ifand only iff0x7" is differentiable forcach chart x.Itiseasy toseethat ())afunction f:R"->R'is differentiable asamap between C®manifolds ifandonlyifitisdifferentiable intheusualsense; (2)afunctionf:M—R"isdifferentiable ifandonlyifeachf!:M—R™ isdifferentiable; (3)acoordinate system (x,U)isadiffeomorphism fromUtox(U); (4)afunction f:M—Nisdifferentiable ifandonlyifeachy/ofis differentiable foreach coordinate system yofN; (5)adifferentiable function f:M—Nisadiffeomorphism ifandonlyiffisone-one ontoandf~!:N>Misdifferentiable. The differentiable structures onmany manifolds aredesigned tomake certain functions differentiable. Consider first theproduct M,xM2oftwo differen- 32 Chapter 2 tiable manifolds M;, and thetwo“projections” ;:M,xMz—>M;defined by7;(p1, P2)=pi.Itiseasy todefine adifferentiable structure onM;xMz which makes each m;differentiable. Foreach pair (x;,Us)ofcoordinate systems onM;,weconstruct thehomeomorphism XyxXx2:U,xU2>R42 defined by 1 X2(P1s P2)=Or(P1),X2(p2)), ee, 1XXz=(410-1, X20-72). Then weextend this atlas toamaximal one. Similarly, there isadifferentiable structure onP”which makes themap f:S"+P"(defined byf(p) =[p]={p,—p}) differentiable. Consider anycoordinate system (x,U) forS",where Udoes nolcontain —pifitcon- tains p,sothatf|U isone-one. The map xo(f|U)~? isahomeomorphism onf(U) CP*,and anytwo such areC®-related. The collection ofthese homeomorphisms canthen beextended toamaximal atlas. Toobtain differentiable structures onother surfaces, wefirst note that aC° manifold-with-boundary canbedefined inanobvious way. Itisonly necessary toknow when amap f':H” >R”istobeconsidered differentiable; wecallf differentiable when itcanbeextended toadifferentiable function onanopen neighborhood ofH”. A“handle” isthen aC® manifold-with-boundary. Adifferentiable structure onthe2-holed toruscanbeobtained by“matching” thedifferentiable structure ontwohandles, The details involved inthisprocess are reserved forProblem 14. Todealwith C®functions effectively, oneneeds toknow thatthere arelotsof them. The existence ofC™functions onamanifold depends ontheexistence of C® functions onR”which are 0outside ofacompactset.Webrieflyrecallhere thenecessary facts about such C®functions (c.f.Calculus onManifolds, pg.29). Differentiable Structures 33 (1)The function :R->Rdefined by mix? hA(x)=e x#0 0 x=0 -1 1 isC®,andh“(0) =0forall7. (2)Thefunction j:R>Rdefined by mann?“Get? wo={eGenny?eGx|(1,1) i 0) x¢(-1,1) 4 ; isC%. Similarly, there isaC®function k:R+Rwhich ispositive on(0,5) and0 elsewhere, jus3 (3)The function /:R>Rdefined by 1 H x 5 : 0 ‘0 : 3 isC®; itis0forx<0,increasing on(0,6), and 1forx26. (4)The function g:R”->Rdefined by | =0 8(8) =Jla"/e)--- f(a"/e) ahs = isC%; itispositive on(—e,2) x---x(~e,8) and0elsewhere. OnaC®manifold Mwecannow produce many non-constant C®func- tions, Theclosure {x:f(x) #0)iscalled thesupport off,anddenoted simply bysupport f(orsometimes supp/). 2,LEMMA. LetCcUCMwith Ccompact and Uopen. Then there isaC® function f:M—>[0,1] such that f=1onCand support fcU. (Compare Case2ofthe proofofTheorem 15,} 34 Chapter 2 PROOF. Foreach p€C,choose acoordinate system (x,V)withVCUand x(p) =0.Then x(V) >(~e,8) x---x(—6,€) forsome ¢>0.The function gx (where gisdefined in(4))isC° onV.Clearly itremains C™ ifweextend | ittobe0outside ofV.Letfybetheextended function. The function fpcanbe constructed foreach p,andispositive onaneighborhood ofpwhose closure is contained inU.Since Ciscompact, finitely many such neighborhoods cover C, andthesum, fp,+---+-fp,, ofthecorresponding functions hassupport CU. OnCitispositive, soonCitis>6forsome 8>0.Letf=10(fp+:-*+Spm)» where /isdefined in(3). Bytheway,wecouldhavedefined C’manifolds foreachr>1,notjustfor“p=00”.(Afunction f:R”>RisC’ifithascontinuous partialderivatives uptoorder r).A“C° function” isjustacontinuous function, soaC°manifold is justamanifold inthesense ofChapter 1.Wecanalsodefine analytic manifolds (afunction f:R">Risanalytic ata€R”iffcanbeexpressed asa power series inthe(x!—a‘)which converges insome neighborhood ofa).The symbol C®stands foranalytic, anditisconvenient toagree thatr<co<@ foreach integer r>0.Ifa<B,then thecharts ofamaximalCatlasareall C*-related, butthisatlas canalways beextended toabigger atlas ofC*-related charts, asinLemma 1.Thus, aC8structure onMcanalways beextended toaC®structure inaunique way; thesmaller structure isthe“stronger” one, theC°structure (consisting ofallhomeomorphisms x:U—R”)being the largest. The converse ofthis trivial remark isahard theorem: Fora>1, every C®structure contains aC8structure foreachB>a;itisnotunique, of course, butitisunique uptodiffeomorphism. This willnotbeproved here.* Infact, C*manifolds for«#cowillhardly ever bementioned again. One remark isinorder now; theproof ofLemma 2produces anappropriate C* fimction fonaC®manifold, for0<a<00.Ofcourse, for¢=wtheproof *For aproof seeMunkres, Elementary Differential Topology. Differentiable Structures 35 fails completely (and theresult isfalse—an analytic function which is0onan open setis0everywhere). With differentiable functions now atourdisposal, itisfitting thatwebegin differentiating them. What weshall define arethepartial derivatives ofadif- ferentiable function f:M—R,with respect toacoordinate system (x,U). At thispoint classical notation forpartial derivatives issystematically introduced, soitisworth recalling alogical notation forthepartial derivatives ofafunction Jf:R"=R.Wedenote byD;f(@) thenumber _f(a',....a' +h,..-.4") -f@ im $+. ho h The Chain Rule states that ifg:R™—R”and f:R”>R,then 1 Dj(fog)la)=D>Dif(g(a)-Djs!(a). izt Now, forafunction f:M>Rand acoordinate system (x,U)wedefine of af = dxt(P)=9577=Difox™)(x(p)), .ar i . . (orsimply 35=Di(f©x7') 0.x,asanequation between functions). Ifwe define thecurve cj:(—e,£) >Mby cx(h) =x7'(x(p) +O,---5hy--50)), 2 CYS then thispartial derivative isjust im16) =LP) ho h soitmeasures theratechange offalong thecurve ¢;;infactitisjust(foc;)'(0). Noticethat ax! 1fie;xt fi=jagP)=8={oiiotf. Ifxhappens tobetheidentity mapofR”,thenD;f(p) =2f/x!(p), which istheclassical symbol forthispartial derivative. 36 Chapter 2 Another classical instance ofthisnotation, often notcompletely clarified, is theuseofthesymbols 9/9” and8/96 inconnection with “polar coordinates”. Onthesubset AofR?defined by A=R?—{(x,y)€R?:py=0andx20} | =R?-L | wecanintroduce a“coordinate system” P:A+R?by P(x, y)=7, »),6(%, »)), where r(x,y)=Vx?+y?and6(x,y)istheunique number in(0,2)with ) x=r(x,py)cosO(x,y) A ey) y=r(x,y)sinB(x,y). ‘ao This really isacoordinate system onAinoursense, with itsimage being the set{r:r>0}x(0,27). (Ofcourse, thepolar coordinate system isoften Wnt ~---------- (7,8) “@-axis” ——a| “y-axis” notrestricted tothesetA.One candelete anyrayother than Lif@(x,y) isrestricted tolieintheappropriate interval (6,4 +2m); 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) =(r-cos6,r sin). Differentiable Structures 37 From thisformula wecancompute af/@r explicitly: (foP™)(r, 8)=f(rcos6,r sin8), so a, =Ley =Di(foP™)(P(x, ¥) =Dif(P7'( P(x, ¥))-Di[PV (Py) +Daf(P\(P(x,9)-DilPPP,»)) bythe Chain Rule =Dif(x, )-cosO(x, y)+Daf(x,y)+sin(x,¥). Thisformula justgivesthevalueofthedirectional derivative offat(x,¥), along aunit vector v=(cos@(x,y),sin6(x,y))pointingoutwardsfromthe origin to(x,y).This istobeexpected, because ¢1,theinverse image under P 4}snot.(9) ley,” ©0800.) 8x9) ofacurvealongthe“r-axis”, isjustalineinthisdirection. Asimilar computation gives of p9g9)=DSC, Wr y)sinBC,y)]+DoF(X,lrOr,¥)cosO(x,Y)]- Thevector w=(—sin6(x,y),cos@(x,’))isperpendicular tov,andthusthe direction, atthepoint (x,),ofthecurve ¢zwhich istheinverse image under P ofacurvealongthe“6-axis”. Thefactorr(x,})appears because thiscurve w : Le « 38 Chapter 2 goesaroundacircleofthatradiusas6goesfrom0to2x,soitisgoingr(x,y) times asfast asitshould goinorder tobeused tocompute thedirectional derivative offinthedirection w,Note that 8f/86 isindependent ofwhich lineisdeleted from theplane inorder todefine thefunction @unambiguously. Usingthenotation 8f/8x forDyf,etc.,andsuppressing theargument (x,}') everywhere (thus writing anequation about functions}, wecanwrite theabove equations as af_af af Set = sindorOxcos+aysin af af. of5a=ag +acos6. Inparticular, these formulas also telluswhat 4x/dr etc., are, where (x,y) denotes theidentity coordinate system ofR?.Wehave 4x/8r =cos6, etc.,so ourformulas canbeputintheform af_afax,afayar~ axBr*ByOr af_afax,afay 00 «8x00” ay36° 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 Jt:M—Ris differentiable, then on UNV wehave af eyafaxt 1se sa aayDsaxdytja PROOF. It’stheChain Rule, ofcourse, ifyoujustkeep your cool: a =Lo=Difoy)(p)) =Dif ox™]o [xoy")(p) a =YODifox(eoye) -Dilx0HFOCD) j= Differentiable Structures 39 n =DDS 0x7") -Dilx!oy") ja naf. ax!=La: Br %ja Atthispoint wecould introduce the“Einstein summation convention”. No- ticethatthesummation inthisformula occurs fortheindex j,which appears both“above” (inax!/ay')and“below” (in9f/8x/). Therearescadsoffor- mulas inwhich thishappens, often with hoards ofindices being summed over, andtheconvention istoomit the>sign 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, andbecause bydo- ingthings “right”, onecanavoid what ElieCartan hascalled the“debauch of indices”. Wewilloften write formula (1)intheform a ax! a here /@y! isconsidered asanoperator taking thefunction ftoaf/ay'. The operator taking ftof/8y!(p) isdenoted by a| a|“.axJ a=a]; thus =] =) —W)s] -ay'|, ay,xaye?Oxi|, Forlater usewerecord aproperty of£=/x'|p: itisa“point-derivation”. 4.PROPOSITION. Foranydifferentiable f,g:M—R,andanycoordinate system (x,U) with p€U,theoperator £=9/8!|psatisfies (fa) =F(pye(a) +E(P)8(e)- PROOF. Left tothe reader. If(x,U) and (x',U") aretwocoordinate systems onM,the1x1matrix axti(Fo) 40 Chapter 2 isjust theJacobian matrix ofx‘0x7)atx(p). Itisnon-singular; infact, its inverse isclearly Ox!(So) : Nowif f:M">N"isC® and (y,V)isacoordinate systemaround f(p), therankofthe mx7matrix aie I(3370) 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 g€N isaregular value ifand only ifpisaregularpointoffforeveryp€f-"(q)- This istrue, inparticular, ifg¢{(M)—a non-value offisstilla“regular value”. Iff:R>R,thenxisacritical pointoffifandonlyiff'(x)=0.It ispossible forallpoints oftheinterval [a,6]tobecriticalpoints,although this canhappen only iffisconstant on[a,d]. Iff:R?->Rhasallpoints as critical values, then Dif=D2f=0everywhere, sofisagain constant. On theother hand, afunction f:R?>R?mayhave allpoints ascritical points without being constant, forexample, f(x,))=x.Inthiscase,however, the image f(R2) =Rx{0}CR?isstilla“small” subset ofR?.Themost important theorem about critical points generalizes thisfact. Tostateit,wewillneedsome terminology. Recall that aset ACR"has“measure zero” ifforevery €>0there isa sequence By,Bz,B3,... of(closed oropen)rectangles with oo AcUBn n=l and co vB) <e, n=l where v(B,) isthevolume ofBy. Wewant todefine thesame concept fora subset ofamanifold. Todothisweneedalemma, which inturndepends on alemma from Calculus onManifolds, which wemerely state. Differentiable Structures 41 5.LEMMA. Let4CR"bearectangle andletf:A>R”beafunction suchthat|Djf#| <KonAfori,j=1,...,". Then If) -SO) <0?Kix —y| forallx,y €A. 6.LEMMA. Iff:R"—R"isC}andACR"hasmeasure 0,then f(A) has measure 0. PROOF. Wecanassume that Aiscontained inacompact setC(ince R"isa countable union ofcompact sets). Lemma 5implies that there issome Ksuch that IF) -FO) <0?Kix =yl forallx,y €C.Thus ftakes rectangles ofdiameter dinto setsofdiameter <n’Kd. This clearly implies thatf(A) hasmeasure 0ifAdoes. Asubset AofaC®n-manifold Mhasmeasurezeroifthereisasequence ofcharts (x;,U;), with ACU;Uj,such that each setxj(A 1Uj)CR"has measure 0.Using Lemma 6,itiseasy toseethat ifACMhasmeasure 0,then x(A NU) CR"hasmeasure 0foranycoordinate system (x,U). Conversely, ifthiscondition issatisfied andMisconnected, orhasonly countably many components, then itfollows easily from Theorem 1-2that Ahasmeasure 0.(But ifMisthedisjoint union ofuncountably many copies ofR,and Aconsists of onepoint from each component, then Adoes nothave measure 0}.Lemma 6 thus implies another result: 7.COROLLARY. Iff:M>NisaC'function between twon-manifolds andACMhasmeasure 0,then f(A) CNhasmeasure 0. PROOF. There isasequence ofcharts (x;,U;) with ACU;Ujand each set x;(A NU;) ofmeasure 0.If(y,V)isachart onN,then f(A)NV =U; f(AN U;)OV. Each set VFA NU) AV) =yofox"(x(A NU;)) hasmeasure 0,byLemma 6.Thus(f(A) NV)hasmeasure 0.Sincef(U,Ui) iscontainedintheunionofatmostcountably manycomponents ofNy,itfollows that f(A) hasmeasure 0.4 42 Chapter 2 8.THEOREM (SARD’S THEOREM). Iff:M>NisaC’map between n-manifolds, and Mhasatmost countably many components, then thecritical values offform asetofmeasure 0inN. PROOF. Itclearly suffices toconsider thecase where MandNareR".But thiscase isjustTheorem 3.14 ofCalculus onManifolds. The stronger version ofSard’s Theorem, which wewillnever use(except once, inProblem 8-24), states* that thecritical values ofaCémapf:M">N™ areasetofmeasure 0ifk>1+max(# —m,0). Theorem 8istheeasycase, andthecasem>1isthetrivial case(Problem 20).Although Theorem 8will bevery important later on,forthepresent wearemore interested inknowing whattheimage off:M—Nlookslikelocally, intermsoftherankkoff atp€M.More exact information canbegiven when factually hasrank k inaneighborhood ofp.Itshould benoted thatfmust have rank >kin some neighborhood ofp,because some kxksubmatrix of(A(y! of)/ax!) hasnon-zero determinant atp,and hence inaneighborhood ofp. 9.THEOREM. (1)Iff:@"—N™hasrankkatp,thenthereissomecoor- dinate system (x,U)around pandsome coordinate system (y,V)around f(p) with yofx7 intheform vofoxa',...,a") =(a',...,a*, w@),....~™a)). Moreover, given anycoordinate system ),theappropriate coordinate system onNcanbeobtained merely bypermuting thecomponent functions ofy. (2)Iffhasrank kinaneighborhood ofp,then there arecoordinate systems (x,U)and (y,V)such that yofoxta',...,a") =(a',...,a*,0,...,0). Remark: The special case M=R",N=R™isequivalent tothegeneral theo- rem, which gives only local results. Ifyistheidentity ofR’”,part (1)says that byfirstperforming adiffeomorphism onR”,andthen permuting thecoordi- nates inR™,wecaninsure that fkeeps thefirst kcomponents ofapoint fixed. These diffeomorphisms onR"andR™areclearly necessary, since fmay not even beone-onc onR*x{0}CR”,anditsimage could, forexample, contain only points with firstcoordinate 0. *For aproof, seeMilnor, Topology From theDifferentiable Viewpoint orSternberg, Lectures on Differential Geometry. Differentiable Structures 43 Inpart (2)wemust clearly allow more Jeeway inthechoice ofy,since f(R”) may notbecontained inanyk-dimensional subspace ofR”. PROOF. (1)Choose some coordinate system uaround p.Byapermutation of thecoordinate functions u!andy!wecanarrange that ayo /)_ () de(Dm) #0a,P=1,...,k. Define x= yTof a=,..,k x=” rok+l,...,n, Condition (1)implies that aor/)x ax! _aub Qdet(Eo)=det T#0.0 4 This shows thatx=(xou7!) owisacoordinate system insome neighbor- hood ofp,since (2)andtheInverse Function Theorem show thatxou isa diffeomorphism inaneighborhood ofu(p). Now g=x(a',...,a") means x(q) =(a',...,a"), hence x!(q) =a’, h Yofqgy=a a=l,...,k neence u’(q)=a” r=k+1,....m, so yofoxtal,...ja)=yof(g) forg=x7(a',...,a") =(a',...,a*,__). (2)Choose coordinate systems xandvsothatvof0x7?hastheformin(1). Since rank f=kinaneighborhood ofp,thelower square inthematrix | O auto f) iiAL) = D (ys]YX1 Dm 44 Chapter 2 must vanish inaneighborhood ofp.Thus wecanwrite wa=Wial,...a*) r=k4+1,....m. Define ytaot yay o(v,...,v*). Since (8)you'(,...,8") =yg) forv(g)=(b',...,5") =(b',...,0K, BET —PH. bk),b™=(BI... BK), theJacobian matrix 10 ay) fa AX [el has non-zero determinant, soyisacoordinate system inaneighborhood off(p). Moreover, yofoxtal,...,a") =you love fox(al,...,a") =you(a’,...,a%, yk(a),...,¥"(a)) =(a),...ak, wa) -HHa!,..,0%), wa)—Ha, a) by@) =(a',...,a*,0,...,0). & Theorem 9acquires aspecial form when therank offisnorm: 10.THEOREM. (I)Ifm<nand f:M">N™ hasrank matp,then for any coordinate system (y,V) around f(p), there issome coordinate system (x,U)around pwith yofox(a,...,a") =(a',...,a7). Differentiable Structures 45 (QIn <mand f:M" >N” hasrank natp,then forany coordinate system (x,U) around p,there isacoordinate system (y,V)around f(p) with yofoxal,...,a") =(al,...,a",0,...,0). PROOF. (1)This ispractically aspecial case of(1)inTheorem 9;itisonly necessary toobserve that when k=m,itisclearly unnecessary, intheproof of thiscase, topermute they!inorder toarrange that aye f)_ . det(ab(p))#9a,B=1,...,m; onlytheu!need bepermuted. (2)Since therank offatany point must be<7,therank offequals 7 insome neighborhood ofp.Itisconvenient tothink ofthecase M=R” and N=R”andproduce thecoordinate system yforR”when wearegiven theidentity coordinate system forR”.Part (2)ofTheorem 9yields coordinate systems @forR”andwforR”suchthat vofod(a',...,a") =(a',...,a",0,...,0). Even ifwedonotperform $7?first, themap /fstilltakes R”intothesubset ee R" S(R") =£(G(R")) R” ¢ f v es 7 wife") {(R") which ytakes toR"x{0} CR™—the points ofR”just getmoved to thewrong place inR”x{0}. This canbecorrected byanother map onR”. Define 4by 2(b', ...,b™) =(G10, ...,b"),b" 1,....b™). Then howpof(al,...,a") =lopo fog"(b},...,6") for(b',...,") =(a) =A(b),...,67,0,...,0) =(¢'(1,...,0"),0,...,0) =(a',...,a7,0,...,0), so1.0Wisthedesired y.Ifwearegivenacoordinate systemxonR”other than theidentify, wejust define (BE, b™) =(xO, 6"), B",..., 6); itiseasily checked that y=0wyisnow thedesired y. 46 Chapter 2 Although pisaregular point of incase (1)ofTheorem 10andacritical point incase(2)(if2<m),itiscase(2)which most interests us.Adifferentiable function f:M" —N” iscalled animmersion ifthe rank offisn,the dimension ofthedomain M,atallpoints ofM.Ofcourse, itisnecessary that m=n,anditisclear from Theorem 10(2) that animmersion islocally one-one (60itisatopological immersion, asdefined inChapter 1).Ontheother hand, adifferentiable map fneed notbeanimmersion even ifitisglobally one-one. Thesimplest example isthefunction f:R>Rdefined byf(x) =x3,with f'(0) =0.Another example is er x0 J g()= 0 x=0 ex? x<0. Amore illuminating example isthefunction h:R—R?defined by 5 hh(x) =(g(x), lex)))s .4 although itsimage isthegraph ofanon-differentiable function, thecurveitself manages tobedifferentiable byslowing down tovelocity 0atthepoint (0,0). One caneasily define asimilar curve whose image looks likethepicture below. Three immersions ofRinR?areshown below. Although thesecond and third immersions 8;andf2areone-one, their images arenothomeomorphic B2(R) Differentiable Structures 47 toR.Ofcourse, even iftheone-one immersion f:P—Misnotahomeo- morphism onto itsimage, there iscertainly some metric andsome differentiable structure onf(P) which makes theinclusion map i:f(P) >Manimmer- sion,Ingeneral, asubset M,CM,withadifferentiable structure (notnec- cssarily compatible with themetric Mjinherits asasubset ofM),iscalled an immersed submanifold ofMiftheinclusion mapi: M;>Misanimmersion. Thefollowing picture, indicating theimage ofanimmersion B3:R>S!xS!, Bs(®)COG firsttimearoundAt second time around shows that M;may even beadense subset ofM. Despite these complications, ifMjisak-dimensional immersed submanifold ofM”and Ujisaneighborhood inM;ofapointp€M,,thenthereisa coordinate system (y,¥)ofMaround p,such that UNV=tgeM:yg)=~=y"(q)=0}; a this isanimmediate consequence ofTheorem 10(2), with f=i.Thus, if g:My>NisC®(considered asafunction onthemanifold Mj)inaneigh- borhood ofapoint p€M,,then there isaC®function gonaneighborhood VCM ofpsuch that g=gofonVMMj—we candefine = , Iq’) =YQ) =1,..0k =ay 8g)=8(),whereCronce r=k+l,....n 48 Chapter 2 Ontheother hand, even ifgisC® onallofM,wemay notbeable to define onM.Forexample, thiscannot bedone ifgisoneofthefunctions B;':Bi(M) >R. One other complication arises with immersed submanifolds. IfMy,CMis animmersed submanifold, and f: P>MisaC®™ function with f(P) CMi, itisnotnecessarily truethatfisC®whenconsidered asamapintoM,,withits C®structure. Thefollowing figure shows that/might noteven becontinuous S(P) tf )p M,C M=R My asamap intoMj. Actually, thisistheonly thing that cangowrong: 11.PROPOSITION. IfM; CMisanimmersed manifold, f: P>Mis aC®™ function with f(P) CMj, and fiscontinuous considered asamap intoMj,then fisalsoC®considered asamap into Mj. PROOF. Leti:M,>Mbetheinclusion map. Wewant toshow thati~0f isC® ifitiscontinuous. Given p€P,choose acoordinate system (y,V)forM around f(p) such that Ur=tgeV:yg) =. =y"(q)=0) isaneighborhood off(p) inMyand(y"|U;,...,.»*|Ui) isacoordinate system ofM;onUj. Differentiable Structures 49 Byassumption, i—!©fiscontinuous, so 7!ci(open set)isanopen set. Since U;isopen inMj,thismeans thatf~1(U,) CPisopen. Thus ftakes some neighborhood ofp€PintoUj.Since ally/ofareC®,andy?,..., y* areacoordinate system onUj,thefunction fisC® considered asamap into My, Most ofthese difficulties disappear when weconsider one-one immersions Jf:P>Mwhich arehomeomorphisms onto their image. Such animmersion iscalled animbedding (“embedding” fortheEnglish), Animmersed subman- ifold MyCMiscalled simply a(C®) submanifold ofMiftheinclusion map i:M,>Misanimbedding; itiscalledaclosed submanifold ofMifM,is also aclosed subset ofM. Lo aXeow submanifold ‘There isone way ofgetting submanifolds which isvery important, and gives thesphere S"-! CR" —{0}CR®,defined as{x:}x|?=1},asaspecial case. 12,PROPOSITION. Iff:M”—Nhasconstant rank kona neighborhood off-()), then f~!(y) isaclosed submanifold ofMofdimension n—k(or isempty). Inparticular, ifyisaregular value off:M">N™,then f(y) isan(n—m)-dimensional submanifold ofM(orisempty). PROOF, Left tothe reader. 4 Itistobehoped that however abstract thenotion ofC®manifolds may appear, submanifolds ofR%willscem likefairly concrete objects. Now itturns outthatevery(connected) C®manifold canbeimbedded insome R%,sothat manifolds canbepictured assubsets ofEuclidean space (though thispicture isnotalways themost useful one). Wewillprove thisfactonly forcompact manifolds, butwefirstdevelop some ofthemachinery which would beused in 50 Chapter 2 thegeneral case, since wewillneed itlater onanyway. Unfortunately, there are many definitions andtheorems involved, If©isacover ofaspace M,acover(’ofMisarefinement of©(or “refines 0”)ifforeveryUin@’thereissomeVinOwithUCV(thesetsof0” are“smaller” than those of@)—a subcover isavery special case ofarefining cover. Acover(iscalledlocally finiteifeveryp€Mhasaneighborhood W which intersects only finitely many setsin0. 13,THEOREM. If@isanopen cover ofamanifold M,thenthereisanopen cover ofMwhich islocally finite and which refines @.Moreover, wecan choose allmembers of(’tobeopen setsdiffeomorphic toR”. PROOF. Wecan obviously assume that Misconnected. ByTheorem 1.2,there arecompact setsC1,C2,C3,... with M=C,UGUGU---. Clearly C;has anopen neighborhood U;with compact closure. Then UjUC2hasanopen neighborhood U2with compact closure. Continuing inthisway, weobtain open sets U;,with Ujcompact and Uj¢U;41, whose union contains allC;, and hence isM.Let U-; =Up=9. V4 CS),@Q»yAgLip NowMistheunionfori>|ofthe“annular” regions4;=U;—U;-1. Since cachAjiscompact,wecanobviouslycoverA;byafinitenumberofopensets, each contained insome member of@,andeach contained inV;=Uj4; —Uj-2. Wecanalsochoose these open setstobediffeomorphic toR”.Inthisway we obtain acover ’which refines @andwhich islocally finite, since apoint inU; isnotinVjforj>2+i.& Differentiable Structures 51 Notice that if©isanopen locally finite cover ofaspaceMandCCMis compact, thenCintersects onlyfinitely manymembers [email protected] that anopen locally finite cover ofaconnected manifold mustbecountable (likethe cover constructed intheproof ofTheorem 13). 14,THEOREM (THE SHRINKING LEMMA). Let beanopen locally finite cover ofamanifold M.Then itispossible tochoose, foreachUin@,an opensetU’withU’¢Uinsuchawaythatthecollection ofallU’isalsoan open cover ofM. PROOF. Wecanclearly assume that Misconnected. Let©={Uj, U2,U3,...}. Then Cy=U, —(U2UU3U--) isaclosed setcontained inUj,and M=C; UU,UU3U---. LetUjbean open setwith CyCUjCUJCU). Now =U, ~(U{UU3U---) isaclosed setcontained inU2,and M=UjU C,UU3U---. LetUjbean open setwith CzCUjCcU3CU2.Continue inthisway. Foranyp€Mthere isalargest nwith p€U,,because @islocally finite. Now peU/UUZU--- UU,U(UpsUUngaUs++)5 itfollows that peEUsUUzU+, since replacing Un4; byUs, cannot possibly eliminate p.4 15.THEOREM. Let@beanopen locally finite cover ofamanifold M.Then thereisacollection ofC®functions gu:M—[0,1],oneforeachUinO, such that (1)support ¢yCUforeachU, (2)Sdu(p) =|forallp€M(thissumisreallyafinitesuminsome U neighborhood ofp,by(1)). 52 Chapter 2 PROOF. Case1,EachU_in ©hascompact closure. Choose theU’asinTheorem 14.Apply Lemma 2toU’CcUCMtoobtain aC™function wy: M=[0,1] which is|onU7andhassupport ¢U.Since theU’cover M,clearly SYwu>0everywhere. veo Define vi_u ee UEco Case 2.General case. This case canbeproved inthesame way, provided that Lemma 2istrueforCCUCMwithCclosed (butnotnecessarily compact) and Uopen. Butthisisaconsequence ofCase /: Foreach p€Cchoose anopen setUpCUwith compact closure. Cover M—C with open sets Vghaving compact closure and contained inM—C. Theopencover{Up;Va}hasanopenlocallyfiniterefinement 0towhichCase applies. Let f=>gu, where 0!={U€@:U CU,forsomep}. Ue This sum isC®, since itisafinite sum inaneighborhood ofeach point. Since Xu¢u(p) =|forallp,and¢u(p) =0whenUCVa,clearly f(p)=1 forallp€C.Using thefactthat@islocally finite, itiseasytoseethat support fCU. & 16.COROLLARY. If@isanyopen cover ofamanifoldM,thenthereisa collection ofC® functions ¢;:M—[0,1] such that (})thecollection ofsets{p:$;(p)¥0}islocally finite, (2)Djdi(p)=5forallp€M, (3)foreach ithere isaU€©such that support; CU. (Acollection {@;: M—[0,1}} satisfying (1)and(2)iscalled apartition ofunity; ifitsatisfics (3),itiscalled subordinate to0.) Itisnow fairly easy toprove thelasttheorem ofthischapter. 17.THEOREM. IfM”isacompact C®manifold, then there isanimled- ding {:M—R%forsome N. Differentiable Structures 53 PROOF. Thereareafinitenumber ofcoordinate systems (x1,U;),--- ,(XsUk) withM=UjU---UU,, Choose U;asinTheorem 14,andfunctions yj:M> [0,1]which are|onU/andhave support CU;.Define f:M—RY,where N=nk +k, by SW Xt WheXkWis Wi)» This isan immersion, because anypoint pisinU/forsome i,andonU/,where Wi=1,theNxnJacobian matrix ase Oxexrcontainsthe»xnmatrix =~=].ax? ax} Itisalso one-one. Forsuppose that f(p) =f(g). There issomeisuchthat p€U!, Then Wi(p) =I,soalso ¥i(q) =1.This shows that wemust have q€U;. Moreover, Wai(P) =Wi-Xi), sop=gq,since x;isone-one onU;.#% Problem 3-33 shows that, infact, wecanalways choose N=2n+1. PROBLEMS 1.(a)Show that being C®-related isnotanequivalence relation. (b)Intheproof ofLemma 1,show that allcharts inA’areC%-related, as claimed, 2.(a)IfMisametric spacetogether withacollection ofhomeomorphisms x:U—R"whose domains cover Mand which are C°-related, show that thenatcach point isunique wilhoul using Invariance ofDomain. (b)Show similarly that07iswell-defined foraC®manifold-with-boundary M. 3.(a)AllC®functions arecontinuous, andthecomposition ofC®functions isc™. (b)Afunction f:M—NisC™ ifand only ifgofisC® forevery C? function g:N>R. 4.How many distinct C™ structures arethere onR?(There isonly one upto diffeomorphism; thatisnotthequestion being asked.) 54 Chapter 2 5.(a)IfNCMisopenandA’consists ofall (x,U) inAwith UCN,show that 4’ismaximal for NifAismaximal for M. (b)Show that A’can also bedescribed asthesetofall(x]V O.N,V ON) for (x,V)inA. (c)Show that theinclusion i:N>MisC®, andthat A’istheunique atlas with thisproperty. 6.Check thatthetwoprojections P;andP2onS"-! areC®related tothe2n homeomorphisms f;andg;. 7.(a)IfMisaconnected C®manifold and p,q €M,then there isaC® curve ¢:[0,1] >Mwith c(0) =pand c(1) =q. (b)Itiseven possible tochoose ¢tobeone-one. 8.(a)Show that(M; xM2) xMsisdiffeomorphic toMyx(MzxM3) and that M;xMzisdiffeomorphic toMzxMy. (b)The diffcrentiable structure onM,xM2makes the“slice” maps Pit (Pr, P2) P2*> (Bri,pa) ofMy, Mz>M,xMpdifferentiable forallj;€My, 2€Mo. ()More generally, amap f:N>My,xMzisC®ifandonly ifthecompo- sitions x,0f:N>Myand m20f:N—M2areC®. Moreover, theC® structure wehave defined forMj;xMzistheonly onewith thisproperty. (@)Iffi:N—M;areC™(=1,2),canonedetermine therankof (fi,fa):N—>MyxMzatpintermsofthe ranks off;atp?Forfi:Ni>Mz,show that Sixfa:NyxNo>MyxMa, defined byfixfo(pi, p2)=(AC);fCp2)), isC®anddetermine itsrankintermsofthe ranks offi. 9.Letg:S"-» P"bethemap p++ [p]. Show that f:P”>MisC® ifand onlyiffog:S"-»MisC®.Compare therankoffandtherankoffog. 10.(a)IfUCR®isopenandf:U>Rislocally C®(everypointhasa neighborhood onwhich fisC®), then fisC®°. (Obvious.) (b)Iff:H">RislocallyC®,thenfisC®,ie,fcanbeextended toa C®function onaneighborhood ofH”. (Not soobvious.) 11.Iff:H">Rhastwoextensions g,htoC®functions inaneighborhood ofH”,then DjgandDjharethesame atpoints ofR”—" x{0}(sowecanspeak ofDjfatthese points). 12.IfMisaC®manifold-with-boundary, then there isaunique C®structure on8Msuchthattheinclusion mapi:8M—Misanimbedding. Differentiable Structures 55 13.(a)Let UCM”beanopen setsuch that boundary Uisan(w—1)-dimen- sional (differentiable) submanifold. Show that7isann-dimensional manifold- with-boundary. (Itiswell tobear inmind thefollowing example: ifU={x€ R":d(x,0) <lor]<d(x,0) <2},thenTisamanifold-with-boundary, but aU#boundary U.) (b)Consider thefigure shown below. This figure may beextended byputting ahC$ a smaller copies ofthetwoparts ofS’into theregions indicated byarrows, and then repeating thisconstruction indefinitely, The closure Softhefinal resulting figureisknownasAlexander’s HornedSphere.ShowthatSishomeomorphic toS?. (Hint: The additional points intheclosure arehomeomorphic totheCantor set.)IfUistheunbounded component ofR?—S,then S=boundary U,but Uisnota2-dimensional manifold-with-boundary, sopart(a)istrueonly for differentiable submanifolds. 14.(a)There isamapf:R?>R?suchthat ()£0,0)=(2,0)forallx, (2)S(x,y) CH?fory>0, (3)f(x,y) CR?-H?fory<0, 56 Chapter 2 (4)frestricted totheupper half-plane orthelower half-plane isC, butf itselfisnotC™. . (b)Suppose Mand NareC® manifolds-with-boundary and f:8M —dN isadifleomorphism. LetP=MUyNbeobtained fromthedisjoint union ofMand Nbyidentifying x€2Mwith f(x) €aN.If(x,U) isacoordinate system around p€3M and (y,V) acoordinate system around f(p), with S(U 18M) =VNAN, and (yo f)|U NAM =x|U NOM, wecan define a homeomorphism fromUUVCPtoR"bysending UtoH”byxandVto thelower half-plane bythereflection ofy.Show that thisprocedure does not ao Ley 1rca Nw CO ofy define aC™ structure on P. ()Nowsuppose thatthereisaneighborhood UofaMinMandadiffeo- morphism a:U—8Mx(0,1), such that «(p) =(p,0) forallp€9M, anda similar diffeomorphism B:V>9Nx(0,1). (Wewillbeable toprove later that such diffeomorphisms always exist). Show that there isaunique C®structure a \aVv = as onPsuchthattheinclusions ofMandNareC®andsuchthatthemapfrom UUY toaMx(I, 1)induced by@andfisadiffeomorphism. (®)Byusing twodifferent pairs («,8),define twodifferent C®structures onR?, considered astheunion oftwocopies ofH?withcorresponding points on0H? identified. Show that theresulting C®manifolds arediffeomorphic, butthat thediffeomorphism cannot bechosen arbitrarily close totheidentity map, Differentiable Structures 57 15.(a)Find aC®structure onH!xH!which makes theinclusion intoR? 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 f:R>Rdefined by ee x>0 x) =Ad{sx<0 isC®(theformula e~/" isusedjusttogetafunction which is>0forx<0, ande~/l*! could beusedjustaswell). 17.Lemma 2(asaddended bytheproofofTheorem 15)showsthatifCyandC2 aredisjoint closed subsets ofM,then there isaC®function f:M—>[0,1] such thatC;Cf~'(0) and@cf7'(1). Actually, wecaneven findfwith Cy=f~'0) andCz=f~!(1). Theproofturnsouttobequiteeasy,onceyou know the trick. (a)Itsuffices tofind,foranyclosed CCM,a C®function fwith C=f~'(0). (b)Let{U;} beacountable cover ofM—C,where each U;isoftheform U;=x7"({fa €R":Jal<1) forsome coordinate system xtaking anopen subset ofM—C onto R”, Let fi:M—[0,1] beaC®function withf;>0onU;andfi=0onM—Uj. Functions like ah Bf Oxi? Axsaxk? willbecalled mixed partials offi,oforder 1,2,.... Let @;=supofallmixed partials offi,..., f;ofallorders <i. Show that soSi f=Lamist isC®, and C=f~'(0). 18.Consider thecoordinate system (y',y?)forR?defined by y'(a,b)=a (a,b) =a+b. 58 Chapter 2 (a)Compute 8f/8y!(a, b)fromthedefinition. (b)Also compute itfrom Proposition 3(tofind8///ay/, write each J!interms ofy!andy?). Notice that8f/dy! #8f/81' eventhough y!=/';theoperator 8/8y‘ depends onyand i,notjust ony’. 19.Compute the“Laplacian” a Ee intermsofpolar coordinates. (Firstcompute 8/8xintermsof8/drand0/80; thencompute 87/9x? fromthis).Answer: 1[2(¢72)+ &G§)). 20.Iff:M">N™isC!andm>n,then f(M) hasmeasure 0(provided that Mhasonly countably many components). 21.The following pictures show, forn=1,2, and3,asubdivision of[0,1] x (0,1]into2%"squares, An,1,--.sAn,22"3 Square An,x islabeled simply k.The numbering isdetermined bythefollowing conditions: (a)The lower leftsquare isAn,1- (b)Theupper leftsquare isA,,2n. (c)Squares Ap,x andAp,k41 have acommon side. (d)Squares Ay,4141, Anal42, An,4l+3 An,al44 arecontained inAy—1141- 3 LEPPrPr)GEAH2 BBR ococeee CiTT) RREPEEEGttTTTT ty Define f:[0,1] >[0,1]x(0,1)bythecondition k-\ k SDE Anxforallomst<a Show that /iscontinuous, onto [0,1]x[0,1], and notone-one. Differentiable Structures 59 22.Forp/2" €(0,1],define f(p/2") €R?asshown below. ASG) I £Qf fee) 1) 4=f0)/“SE FO SY f@ (a)Show that fisuniformly continuous, sothat ithasacontinuous extension g:[0,1] >R?, Show thatgisone-one, andthatitsimage willnothave measure 0iftheshaded triangles arechosen correctly. (b)Consider thehomeomorphic image ofS!obtained byadding, below the image ofg,asemi-circle with diameter thelinesegment AB, What does the inside ofthis curve look like? 23,Letc:[0,1] >R"becontinuous. Foreach partition P=(lo,...,tx}of (0,1],define k £(6,P)=Jd(elti),c4-1)). isl Thecurvecisrectifiable if{£(c,P)}isbounded above(withlengthequaltosup(£(c, P)}).Showthattheimageofarectifiable curvehasmeasure 0. 24,(a)IfMisaC™manifold, asetMy;CMcanbemade into ak-dimen- sional submanifold ofMifandonly ifaround each point inM;there isa coordinate system (x,U) onMsuch thatMyQU ={p:x*#'(p) =.) = x"(p) =0}. (b)The subset 44;canbemade into aclosed submanifold ifand only ifsuch coordinate systems exist around every point ofM. 25.Theset{(x,x1):x€R}isnottheimageofanyimmersion ofRintoR?. 26.(a)IfUcRFisopen andf:U>R"-* isC™, then thegraph of Sf=p, f(p)) €R": p€U}isasubmanifold ofR". (b)Every submanifold ofR”islocally ofthisform, after renumbering coor- dinates. (Neither Theorem 9nor10isquite strong enough. You will need 60 Chapter 2 theimplicit function theorem (Calculus onManifolds, pg.41).Theorem 10ises- sentially Theorem 2-13 ofCalculus onManifolds; comparison with theimplicit function theorem willshow how some information hasbeen allowed toescape.) 27.(a)Animmersion from one n-manifold toanother isanopen map (the image ofanopen setisopen). (b)IfMand Naren-manifolds with Mcompact and Nconnected, and Jf:M>Nisanimmersion, thenfisonto. 28,Prove Proposition 12:Iff:M" —Nhasconstant rank konaneigh- borhood off—!(y), then f—'(y) isa(closed) submanifold ofMofdimension n~k (orisempty). 29,Letf:P?+R3bethemap g([x, y,2]) =(yz, x2,xy) defined inChapter 1,whose image istheSteiner surface. Show thatgfailsto beanimmersion at6points (theimage points arethepoints atdistance +1/2 oneach axis). There isawayofimmersing P?inR?,known asBoy’s Surface. SeeHilbert and Cohn-Vossen, Geometry andtheImagination, pp.317-321. 30.Acontinuous function f:X>Yisproper iff~'(C) iscompact for every compact CCY.The limit setL(f) offisthesetofally€Ysuch that y=limf(xn) forsome sequence x1,X2,x3,... €Xwith noconvergent subsequence. (a)L(f) =if'and only iff isproper (b)$(X) CY isclosed ifandonly ifL(f) Cf(X). (c)There isacontinuous f:R+R?with f(R) closed, butL(/) #9. (d)Aone-one continuous function {: ¥>Yisahomeomorphism (onto its image)ifandonlyifL(f)0f(Y)=9. (e)Asubmanifold M,CMisaclosed submanifold ifandonlyiftheinclusion mapi:Mi>Misproper. (f)IfMisamanifold, there isaproper map f:M—R;thefunction fcan bemade C®™ ifMisaC® manifold, 31.(a)Find acover of[0,1] which isnotlocally finite butwhich is“point- finite”: every point of(0,1]isinonly finitely many members ofthecover. (b)Prove theShrinking Lemma when thecover 0)ispoint-finite andcountable (notice thatlocal-finiteness isnotreally used). (©)Prove theShrinking Lemma when 0isa(not necessarily countable) point- finite cover ofanyspace. (You willneed Zorn’s Lemma; consider collections C Differentiable Structures 61 ofpairs(U,U’)whereU€@,U’CU,andtheunionofall U’for(U,U") €@, together with allother U€©covers thespace.) 32.(a)IfMyCMisaclosedsubmanifold, U>M;isanyneighborhood, andf:M;>RisC®, then there isaC®function f:M>Rwith f=f onM,,andwithsupport fCU. (b)This isfalseifAf=RandM;=(0,1). (©)This isfalse ifRisreplaced byadisconnected manifold N. Remark: Itisalsofalse ifM=R?,M,=N=S',andf=identity;infact,in thiscase, fhasnocontinuous extension toamapfrom R?toS!,buttheproof requires some topology. However, fcanalways beextended toaC®function inaneighborhood ofMj(extend locally, andusepartitions ofunity). 33.(a)The setofal]non-singular nxmatrices with real entries iscalled GL(n, R),thegeneral linear group. ItisaC®manifold, since itisanopen subset ofR".Thespecial linear group SL(n,R), orunimodular group, isthe subgroup ofallmatrices with det=1.Using theformula forD(det) inCalculus onManifolds, pg.24,show thatSL(n, R)isaclosed submanifold ofGL(n,R) of dimension n?—1, (b)The symmetric 2xnmatrices may bethought ofasR™"+"/?_ Define w:GL(n,R) >(symmetric matrices) by¥(A) =A-At,where Aisthe transpose ofA.The subgroup y~!(I) ofGL(n,R) iscalled theorthogonal group O(n). Show that A€O(n) ifand only iftherows [orcolumns] ofAare orthonormal. (c)Show that O(”) iscompact. (d)Forany A€GL(n,R), define Ra: GL(n,R) >GL(n,R) byRa(B) =BA. Show thatRyisadiffeomorphism, andthatyoR4=yforallA€O(n). By applying thechain rule, show thatforA€O(n) thematrix iy ij(Fa) hasthesamerankas(Fa): (Here x*!arethecoordinate functions inR™,andy/then(n-+1)/2 component functions ofy.)Conclude from Proposition 12thatO(n) isasubmanifold of GL(n,R). (€)Using theformula VA) =Yanan (A=(ay), k 62 Chapter 2 show that .ay k=ifs ay! ay kaj#i=(ay=d@ jaax’ 2a)k=i=j 0 otherwise. Show that therank ofthismatrix isn(n+1)/2atI(and hence atAforall A€O(n).) Conclude that O(7) hasdimension n(n—1)/2. (f)ShowthatdetA=1forallA€O(n).ThegroupO(n)MSL(n,R)iscalled thespecial orthogonal group SO(n), ortherotation group R(n). 34. LetM(m,n) denote thesetofallmxnmatrices, and M(m,n;k) theset ofallmx nmatrices ofrank k. (a)Forevery Xo€M(m,n;k) there arepermutation matrices Pand Qsuch that AgoB 5 PXoQ=(Gae whereAoiskxkandnon-singular. (b)There issome ¢>0such that Aisnon-singular whenever allentries of A- Agare <e. (If AB PXQ=(Fa3) where theentries ofA—Aoare<€,then Xhasrank kifandonly ifD= CA7"B. Hint: IfI,denotes thekxkidentity matrix, then kk 0 \(AB\_{ A B XIpx)\C D)-\XA4+C xXB+D)° ()M(m,n:k) CM(m,n) isasubmanifold ofdimension k(m+n—k)forall k<m,n. CHAPTER 3 THE TANGENT BUNDLE a Arent v€R”isfrequently picturedasanarrowfrom0tov.Butthereare many situations where wewould liketopicture thissame arrow asstarting _y atadifferent point p€R": —™ P Leu Forexample, suppose c:R—R”isadifferentiable curve. Then c'(t) = (c(t), -..,¢"(t)) isjustapointofR",butthelinebetween (1)andc(t)+¢"(¢) istangent tothecurve, and the“velocity vector” or“tangent vector” c’(/) of thecurveciscustomarily pictured asthearrow from¢()toe()+¢’(t). c(t) +c'() ePrz0) 63 64 Chapter 3 Togive thispicture mathematical substance, wesimply describe the“arrow” from ptop+vbythepair (p,v). The setofallsuch pairs isjustR”xR”, which wewillalsodenote byTR", the“tangent space ofR””; elements ofTR” arecalled “tangent vectors” ofR”.Wewilloften denote (p,v) €TR" byvp (“the vector vatp”); inconformity with thisnotation, wewilldenote theset ofall(p,v) forv€R”byR",, Attimes, itismore convenient todenote a member ofTR" byasingle letter, likev.Torecover thefirstmember ofapair v€TR", wedefine the“projection” map x:R"xR”>R"byx(a,b) =a. Foranytangent vector v,thepoint 7(v) is“where it’sat”. The setx~!(p) may bepictured asallarrows starting atp.Alternately, a8 itcanbepictured more geometrically asaparticular subset ofR"xR",the onevisualizable case occurring when n=1.This picture gives risetosome Wi | || TR" | |\ || |a"(p)|| | Hl fr R" P terminology—we callx~!(p) thefibre over p.This fibre canbemade intoa TheTangent Bundle 65 vector space inanobvious way: wedefine (p,v)®(p,w) =(p,v+wv) ae(p,v) =(p,a-v). (The operations @and «should really bethought ofasdefined on Up) xp), and RxTR", respectively. peR" Usually wewilljustuseordinary +and-instead of©and+.) Iff:R">R”isadifferentiable map,andp€R",thenthelineartrans- formation Df(p): R”>R"™may beused toproduce alinear map from R",>Ry) defined by Up>[Df(PM fp). This map, whose apparently anomalous features willsoon bejustified, isde- noted byfxp; thesymbol f,denotes themap f.:TR" >TR™ which istheunionofallf.p.Sincef.p(v)isdefined tobeavector€R"scpy, thefollow- ingdiagram “commutes” (thetwopossible compositions from TR" toR”are equal), fTR’ —*— TR" *| le POsaitEp/p Thus, f,hasthemap /f,aswell asallmaps Df(p), built into it. This isnottheonly reason fordefining f,inthisparticular way, however. Suppose thatg:R™+R*isanother differentiable function, sothat, bythe chain rule, @ D(g0 f)(p) =Dg(f(p)) ©Df(p)- Byourdefinition, &(DIDO) =(DESPM)(PMOM))ecrepn: This looks horribly complicated, but,using (1),itcanbewritten Bx(fa(Up))=(82S)x(Yp)s 66 Chapter 3 thus wehave B40fe=(80fae Thisrelation wouldclearlyfallapartcompletely iff,(vp)werenotinR™spy; with ourpresent definition off,,itismerely anelegant restatement ofthe chain rule. Henceforth, wewillstate almost allconcepts about Jacobian matrices, like rank orsingularity, interms off,,rather than Df.The “tangent vector” ofa curve c:R+R"canbedefined interms ofthisconcept, also, The tangent vector of¢at¢may bedefined as ew €Rew: [If¢happens tobeoftheform io eO=OSOA---.! c(t) =(4f() forf: RAR i then Mew =0SOc thisvector liesalong thetangent linetothegraph offat(,f(t)).] Notice thatthetangent vector of¢at/isthesame as eo(1e) =[Dee =(CMO, ++. Dew where 1,=(/,J)isthe“unit” tangent vector ofRat¢. 0 i t ly——_+___+ ____._1 , Ifg:R">R”isdifferentiable, then gocisacurve inR™, The tangent TheTangent Bundle 67 vector ofgoc at/is (g0c)a(Iz) =Ba(Co(I1)) =g,(tangent vector of¢at1), y e() _s, B(c()) &(v) Consider nowann-dimensional manifold Mandanimbedding i:M>RY. Suppose wetakeacoordinate system (x,U) around p,Then iox7" isamap fromR”toR¥withrank2.Consequently, (fox~")4(R"x(p)) isann-dimen- sional subspace ofR™i,‘p):This subspace doesn’t depend onthecoordinate RY a, SS =(P) . system x,forifyisanother coordinate system, then (oy) =(fox oxoy')y =(Lox), o(x0y's and (oy aya? Ry) >B"xcey isanisomorphism (with inverse (y0x~")ax(py)- 68 Chapter 3 There isanother way toseethis, which justifies thepicture wehave drawn. Ifc:(—e,e) >R"isacurve with c(0) =x(p), thena =ioxocisa curve inRYwhich liesin(M4), andevery differentiable curve ini() isofthis Oxy) (Pp) x PWaaS/S form (Proof). Now (lo) =(F0X7")e 0Cao), sothetangent vector ofevery @isin(0x7), (R"x(p)). Moreover, every vector inthissubspace isthetangent vectorofsomea,sinceeveryvectorinR"x¢p) 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,/)p. Wenow want toJook atthe(disjoint) union TM) =(JMip ¢i)xRYCTR. peM Wecandefine a“projection” map x:T(M,i) >M by x(v)=p ifvE(M,i)p. AsinthecaseofTR", each “fibre” x~'(p) hasavector space structure also, Beyond thiswehavetolookalittlemorecarefully atsomespecific examples. Consider first the manifold M=S'and the inclusion i:S'>R*, The curvec(6)=(cos6,sin6)passesthrough everypointofS',and c'(0) =(—sin@, cos8)40. Foreach p=(cos6,sin6) €S!,letup=(—sin9,cos 6),(itclearlydoesn’t matter which oftheinfinitely many possible 6’swechoose). Then (S!,i)con- TheTangent Bundle 69 sistsofallmultiples ofthevector u».Wecantherefore define ahomeomorphism up A 2 up Si:T(S',i) >S!xR! byfi(Au,) =(p,d), which makes thefollowing dia- gram commute. T(S!,i) 4.51 xR! INoe [x'"(a,b)=a] Sy Ifwedefine the“fibres” ofx’tobethesetsx’~'(p), theneachfibrehasavector space structure inanatural way, Commutativity ofthediagram means thatfi takes fibres intofibres; clearly f;restricted toafibre isalinear isomorphism onto theimage. Now consider themanifold M=S?and theinclusion i:S?CR3, Inthis casethereisuomapf2:T(S?,i) >S?xR?withtheproperties ofthemapfi. Ifthere were, then, forafixed vector v¥0inR?,thesetofvectors (up) :p€S?} would beacollection ofnon-zero tangent vectors, oneateach point ofS?, which varied continuously, Itisawell-known (hard) theorem oftopology that thisisimpossible (you can’t comb thehaironasphere). ByAesWeASS 4ANNSWess Neen 70 Chapter 3 There isanother example where wecanprove thatnoappropriate homeo- morphism T(M,i) >MxR?exists, without appealing toahard theorem of topology. Themapfwilljustbetheinclusion M—RR?where MisaMobius strip, tobeprecise, theparticular subset ofR?defined inChapter 1—M isthe image ofthemap J:[0,2] x(—1,1) >R?defined by £06,1)=(2088+1cos§cos6,2sin@+cos$sin8,sin$). Ateach point p=(2cos6,2sin 6,0)ofM,thevector 1 I+ re aSfof TN Up=(—2sin 8,2c0s0,0), =fx((1,0).0,0)) isatangent vector. The same istrueforallmultiples off,((0, 1)@,0)), Shown asdashed arrows inthepicture. Notice that L(O,Noo.9) =(2/000, Ylea,0.0 a,=[Foo] =(1,0,0)¢2,0,0), d (2,0,0) while a, HONer») =[Lame] =(10.0200: ar (2,0,0) This means that wecannever pick non-zero dashed vectors continuously onthe setofallpoints (2cos 6,2sin6,0): Ifwecould, then each vector would be Su(0,4.(8))0,0)) forsome continuous function A:[0,22] >R.This function would have tobe non-zero everywhere andalsosatisfy A(2x) =—4(0), which itcan’t (byaneasy theorem oftopology). The impossibility ofchoosing non-zero dashed vectors continuously clearly shows thatthere isnoway tomap T(M,/i), fibre byfibre, TheTangent Bundle 7 homeomorphically onto MxR?,Wethushave another casewhere T(M,i) does not“look like” aproduct MxR". Foranyimbedding i:M>R¥,however, thestructure ofT(M,i) isalways simple locally; if(x,U) isacoordinate system onM,then x~'(U), thepart ofT(M,ji) over U,canalways bemapped, fibre byfibre, homeomorphically onto UxR". Infact, foreach p€U,thefibre (M,i)p equals (10X™")ax¢p) (R"x¢p)) =Mp(R"x¢py) 5 where theabbreviation m,hasbeen introduced temporarily; wecantherefore define fix7\(U) >UxR" by S(Mp(xips)) =(P»)- Instandard jargon, T(M, i)is“locally trivial”. This additional feature qualifies T(M, i)tobeincluded among anextremely important class ofstructures: Ann-dimensional vector bundle (orn-plane bundle) isafive-tuple &=(E,x, B,®,0), where ()Eand Barespaces (the“total space” and “base space” of&, respectively), (2)x:E> Bisa continuous map ontoB, (3)®and©aremaps @:Un) xm"(p)>EO: RxE>E, pes with ®(x7!(p) xx7"(p)) Cx7"(p) and@(R xx7"(p)) Cc x~'(p), which make each fibre x~!(p) intoann-dimensional vector space over R, such thatthefollowing “local triviality” condition issatisfied: Foreach p€B,there isaneighborhood Uofpanda homeomor- phism 1:x~'(U) +UxR"which isavector space isomorphism from each x~'(g) onto qxR",forallg€U. 72 Chapter 3 Because thislocal triviality condition really isalocal condition, each bundle £=(E,x, B,®,©) automatically gives risetoabundle &[4 over anysubset Ac B;tobeprecise, ELA=(xN(A), le"(A),A,|Upea '(p)x"(p), ORx71(4)). Notation ascumbersome asallthisinvites abuse, and weshall usually refer simply toabundle x:E—B,oreven denote thebundle byEalone. For vectors v,w€x~!(p) anda €R,wewilldenote @(v, w)and©(a, v)byv+u, anda-vorav,respectively., The simplest example ofann-plane bundle isjust¥xR"with x:XxR" > Xtheprojection onthefirstfactor, andtheobvious vector space structure on each fibre. This iscalled thetrivial n-plane bundle over Xand willbedenoted bye"(X). The “tangent bundle” TR" isjust e”(R”). The bundle T(S',i) considered before isequivalent toe'(S). Equivalence ishere atechnical term: Two vector bundles |=2): EF)>Band &= 12:E,>Bareequivalent (&~&2)ifthere isahomeomorphism h:E)>E2 which takes each fibre x;~'(p) isomorphically onto x27!(p). The map his called anequivalence. Abundle equivalent toe”(B) iscalled trivial. (The local triviality condition forabundle &just says that &|U istrivial forsome neighborhood Uofp.) Thebundles T(S?,’) andT(M,j) arenottrivial, butthere isanevensimpler example ofanon-trivial bundle. The Mébius strip iseif (notT(M,i)) canbe ronsidered asa1-dimensional vector bundle over S',forMcan beobtained irom (0,1]xRbyidentifying (0,a)with(1,—a),whileS!canbeobtained from = = 0 =I = = . 2e= A = |. TheTangent Bundle 73 {0,1]byidentifying 0with1;themapxisdefined byx(t,a)=ffor0<! <1 and x({(0,4), (1,—@)}) ={0,1}. The diagram above illustrates local triviality nearthepoint {0,1} ofS?.Suppose thats:S!+Misacontinuous function with 205=identity ofM(such afunction iscalled asection). Such amap M 3 s corresponds toacontinuous function $:[0,1] >Rwith 5(0)=~5(1). Since 5 must be0somewhere, thesection smust be0somewhere (thatis,5(@)€~'(0) must bethe0vector forsome 6€S?).This surely shows that Misnotatrivial bundle. Anequivalence isobviously theanalogue ofanisomorphism. The analogue ofahomomorphism isthefollowing* Abundlemapfrom&to&isapairof continuous maps (f,f),with f:Ey>Ezandf:By—Ba,such that (1)thefollowing diagram commutes 5 & on 3B, Be, (2)f:m7(p) >w2-"(f(p)) isalinear map. Thepair(f.,f)isabundle map from TR‘ toTR!foranydifferentiable Sf:R*>RYIfM*™CRE andNC R!aresubmanifolds, i:Mf>R*and j:N>R'aretheinclusions, andthemap _/satisfies {(M) CN,then f, *There areactually several possible choices, depending onwhether oneisconsider-ingallbundles atonce,fixedbundles overvariousspaces,orafixedbasespacewith varying bundles. Thus fmay berestricted tobeanisomorphism onfibres andfto betheidentity, orahomeomorphism. The relations between some ofthese cases are considered intheproblems. 74 Chapter 3 takesT(M,i) toT(N,j);toseethis,justremember thatv€T(M,ji),isthe tangent vector ofacurve ¢inM,sof.(v) isthetangent vector ofthecurve fecin N,andconsequently f,(v) €T(N, j).Inthisway weobtain abundle map from T(M,i) toT(N, j).Actually, itwould have sufficed tobegin with aC™function f:M—N,since fcanbeextended toRélocally. Infact, thisconstruction could begeneralized much further, tothecase where iand j aremerely imbeddings oftwoabstract manifolds MandN,andf:M>N isC®; wejustconsider thefunction jofoi7!: i(M) >i(N) andextend it locally toRX.Thecasewhich wewant toexamine most carefully isthesimplest: whereM=Nand/istheidentity, while/and/aretwoimbeddings ofM inR*andR’,respectively. Elements ofT(M,i)p areoftheform (0x7), (w) yar v uy forw€R"x(p), while elements ofT(M, j)pareoftheform (j0x~1),(w) for w€R"x(p). Ifwemap (60x), (w) (fox), (w) weobtain abundle map from T(M,i)|U toT(M, j)IU, which isobviously an equivalence. The map (M,i)y >(M,/)pinduced onfibres isindependent of thecoordinate system x,forif(y,V)isanother coordinate system, then (£0ypa(w)=(60x(x 0yx(w)) (Joy alw)=fox )a(Ceoy)a(w)). Wecantherefore putallthese maps together, and obtain anequivalence from T(M,i) toT(M,j).Inotherwords,thedependence ofT(M,i) oniisal- most illusory; wecould abbreviate T(M,i) toTM, ifweagreedthatTMreally denotes anequivalence class ofbundles, rather than onebundle. That isthe TheTangent Bundle 75 sortofthing analgebraist might do,anditisundoubtedly ugly. What wewould liketodoistogetasingle bundle foreach M,insome natural way, which has alltheproperties anyoneoftheseparticular bundles 7(M,7)has.Canwedo this? Yes, wecan. When wedo,TR" will bedifferent from our olddefini- tion (namely, e"(R")), andsowillf.forf:R"+R", soinstating ourresult precisely wewillwrite “old f,”when necessary. 1,THEOREM. Itispossible toassign toeach n-manifold Mann-plane bun- dleTMoverM,andtoeachC®mapf:M—Nabundle map(f,,f),such that: ()If1:M—Mistheidentity, then1.4:TM—TMistheidentity. If g:N—P,then(gof)s=g%ofre (2)There areequivalences ":TR" >e"(R") such thatforevery C®func- tion £:R">R™thefollowing commutes. rR. 7R™ a| i ; 2r(Rty OL, omegmy (3)IfUCMisanopen submanifold, then TU isequivalent to(7M)|U, andforf:M>Nthemap(f|U)«: TUTNisjusttherestriction off,.More precisely, there isanequivalence TU~(TM)|U such that thefollowing diagrams commute, wherei:U>Mistheinclusion.* Tu iy TM TU (SIU)«TN (TM)|U ™ PROOF. The construction ofTM isaningenious, though quite natural, sub- terfuge. Wewillobtain asingle bundle forTM, buttheelements ofTMwilleach belarge equivalence classes. *When using thenotation /f,,itmust beunderstood thatthesymbol “f”really refers to atriple(f,M@, N)where f:M—N.Theidentitymap|ofUtoitself'and theinclusion mapi:U+Mhavetobeconsidered asdifferent,sincethemaps1,:TU>TUand ig:TU+TMarecertainly different (they map TUintotwodifferent sets). 76 Chapter 3 The construction ismuch easier tounderstand ifwefirstimagine thatweal- ready hadourbundles TM. Then if(x,U) isacoordinate system, wewould have amap x4:TU —T(x(U)), and thiswould beanequivalence (with inverse (x7!),). Since TUshould beessentially (TM)|U, andT(x(U)) should essentially bex(U) xR",apoint e€x~!(p) would betaken byx»tosome (x(p),v). Herevisjustanelement ofR”(andeveryvwould occur, sincex» maps ~!(p) isomorphically onto{p}xR"). Ifyisanother coordinate system, then »(e) would be(y(p), w)forsome w€R”.Wecaneasily figure outwhat therelationship between vandwwould be;since (x(p), v)istaken to(y(p), w) byy.0%47! =(yox™!),, and(yox~!), issupposed tobetheold(y0x7!)s, wewould have @) w=Diyox!)(x(p))(o). : This condition makes perfect sense without anymention ofbundles. Itisthe clue which enables ustonow define TM. Ifxand yarecoordinate systems whose domains contain p,and v,w€R", wedefine (x,0)7Ow) if(a)issatisfied. Itiseasytocheck (using thechain rule)that+isanequivalence relation; the equivalence class of(x,v)willbedenoted by[x,vl. These equivalence classes willbecalledtangent vectors atp,andTMisdefined tobethesetofalltangent vectors atallpoints p€M;themapxtakes %equivalence classes top.We define avector space structure onx~!(p) bytheformulas [yulp +be,wy=bx,u +wy a-[x,v]p =[x,a- vp; thisdefinition isindependent oftheparticular coordinate system xory,because D(yox7})(x(p)) isanisomorphism fromR”toR™. Our definition ofTMprovides aone-one onto map (b) tein7\(U) >UxR", namely [x,v]q>(9,0). Wewant thistobeahomeomorphism, sowewant tx~!(A)tobeopen forevery openACUxR’,andthuswewantanyunionofsuchsetstobeopen. There isametric with exactly these setsasopen sets,butitisalittle ticklish toproduce, soweleave thisonepart oftheproof toProblem 1. Wenow have abundle x:TM —M. Wewilldenote thefibre x~1(p) byM,,inconformity with thenotation R",, though TM, might bebetter. If TheTangent Bundle 77 f:M>N,and (x,U) and (y,V)arecoordinate systems around pandf(p), respectively, wedefine ©) Sal wp)=1,DOe foxx PW epy- Ofcourse, itmust bechecked that thisdefinition isindependent ofxandy (thechain rule again), Condition (1)ofourtheorem isobvious. Toprove (2),wedefine t”tobetz,where /istheidentity map ofR”andty isdefined in(b);itistrivial, though perhaps confusing tothenovice, toprove commutativity ofthediagram. Condition (3)ispractically obvious also. Infact, thefibre ofTUover p€U isalmost exactly thesame asthefibre ofTM over p;theonly difference isthat each equivalence class for Mcontains some extra members, since inMthere aremore coordinate systems around pthan there areinUCM.& Henceforth, thebundlex:TM—>MwillbecalledthetangentbundleofM. Ifi:M—RXisanimbedding, then TMisequivalent toT(M,i). Infact,if (x,U)isacoordinate system around p,and/istheidentity coordinate system ofR¥,then f(lx, Up)=U, DGox)(x(P)Mipy by(©) Mat I (p), Diox™")x(P))()) €(Mp; thecomposition 1",iseasily seen tobeanequivalence. ButT(M,i) willplay nofurther role inthisstory—the abstract substitute TM willalways beused instead. Having succeeded inproducing abundle over each M,which isequivalent to T(M,i), wenext askhow fortuitous this was. Can one find other bundles with thesame properties? The answer isyes,andweproceed todefine twodifferent such bundles. Forthefirst example, weconsider curves c:(—,£) —M,each defined onsome interval around 0,with c(0) =p.If(x,U) isacoordinate system around p,wedefine «x 0¢andxoz,mapping RtoR", crea ifandonlyif thesamederivative at0. The equivalence classes, forallp€M,willbetheelements ofournew bun- dle,T’M. Forf:M+Nthere isamapfjtaking the*equivalence class 78 Chapter 3 of¢tothe/%,equivalence classoffoc. Without bothering tocheck details, wecanalready seethat thisexample is“really thesame” asTM— the%equivalence classofx7! y, x,dsto: P Fx,¥]pcorresponds to:eteyisacurveinR"with"(0)=»5 under thiscorrespondence, fycorresponds tof,. Inthesecond example, things arenotsosimple. Wedefine atangent vector atptobealinear operator £which operates onallC™functions fandwhich isa“derivation atp”: £(fg) =S(p)e(g) +g(p)e(s)- Wehavealready seenthattheoperators £=2/8x!l,havethisproperty. For these operators, clearly £(f) =£(g) iff=ginaneighborhood ofp.This condition isactually true foranyderivation £.For, suppose that f=0ina neighborhood ofp.There isaC®function h:M>Rwith h(p) =1and support hCf~'(0). Then 0=£(0) =£(fh) =f(0)E(h) +HOLL) =0+4(f). Thus, iff=ginaneighborhood of0,then 0=£(f —g)=£(f)—£(g). Iff isdefined only inaneighborhood ofp,wemay usethistrick todefine £(f): choose htobe1onaneighborhood ofp,withsupport hCf~1(0), anddefine £(f) as&(fh). The setofallsuch operators isavector space, butitisnot@priori clear what itsdimension is.This comes outofthefollowing. 2.LEMMA. LetfbeaC®function inaconvex open neighborhood Uof0 inR®,with f(0) =0.Then there areC® functions g;: U>Rwith ()S004, 2.52) =Diy xigiel,...,2") forx€U, (2)gi(0)=Dif(0). (The second condition actually follows from thefirst.) PROOF. Forx€U,lethx(t) =f(tx); this isdefined for0<1<1,since Uis convex. Then 1 1m |fox)=fo)f=fbnar= [>d.ses)-x! at0jst Therefore wecanletg(x)=fyDifltx) dt. TheTangent Bundle 79 3.THEOREM. The setofalllinear derivations atp€M” isann-dimen- sional vector space. Infact, if(x,U)isacoordinate system around p,then a a ant], ax, span thisvector space, andanyderivation £canbewritten tn i,2 f=)rel).|> ist ax!|, (s0€isdetermined bythenumbers £(x!)). PROOF. Notice that £Q =£01-1) =1-£(1) +1-£01), so€(1) =0.Hence £(c) =¢-£(1) =0foranyconstant function ¢onU. Consider thecasewhere M=R”andp=0.Assume Uisconvex. Given f onU,choose g;asinLemma 2,forthefunction f~f(0). Then n ; _ _ _ 1 (Idenotes thei ka7 : =Pegi) +14g) ist tn af =i=hedg +0. ist This shows that3/9/'Ip span thevector space; they areclearly linearly inde- pendent. Itisasimple exercise tousethecoordinate system xtotransfer this result from R”toM. FromTheorem 3wecanseethat,onceagain, abundle constructed fromall derivations atallpoints ofMis“really thesame” asTM. Wecanlet 8l=Lexl,correspondto[x,a]p; the formula , a|>ay8 sal=2mrPazG| > ax], Syax! ay], 80 Chapter 3 derived inChapter 2,shows that ra2]=oe] epOY ai—|=bi] ifandonlyif bf=Sra(yp), >ax"|,xay'|, >ax! andthisisprecisely theequation which saysthat(x,a) >(y,b). Itiseasily checked that under thiscorrespondence, themap which corresponds tof,can bedefined asfollows: [fe(O)(g) =£(g0f). 1 Noticethatifxdenotes theidentity coordinate systemonR",then>a!s|corresponds toa,when weidentify TR"withe”(R"). inl cd Wewill usually make nodistinction whatsoever between atangent vector v€M,andthelinear derivation itcorresponds to,thatis,between [x,a@], and n a——|; » ax'|, consequently, wewillnothesitate towrite v(/) 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 (fev) (8)=vee f)- Itiscustomary todenote theidentity coordinate system onR!by1,andto write a forar dt|,, atl,” thisisabasis forRyy. Ifc: R->Misadifferentiable curve, then d iscalled thetangent vector to¢atto.Wewilldenote itbythesuggestive symbol dc thi TheTangent Bundle 81 This symbol willbesubjected tothestandard abuses onefinds (unexplained) in calculus textbooks: thesymbol de * de3Willoftenstandfor|. thesubscript “t”now denoting aparticular number¢€R,aswellastheidentity coordinate system. Asyoumight well expect, itisnoaccident that oursecond and third 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. Thetangent bundle 7MofaC®manifold hasalittlemorestructure than anarbitrary n-plane bundle. Since TMlocally looks likeUxR”,clearly TM isitself amanifold; there is,moreover, anatural way toput aC® structure onTM. Ifx:U+R”isachart onM,then every element v€(TM)|U is uniquely oftheform ngv=Dea ;p=nxtv). ist ax!Pp Letusdenote a!byX/(v). Then themap Ves(XECOD),px"GC(u)),FY), X"(W))€RM isahomeomorphism from (TM)|U tox(U) xR". This map, (xo2,2), is simply themap x,when weidentify TUwith UxR”inthestandard way. If (y,V)isanother coordinate system, and 7.8v=ooaT°Wy then, aswehave already seen, 2ay! Ln Je x yf ox} =Lea” =Lanw ox*)(x(p))- This shows thatif(t,a)=(,...,",a!,...,a") €R,then Ya0(Xx) a) =(vox), Diep! Di(yl ox), --.5Liar a!Dily"0x7")). This expression shows thaty0(x,)7! isC®. 82 Chapter 3 Wethus have acollection ofC°-related charts onTM, which can beex- tended toamaximal atlas. With thisC®structure, thelocal trivializations x,areC°. Ingeneral, a vector bundle x:E—Biscalled aC™vector bundle ifEandBareC° manifolds and there areC® local trivializations inaneighborhood ofeach point. Itfollows thatx: E>BisC™. Recall thatasection ofabundle x:E>Bisacontinuous functions: B>E such thatxos=identity ofB;forCvector bundles wecanalsospeak ofC° sections. Asection ofTM iscalled avector field onM;forsubmanifolds M ofR”,avector field may bepictured asacontinuous selection ofarrows tangent toM.The theorem thatyoucan’t comb thehaironasphere juststates that GeyWeis there isnovector fieldonS?which iseverywhere non-zero. Wehaveshown that there donotexist twovector fields ontheMobius strip which areeverywhere linearly independent. Vector fields arecustomarily denoted bysymbols like X,Y,orZ,and the vector X(p) isoften denoted byXp(sometimes X¥may beused todenote a single vector, insome M,). Ifwethink ofTMasthesetofderivations, then for anycoordinate system (x,U),wehave xw=ain2|forallpeuax? . it iP Thefunctions a‘arecontinuous orC®ifandonlyifX:U>TMiscontin- uous orC®. IfXand¥aretwovector fields,wedefine anewvector field¥+Yby (X+Y)(p) =X(p) +¥(p). Similarly, iff:M>R,wedefine thevector field {Xby (FX) (p)=f(P)X(p). TheTangent Bundle 83 Clearly ¥+Yand fXareC® ifX,Y,and fareC*. OnUwecanwrite n aX=Lee ist thesymbol 3/0x/ nowdenoting thevector field weu : peol, Iff:M>RisaC®function, andYXisavectorfield,thenwecandefineanewfunction X(f): M—Rbyletting Xoperate onfateachpoint: ¥(f\(p) =Xp(S)- Itisnothard tocheck thatifXisaC™vector field, then ¥(f) isC®for everyC®function f;indeed, iflocally 2 a x0=L4 zapi=l Pp then , = af= fofXiN=dea which isasumofproducts ofC®functions. Conversely, if¥(f) isC°for everyC®function f,thenXisaC™vector field(since X(x!)=a’). Let ¥denote thesetofallC functions onM.Wehave just seen that aC° vector fieldXgives risetoafunction ¥:¥—F.Clearly, Hf +e)=X(N+ Xe) X(fg)=£X(g)+aX(Sf); thus¥isa“derivation” oftheringF.Often, aC™vector fieldXisidentified withthederivation ¥.Thereason forthisisthatifA:F>¥isanyderiva- tion, then A=Xforaunique C®vector field X.Infact, weclearly must define Xp(f) =ACS)(p); andtheoperator X,thus defined isaderivation atp. 84 Chapter 3 Thetangent bundleisthetruebeginning ofthestudyofdifferentiable mani~ folds, andyoushould notread further until yougrok it.*The next fewchapters. constitute adetailed study ofthisbundle. One basic theme inallthese chap- tersisthat anystructure onecanputonavector space Jeads toastructure on anyvector bundle, inparticular onthetangent bundle ofamanifold. Forthe present, wewilldiscuss justonenewconcept about manifolds, which arises in thisvery wayfrom thenotion of“orientation” inavector space. Thenon-singular linear maps{:V—Vfromafinitedimensional vector spacetoitselffallintotwogroups, thosewithdetf>0,andthosewithdetf<0; linear transformations inthefirstgroup arecalled orientation preserving and theothers arecalled orientation reversing. Asimple example ofthelatter is themap f:R"—> R"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 nxnmatrices, and thus withR”’,thentheorientation preserving andorientation reversing maps are disjoint open subsets ofthesetofallnon-singular maps (those with det#0). 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 twodifferent (but isomorphic) vector spaces Vand W;this clearly makes nosense unless wesupply Vand Wwith more structure. Toprovide thisextra structure, wenote thattwoordered bases (v;,...,Un) and(v'1,...,0'n) forVdetermine anisomorphism f:V>Vwith f(u;) =v4; thematrix A=(a;;) offisgiven bytheequations a vi=Slay. j=l Wecall(vy,-..,0n) and (v,,...,0'n) equally oriented ifdetA>0(ie.,iffis orientation preserving) andoppositely oriented ifdetA<0. The relation ofbeing equally oriented isclearly anequivalence relation, divid- ingthecollection ofallordered bases intojusttwoequivalence classes, Either of these two equivalence classes iscalled anorientation forV.The class towhich (v1,...,Un)belongs willbedenoted by[v1,..., ¥],sothat ifpzisanorientation ofV,then (%,.--,Un) €#ifand only if[m,...,Un] =w.Ifwdenotes one *Acultword ofthesixties, “grok” wascoined, purportedly asaword from theMartian language, byRobert A.Heinlein inhispop science fiction novel Stranger inaStrange Land. Itssense isnicely conveyed bythedefinition inTheAmerican Heritage Dictionary: “Tounderstand profoundly through intuition orempathy”. TheTangent Bundle 85 +++ 0 yw ws idAik wy <q »we Th plane of wyand w2 uy Examples ofequally oriented ordered bases inR,R?,andR°. orientation ofV,theother willbedenotedby —y,andtheorientation [e1,..., n] forR”will becalled the “standard orientation”. Now if(V,4)and (W,v)aretwon-dimensional vector spaces, together with orientations, anisomorphism f:V—Wiscalled orientation preserving (with respect tojtand v)if[f(v),.--, f(n)] =vwhenever [v4,...,Un]=M5ifthis holds foranyone(1,...,Un), itclearly holds forall. Forthetrivial bundle e”(X) =XxR”wecanputthe“standard orientation” [0x,€1),+++(%,en)]oneachfibre{x}xR".Iff:e"(X)>e"(X)isanequiva- lence, andXisconnected, then fiseither orientation preserving ororientationreversing oneachfibre,forifwedefinethefunctions aij:¥>Rby n L(%,e1) =Yraji(x) -(,¢4), jel thendet(ajj): X—Riscontinuous andnever0.Ifx:E>Bisanon- trivial n-plane bundle, anorientation 4ofEisdefined tobeacollection of orientations ppforx~!(p) whichsatisfythefollowing “compatibility condition” foranyopen connected setUCB: Ift:27'(U) +UxR”isanequivalence, andthefibres ofUxR”are giventhestandard orientation, then¢iseither orientation preserving ororientation reversing onallfibres. Notice thatifthiscondition issatisfied foracertain t,and’; x~"(U) >UxR" isanother equivalence, then t’automatically satisfies thesame condition, since 86 Chapter 3 tot~!: UxR" >UxR"isanequivalence. Thisshows thattheorientations py define anorientation ofEifthecompatibility condition holds foracollection ofsets Uwhich cover B. Ifabundle Ehasorientation 4.={#p}, ithasanother orientation —p= {=p}, butnotevery bundle hasanorientation. Forexample, theMébius strip, considered asa1-dimensional bundle over S!,hasnoorientation, For, although theMébius strip hasnonon-zero section, wecanpick twovectors from each fibre sothat thetotality Alooks like two sections. Forexample, wecanletAbe[0,1]x{—1, 1}with (0,a) identified with (1,—-a); then Ajust looks liketheboundary oftheMobius strip obtained from [0,i]x[1,1]. If wehadcompatible orientations 1p,wecould define asection s:S'-»Mby choosing s(p)tobetheunique vector s(p) €ANx~!(p) with[s(p)] =Hp. Abundle iscalled orientable ifithasanorientation, and non-orientable oth- erwise; anoriented bundle isjustapair (§,2)whereyzisanorientation for&. This definition canbeapplied, inparticular, tothetangent bundle TM ofa C® manifold M.Inthiscase, wecall Mitself orientable ornon-orientable de- pending onwhether TM isorientable ornon-orientable; anorientation ofTM isalsocalled anorientation ofM,andanoriented manifold isapair (M,1) where jzisanorientation forTM. The manifold R”isorientable, since TR” ~e”(R"), onwhich wehave the standard orientation. Thesphere $”~! CR"isalsoorientable. Toseethiswe aL Pp=w Ms:ar TheTangent Bundle 87 note thatforeach p€S"~! thevector w=pp€e"(R") =~TR" isnolin i,(S",) €TR", (Problem 21),soforv1,...,Un—-1 €S"7!, wecandefine (v1,--.,Un—1) €fpifand only if(w,/4(%1),--.,/(Yn—1)) isinthestandard orientation ofR",.Theorientation 4={up:p€S"~"} thusdefined iscalled the“standard orientation” ofS$”). Thetorus S?xS!isanother example ofanorientable manifold. This can beseen bynoting thatforanytwomanifolds MyandMzthefibre (MyxM2), ofT(M, xM2) canbewritten asVip©Vapwhere (t))«! Vip>(Mz)p isan isomorphism andthesubspaces Vipvarycontinuously (Problem 26).Since TS! istrivial, thisshows thatT(S! xS!)isalsotrivial, andconsequently orientable. Any r-holed torus isalso orientable—the proof ispresented inProblem 16, which also discusses thetangent bundle ofamanifold-with-boundary, The Mobius strip Mjsthesimplest example ofanon-orientable 2-manifold. Fortheimbedding ofMconsidered previously wehave already seenthatonthe 1z : zxor subset S={(2cos6,2sin @,0)} CMthere arecontinuously varying vectors up, butthat itisimpossible tochoose continuously from among thedashed vectors wy=f.((0, 1)(o,0)) andtheir negatives. Ifwehadorientations tpforp€S, then wecould simply choose wpif[vp,wp]=Hpand —w, otherwise. Theprojective plane P?must benon-orientable also, since itcontains the Mobius strip (foranyorientable bundle £=x:E—B,therestriction &|B’ toanysubset B’CBisalsoorientable). Non-orientability ofP?canbeseen inanother way,byconsidering the“antipodal map” A:S?+S?defined by A(p) =—p. This map isjust therestriction ofalinearmapA:R?>R? defined bythesame formula. Themap As:S2,>S?4(p) isjust(p,v) (A(p), A()), when S2,isidentified withasubspace of{p}xR?.Themap4 isorientation reversing, soifvj=(p,ui) €Sp,thebases (u1,¥2,p) and(A(u1), A(u2), A(p)) areoppositely oriented. This shows thatif1isthestandard orientation ofS? and[v4,v2]€fp,then[Ayv1,Asv2]€—Ha(p)- ThusthemapA:S?>S?is 88 Chapter 3 “orientation reversing” (the notion ofanorientation preserving ororientation reversing map f:M—Nmakes sense foranyimbedding fofoneoriented manifold into another oriented manifold ofthesame dimension). From thisfact itfollows easily thatP?isnotorientable: IfP?hadanorientation v={v{p}} andg:S?>P?isthemap p++[p],then wecould define anorientation {ip} onS”byrequiring gtobeorientation preserving; themap Awould then beorientation preserving with respect toji,which isimpossible, since ji=p or —p. Forprojective 3-space P?thesituation isjusttheopposite. Inthiscase, the antipodal mapA:S?>S?isorientation preserving. Ifg:S?>P?isthe map p+[p],weobviously candefine orientations vpforP?byrequiring g tobeorientation preserving. Ingeneral, these same arguments show that P”is orientable for nodd and non-orientable for7even. There isamore “elementary” definition oforientability, which does notuse thetangent bundle ofMatall.According tothisdefinition, Misorientable if there isasubset A’ofthe atlas *forMsuch that (1)thedomains ofall(x,U)€A’coverM, (2)forall(x,U) and (y,V) €A’, idet(2) 20onUnv. Oxd Anorientation 42ofTM allows ustodistinguish thesubset A’asthecollection ofall(x,U) forwhich x.: TM|U >T(x(U)) =x(U) xR”isorientation preserving (when x(U) xR"isgiven thestandard orientation). Condition (2) holds, because itisjustthecondition that (yox7!),: T(x(U)) >T(x(U)) isorientation preserving. Conversely, given A’wecan orient thefibres of TM\U insuch awaythatx,isorientation preserving, andobtain anorientation ofTM. Although ouroriginal definition iseasier topicture geometrically, the determinant condition willbevery important later on. TheTangent Bundle 89 ADDENDUM EQUIVALENCE OFTANGENT BUNDLES The factthatallreasonable candidates forthetangent bundle ofMturn out tobeessentially thesame isstated precisely asfollows. 4.THEOREM*. Ifwehave abundle 7’M over Mforeach M,and abundle map (fj,f)foreachC®mapf:M—Nsatisfying ())ofTheorem 1, (2)ofTheorem ],forcertain equivalences t’”, (3)ofTheorem },forcertain equivalences T’U =(T’M)|U, then there areequivalences em: TM >T'M such that thefollowing diagram commutes forevery C°map f:M—> N. ™—&—.TN T'M#,T'N PROOF. Thedetails ofthisproof aresohorrible thatyoushould probably skip it(and you should definitely quit when you getbogged down); thewelcome symbol +occurs quite aways on.Nevertheless, theidea behind theproof is simple enough. If(x,U) isachart onM,then both (TM)|U and (T’M)|U “Jook like” x(U) xR",sothere ought tobeamap taking thefibres ofone to thefibres oftheother. What wehave tohope isthatourconditions onTMand T'M make them “look alike” inasufficiently strong wayforthisidea toreally work out. Those who have been through thissortofrigamarole before know (.e., have faith) that it’sgoing towork out; those forwhom thissort ofproof is anewexperience should finditpainful andinstructive. *Functorites willnotice thatTheorems |and4saythatthere js,uptonatural equiv- alence, aunique functor from thecategory ofC® manifolds and C® maps tothe category ofbundles and bundle maps which jsnaturally equivalent to(€",old 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 ~,are theequivalences mentioned incondition (3).Letadenote thecomposition Oy=(("1x(U)) 0%0%0(7. (rmyu 2 ru2 rev) 5 reyxv)SEO, eeryix) a Similarly, using equivalence ~!forT’,wecandefine Bx. ~ a a(my 2ru28 ru 25 reix(v)22, e@ryixuy ey Then Bx7! o@x: (TM)|U >(T'M)|U isanequivalence, soittakes thefibre ofTMover pisomorphically tothefibre ofT’Moverpforeachp€U.Ourmaintaskistoshowthatthisisomorphism between thefibresoverpisindependent ofthe coordinate system (x,U). This willbedone inthree stages. (I)Suppose VCUisopenandy=x|V.Wewillneedtonamealltheinclusion maps i:U>M i:VoM ZiVou k:y(V) >x(U). Tocompare @xanday,consider thefollowing diagram. (TM)\U =—ru+T(x(U)) =>(TR")|x(V) LW), on®xv) )le@|.(Q)Jo®le@le = Ya = Jn ly), oO (TMV —=— TV + T(y(V)) —(TR")|y(V) >e"R")Y(V) TheTangent Bundle 91 Eachofthefoursquares inthisdiagram commutes. Toseethisforsquare (1),weenlargeit,asshownbelow.Thetwotriangles ontheleftcommute bycon- dition (3)forTM, andtheoneontheright commutes because i j=7. (TM)|U k.TU a | (TMV Square (2)commutes because koy=x0j.Square (3)commutes forthe samereason assquare (1);theinclusions x(U)>R"andy(V)>R®come intoplay.Square (4)obviously commutes. Chasing through diagram (1)now shows thatthefollowing commutes. (TMU + e(R")|x(U) }} (TMV 25 er(R")Ly(V) This means that forp€V,theisomorphism aybetween thefibres over p isthesame asax. Clearly thesame istrue forBxandBy,since ourproof used only properties (I),(2),and(3),nottheexplicit construction ofTM. Thus By~!o@y=By~! oayonthefibres overp,forevery p€V. (II)Wenow need aLemma which applies toboth TMandT’M. Again, itwill beproved forTM (where itisactually obvious), using only properties (1),(2), and(3),sothatitisalsotrueforT’M. 92 Chapter 3 LEMMA. If4CR" and BCR” areopen, andf: A>BisC®, then the following diagram commutes. Ta=rR)4 4,rey s| [otUP; = m iB mepym TB—— (TR”)|B ———> e(R)"|B PROOF. Case 1.Thereisamapf:R"+R™withf=fonA.Consider the following diagram, where i: A—>R"andj:B>R”aretheinclusion maps. n (TRAIA,Ry]4 a Ir TAlt TR""2" (R") Ta+. rR1"_,emp) A | Icmirye 18,meme Everything inthis diagram obviously commutes. This implies that thetwo compositions ~ nm raTAS rd 24repraSsorpty)LL, emp and ~ mTa$2,78=(rR)BLT, emp BS,om(RM) arecqual andthisproves theLemma inCase |,since themaps “old/,”and “old f,”areequal onA. Case2.General case.Foreach p€A,wewant toshow thattwomaps arethe same onthefibre over p.Now there isamap f:R">R™with f=fonan open setA’,where p€A’CA.Wethen have thefollowing diagram, where every Xcomes from thefactthat some setisanopen submanifold ofanother, TheTangent Bundle 93 andi:A’>Aistheinclusion map. TA5 (TR")|A AAS maya “yc S ~N a aia’ Q c@ c ia v3 Vv S nat (2) ®ra——=—_. rea 24,ceria’ © fe ldfe7a@oldoo° a m18=. (rrp 2, mami Boxes(1),@),and@)obviously commute, and(4)commutes byCase1.To seethatsquare (2)(which hasatriangle within it)commutes, weimbed itin alarger diagram, inwhich j:A>R®istheinclusion map, andother maps have also been named, forease ofreference. TR" Ta=)_.rr4 | lecww Ta!—=L9O__, (PRA! Toprove that A0i,=pox, itsuffices toprove that Vohoiy =vopox, since visone-one. Thus itsuffices toprove js0/,=V040x,which amounts toproving commutativity ofthefollowing diagram. TR" Genf\c TA!——=—— (TR)A’ Since jojisjusttheinclusion ofA’inR",thisdoes commute. 94 Chapter 3 Commutativity ofdiagram (2)shows that thecomposition ~ m 7a. ra= rR”)B1B,omanB coincides, onthesubset (TA)|A’, with thecomposition Ta rea 4, ore 2, omey, andonA’wecanreplace “old/,”by“oldf,”.Inother words, thetwocom- positions areequal inaneighborhood ofanyp€A,and arethus equal, which proves theLemma. (II]) Now suppose (x,U) and (y,V)areanytwocoordinate systems with p€ UAY. Toprove thatBy~! oayandBx! ayinduce thesame isomorphism onthefibre ofTM atp,wecanassume without lossofgenerality that U=V, because part (1)applies toxandx|U1V,aswellastoyandy[UNV. Assuming U=V,wehave thefollowing diagram. x "TexUy) + rRIx(V) “EO earyix(vy x 8) (rmyu Tu (vox), old(0x71), Ny = Ly(U) TOU) >RMU) “VPA, eneryiywy The triangle obviously commutes, and therectangle commutes bypart (II). Diagram (3)thus shows that ay=old(pox), ox. Exactly thesame result holds for7’: By=old(y0x7), 0Bx. The desired result By~! oay=x7! 0axfollows immediately. Nowthatwehaveawell-defined bundlemapTM—T’M(theunionofallBx"o@;),itisclearly anequivalence ey.Theproofthatevof,=fyoem is leftasamasochistic exercise forthereader, TheTangent Bundle 95 PROBLEMS 1.LetMbeanyset,and{(x7, U;)} asequence ofone-one functions x;:U;>R” with U;CMandx(U;) open inR”,such that each xjoxy: xj(U; Uj)>xy(U; NU) iscontinuous. Itwould seem that Mought tohave ametric which makes each U;open andeach x;ahomeomorphism. Actually, thisisnotquite true: (a)LetM=RU{x}, wherex¢R.LetU;=Randx1:U;>Rbe the identity, and letUz=R—{0}U{*},with x2:Uz>Rdefined by x2a)=a, a£0,* xo(*) =0. Show thatthere isnometric onMoftherequired sort, byshowing thatevery neighborhood of0would have tointersect every neighborhood of*,Never- theless, wecanfind onMapseudometric p(afunction p:MxM>Rwith allproperties forametric except that p(p,q) may be0forp¥q)such that p isametric oneach U;and each x;isahomeomorphism: (b)IfACR"isopen,thenthereisasequence Aj,A2,A3,.-.ofopensubsets ofAsuchthateveryopensubset ofAisaunionofcertain Aj’s. (6)There isasequence ofcontinuous functions fj:A—[0,1], with support fi CA,which “separates points and closed sets”: ifCisclosed and p€A~C, then there issome fjwith fi(p) ¢fi(AOC). Hint: First arrange inasequence allpairs (A;,Aj)ofpart(b)withAjCAy. @)Letfi,), 7=1,2,3,-... besuch asequence foreach open setx;(U;). Define 8,3: M>[0,1] by eaw={frm peu; “ 9 peu, Arrange allgi,inasingle sequence G1,G2,G3,..., letdbeabounded metric onR,anddefine ponMby 1 (p.9)=Yd (Gil),Gita). i=l Showthatpistherequired pseudometric. (©)Suppose that forevery p,g €Mthere isaU;and Ujwith p€U;and q€Ujandopen setsByCx;(U;) andBjCx;(Uj) sothat p€x;~'(B;), q€xj7"(By), andx;7}(B;) N.xj7!(Bj) =9.Showthatpisactually ametric on M. 96 Chapter 3 2.(a)Suppose (x,U)and (y,V)aretwocoordinate systems, giving risetotwo maps onTM, tein1(U) >UxR", [x,vg0); yim(VV XR", [yw], +(9,w)- Show thatinx~!(U 9V)thesetsoftheform #,~1(A) forACUxR"open areexactly thesetsoftheform ty~"(B) for B.CVxR"open. (b)Show thatifthere isametric onTMsuch thattx,isahomeomorphism for acollection (x/,U;) with M=U, U;,then alltxarehomeomorphisms. (©)Conclude fromProblem |thatthereisametric on7Mwhichmakes eachtx ahomeomorphism. 3.Show that inthedefinition ofanequivalence itsuffices toassume that the map E;>E2iscontinuous. (Toprove theinverse continuous, note thatlocally itisjustamap UxR">UxR"), 4.Show that inthe definition ofabundlemap,continuityoff:Bi>Bo follows automatically from continuity off:E;>E2. 5.Aweak equivalence between twobundles overthesame base space Bis abundle map (f,f)where fisanisomorphism oneach fibre, and fisa homeomorphism ofBonto itself, Find twoinequivalent, butweakly equivalent, bundles over thefollowing base spaces: (i)thedisjoint union oftwocircles, (i)afigureeight C7><), (ii)thetorus. 6.Given abundle map(/,f),show thatf=gohwhere gandharecontin- uous maps such that/takes fibres linearly tofibres, while gisanisomorphism ‘oncach fibre. 7.(a)Show that foranybundle x:E>B,themap s:B>Ewith s(p) the0vector of27!(p) isasection. (b)Showthatann-plane bundle &istrivial ifandonlyifthereare1sections 51,95 which areeverywhere linearly independent, i.e.51(P),---s5n(P) € x~}(p) arelinearly independent forallp€B. (©)Show thatlocally every -plane bundle has1linearly independent sections. 8.(a)Check that>isanequivalence relation onthesetofpairs (x,v). (b)Check thatthedefinition off,isindependent ofthecoordinate systems x and ywhich areused. (c)Check theremaining details inTheorem 1. TheTangent Bundle 97 9.(a)Show that thecorrespondence between TM and equivalence classes of curves under which [x,»]corresponds tothe#equivalence classofx~!oy, foryacurve inR"with y/(0) =v,makes f.correspond tofi. (b)Show thatunder thecorrespondence [x,a]p +>>;a‘d/ax"|,, themapf, canbedefined by [AOl@) =£0 f). 10.IfVisafinite dimensional vector space over R,define aC®structure onV andahomeomorphism fromVxVtoTVwhich isindependent ofchoice of bases. Asinthecase ofR”,forv,w€Vwewilldenotebyvw€Vwthevector corresponding to(w,v). 11. Ifg: R—>Ris C™ show that B(x) =8(0)+8'(O)x +x7h(x) forsome C® function h:R>R. 12.(a)Let#bethesetofallC®functions f:M>Rwith f(p) =0,and let£:F,>Rbealinearoperator with£(fg)=0forallf,g€Fp.Show that£hasaunique extension toaderivation. (b)LetWbethevector subspace ofF,generated byallproducts fgforf,g € Fp.Show thatthevector space ofallderivations atpisisomorphic tothedual space (F,/W)*. (c)Since (¥,/W)* hasdimension n=dimension ofM,thesame must betrue of¥,/W. Ifxisacoordinate system with x(p) =0,show thatx1+W,..., x"+Wisabasis forF/W (useLemma 2).The situation isquite different for C!functions, asthenext problem shows. 13.(a)LetVbethevector space ofall C?functions f:R>Rwith /(0) =0, andJetWbethesubspace generated byallproducts. Show thatlimf(x)/x? exists forallf€W. x0 (b)For0<e<1,let xi!x>0 x) =Le){3x<0. Show that al]f;areinV,andthat they represent linearly independent elements ofV/W. (c)Conclude that (V/W)* hasdimension ¢®=2°. 14.Iff:M—Nandf,isthe0maponeachfibre,thenfisconstant on each component ofM. 98 Chapter 3 15.(a)Amapf:M—Nisanimmersion ifandonlyiff,isone-one on eachfibreofTM.More generally, therankoffatp€Mistherankofthe linear transformation fx:Mp>Nycp)- (b)Iffog =f,where gisadiffeomorphism, thentherankoffog ata equals therank offatg(a). (Compare with Problem 2-33(d).) 16.(a)IfMisamanifold-with-boundary, thetangent bundle TMisdefined exactly asforM;elements ofMpare~yequivalence classes ofpairs (x,v). Although xtakes aneighborhood ofp€3Monto Hl,rather than R®,the vectors vstillrunthrough R",soM,stillhastangent vectors “pointing inall directions”. Ifp€0Mandx:U>H"isacoordinate system around p,then Pais x47}(R"™1y¢p)) CMpisasubspace. Show thatthissubspace doesnotdepend onthechoice ofx;infact,itis/,(8M)p, where i:8M—Mistheinclusion. (b)Leta€R"™!x{0}CH”.Atangent vector inHqissaidtopoint“in- ward” if,under theidentification ofTH” with e”(H"), thevectoris(a,v)where v">0.Avector v€Mp,which isnotini,(@M)p issaid topoint “inward” if ur ‘award @ outward X4(v) €H"xp) points inward. Show thatthisdefinition does notdepend on thecoordinate system x. (c)Show that ifMhasanorientation y,then 3Mhasaunique orientation dysuch that[v1,...,%n—1) =(f)p ifandonly if[2v,i401, -+548%m—1) =Hpfor every outward pointing w€Mp. (d)Ifyistheusualorientation ofH”,showthatduis(—1)” timestheusual orientation ofR"-! =dH”. (The reason forthischoice willbecome clear in Chapter 8.) (c)Suppose weareinthesetupofProblem 2-14.Define g:9Mx[0,1)> ONx[0,1)byg(p,t) =(f(p),1). ShowthatTPisobtained fromTMUTN TheTangent Bundle 99 byidentifying vE(OM), with (B7')egaoe(v) €(QN)s¢p)- (fIfMand Nhave orientations 4and vand f:(8M,9u) >(aN, av)is orientation-reversing, show thatPhasanorientation which agrees with »andv on Mc PandNCP. (g)Suppose MisS?with twoholes cutout,andNis[0,1] xS!.Letfbe adiffeomorphism from MtoNwhich isorientation preserving ononecopy ofS!andorientation reversing ontheother. What istheresulting manifold P? 17.Show thatTP? ishomeomorphic tothespace obtained from T(S?,i) by identifying (p,v)€(S?,/)p with(~p, -v)€(S?,i)-p. 18.Although there isnoeverywhere non-zero vector fieldonS?,there isone onS?—{(0,0, 1)},which isdiffeomorphic toR?.Show thatsuch avector field canbepicked sothat near (0,0, 1)thevector field looks likethefollowing picture (a“magnetic dipole”): xLf XNwe 19.Suppose wehave a“multiplication” map (a,b) +>a-b from R"xR"toR" thatmakes R”into a(non-associative) division algebra. That is, (a1+42)-b=a,-b+az-b a+(b) +b.) =a-b) +a+by Ma-b)=(Aa)-b=a-(Ab) forkeR @-(1,0,...,0) =a and there are nozero divisors: a,b£0 => ab£0. 100 Chapter 3 (For example, forn=1,wecanuseordinary multiplication, and forn=2 wecanuse“complex multiplication”, (a,6) «(¢,d) =(ac~bd,ad +be).) Let €1,-+-,€n bethestandard basis ofR". (a)Every point inS”~! isa-e;foraunique a€R". (b)Ifa£0, thena-e),...,a@ +enarelinearly independent. (c)Ifp=a-e €S"~', thentheprojection ofa-e2,...,a-@n on(S""!,f)p arelinearly independent. (d)Multiplication byaiscontinuous, (ce)TS"! istrivial. ()TP"! istrivial. The tangent bundles TS? and TS” areboth trivial. Multiplications with therequired properties onR*andR®areprovided bythe“quaternions” and “Cayley numbers”, respectively; thequaternions arenotcommutative and the Cayley numbers arenoteven associative. Itisaclassical theorem that the reals, complexes, andquaternions aretheonly associative examples, Fora simpleproof,seeR.S.Palais,TheClassification ofRealDivision Algebras, Amer. Math. Monthly 75(1968), 366-368. J.FAdams hasproved, using methods of algebraic topology, that n=1,2,4,or8. [Incidentally, non-existence ofzero divisors immediately implies thatfora#0 there issome 6with ab=(1,0,...,0) and ’with b/a =(1,0,...,0). Ifthe multiplication isassociative itfollows easily that b=b’,sothat wealways have multiplicative inverses. Conversely, thiscondition implies thatthere arenozero divisors ifthemultiplication isassociative; otherwise itsuffices toassume the existence ofauniquebwitha-b=b-a=(1,0,...,0)] 20.(a)Consider thespace obtained from [0,1]xR”byidentifying (0,v)with (1,Tv), where T:R"—R”isavector space isomorphism. Show that thiscan bemade into thetotal space ofavectorbundleoverS!(ageneralized Mébius strip). (b)Show that theresulting bundle isorientable ifandonly if7isorientation preserving, 21.Show that forp€S?,thevectorpp€R3,isnotinix(S2,)byshowingthat theinner product (p,¢’(0)) =0forallcurves ¢with ¢(0) =pand |e(/)| =1 forall¢.(Recall that LaYO =SOL 8O) +(FO,8'O, where ¢denotes thetranspose; seeCalculus onManifolds, pg.23.) 22.LetMbea C®manifold. Suppose that(TM)|A istrivial whenever ACM ishomeomorphic toS'. Show that Misorientable. Hint: Anarc¢from TheTangent Bundle 101 po€Mtop€Miscontained insomesuchAso(TM)Jc istrivial. Thusone can“transport” theorientation ofM,, toM,. Itmust bechecked that thisis independent ofthe choice ofc.First consider pairs ¢,c’which meet only atpo andp.The general, possibly quite messy, case canbetreated bybreaking up¢ into small pieces contained incoordinate neighborhoods. Remark: Using results from theAddendum toChapter 9,together with Prob- lem29,wecanconclude thataneighborhood ofsomeS!CMisnon-orientable ifMisnon-orientable. The next twoproblems deal with important constructions associated with vector bundles. 23.(a)Suppose &=7:E>Xisabundle andf:Y>Yisacontinu- ousmap. LetE’CYxEbethesetofall(y,e)withf(y)=(e),define x':E!>Ybyx'(y,e) =y,anddefinef:E’>Ebyf(y,e) =e.Avector space structure canbedefined on ny) ={neieex "(f(y))} byusingthevectorspacestructure onx~!(f(y)). Showthat2:E’>Y¥isa bundle, and(f,f)abundle map which isanisomorphism oneach fibre. This bundle isdenoted byf*(&), andiscalled thebundle induced (from &)by/f. (b)Suppose wehave another bundle &”=x”: E”>Yandabundle map (Jf,f)from &”to&whichisanisomorphism oneachfibre.Showthat&”~ &=f*(). Hint: Mape€E”to(x"(e), f(e)) €E’. (c)Ifg:Z>Y,then (fog)*(&) =g*(/*@)). (d)IfACXandi:A—Xistheinclusion map,theni*(&)=&|A. (e)If&isorientable, then /*(&) isalso orientable. (1)Giveanexample where£isnon-orientable, but/*(€)isorientable. (g)Let =: E>Bbea vector bundle. Since x:E—Bisacontinuous mapfromaspacetothebasespace Bof&,thesymbol x*() makes sense. Show that if&isnotorientable, then x*(€) isnotorientable. 24.(a)Given ann-plane bundle &=x:E>Bandanm-plane bundle 7= x’:E’+B,letE”CEx E’bethesetofallpairs (e,e’)with x(e) =x'(e’). Letx"(e,e') =x(e) =x'(e’). Show that x”: E”>Bisan(n+m)-plane bundle. Itiscalled theWhitney sum &@7of&andn;thefibre of&@7over p isthedirect sumx7!(p) ®x/7}(p). (b)Iff:Y>B,show that f*(E®n)=f*(E)©f*(). 102 Chapter 3 ()Given bundles &=2):E;—>Bj,define x:E;xEx>ByxByby m(e1,€2) =(11(€1), 72(2)). Show that thisisabundle &x&over ByxBp. (d)IfA:B>BxBisthe“diagonal map”, A(x)=(x,x),showthat&@n= at xn). (c)If€and»areorientable, showthat£@7isorientable. (f)If&isorientable, and 7isnon-orientable, show that &®77isalso non~ orientable. (g)Define a“natural” orientation onV®Vforanyvector space V,and use thistoshow that&@&isalways orientable. (h)IfXisa“figure eight” (c.f.Problem 5),find twonon-orientable I-plane bundles &and7over Xsuch that&@7isalsonon-orientable. 25.(a)Ifx:E-»MisaC® vector bundle, then xshasmaximal rank at cach point, andeach fibre x7!(p) isaC®submanifold ofE. (b)The 0-section ofEisasubmanifold, carried diffeomorphically onto Bbyx. 26.(a)IfMandNareC®manifolds, andxy[orxy]:MxN>M[orN] istheprojection onM[orN],then T(M xN)~xaq*(TM) ©xn*(TN). (b)IfMandNareorientable, thenMxNisorientable. (c)IfMxNisorientable, thenbothMandNareorientable. 27.Show thattheJacobian matrix ofyy0(x«)7! isoftheform Djyiox7! @) X Dyyiox tJ” This shows that themanifold TM isalways orientable, i.e.,thebundle T(TM) is orientable. (Here isamore conceptual formulation: forv€TM, theorientation for(TM)y can bedefined as a af ay ayy, agton)|,’ ?Axton,’ dx]? 7?der], ]? theformof40(x)7! showsthatthisorientation isindependent ofthechoice ofx.)Adifferent proof that 7Misorientable isgiven inProblem 29. 28.(a)Let(x,U) beacoordinate system onMwith x(p) =0andletv€Mp beDjsa!4/8x"|, .Consider thecurvecinTMdefined by y=a CW=v4ss, TheTangent Bundle 103 Show that de a =O)=—].a=ae| (b)Findacurve whose tangent vector at0is0/8(x! ox)|,,. 29.This problem requires some familiarity with thenotion ofexact sequences (ccf.ChapterI}),AsequenceofbundlemapsE;+Ey+Eywithf=¢= identity ofBisexact ifateach fibre itisexact asasequence ofvector space maps. (a)If€=x:E>BisaC™ vector bundle, show that there isanexact sequence O- 2°(&) >TE>x*(TB) >0. Hint:(1)Anelement ofthetotalspaceof2*(€) isapairofpoints inthesame fibre, which determines atangent vector ofthefibre. (2)Map ¥€(TE)e to (e,74X). (b)If 0E,;>Ex>E3>0'sexact, then each bundle £;isorientable if the other two are. (0)T(TM) isalways orientable. (d)Ifx:E>Misnot orientable, then themanifold Eisnotorientable. (This iswhy theproof thattheMébius strip isanon-orientable manifold issosimilar totheproof thattheMébius bundle over S!isnotorientable.) The next two Problems contain more information about thegroups intro- duced inProblem 2-33, Inaddition tobeing used inProblem 32,thisinforma- tionwillallbeimportant inChapter 10. 30.(a)Letpo€S"~! bethepoint (0,...,0, 1).Form >2define f:SO(n) > S™' byf(A) =A(po). Show thatfiscontinuous andopen. Show that {~1(po) ishomeomorphic toSO(n —1),andthen show that{~!(p) ishome- omorphic toSO(n —1)forallp€S">!. (b)SO(1) isapoint, soitisconnected. Using part (a),and induction onn, prove that SO(7) isconnected foralln>1. (c)Show that O() hasexactly twocomponents. 31.(a)IfT:R">R®isalinear transformation, T*:R">R",theadjoint ofT,isdefined by(T*v, w)=(v,Tw) (foreach v,themap w+>(v,Tw) is linear, soitisw+ (T*v, w)foraunique T*v). IfAisthematrix ofTwith respect totheusual basis, show thatthematrix of7”isthetranspose A‘. 104 Chapter 3 (b)Alinear transformation T:R">R”isself-adjoint ifT=T*,sothat (Tv,w)=(v,Tw)forallv,w€R".IfAisthematrix ofTwithrespect tothe standard basis,thenTisself-adjoint ifandonlyifAissymmetric, At=A,It isastandard theorem thatasymmetric Acanbewritten asCDC~! forsome diagonal matrix D(forananalytic proof, seeCalculus onManifolds, pg.122). Show thatCcanbechosen orthogonal, byshowing thateigenvectors fordistinct eigenvalues areorthogonal. (0)Aself-adjoint T(orthecorresponding symmetric A)iscalled positive semi- definite if(Tv, v)>0forallv€R",andpositive definite if(Tv, v)>0forall v0.Show that apositive definite Aisnon-singular. Hint: UsetheSchwarz inequality. (d)Show thatAt-Aisalways positive semi-definite. (e)Show thatapositive semi-definite Acanbewritten asA=B?forsome B. (Remember that Aissymmetric.) (1)Show thatevery A€GL(n,R) canbewritten uniquely asA=Ay-Az where A,€O(n) andAzispositive definite. Hint: Consider At.A,andusepart(c). (g)Thematrices A,andAzarecontinuous functions ofA.Hint: IfA>A andA®)=A“, .A, thensome subsequence of{A“);} converges. (h)GL(#,R) ishomeomorphic toO(n) xR™@+)?2 andhasexactly twocom- ponents, {A: detA>0}and{A:detA<0}.(Noticethatthisalsogivesus another wayoffinding thedimension ofO(n).) 32.Twocontinuous functions fo,fi:X¥>Yarecalled homotopic ifthereis acontinuous function H:Xx[0,1] >Ysuch that fix) =HQ,i) 1=0,1. Thefunctions H;:X->Ydefined byH;(x) =H(x,1) may bethought ofasa pathoffunctions fromHo=fotoHi=fi.ThemapHiscalled ahomotopy between foandfj. The notation f:(X,A) >(Y,B), for ACcXand BC Y,means that f:X— Yand f(A) CB.Wecallfo,fi:(X,A)>(Y,B)homotopic (asmaps from (X,A)to(Y,B))ifthere isanHas above such that each H;:(X,A)> (Y,B). (a)IfA:[0,1] >GL(@, R)iscontinuous andH:R”x[0,1]>R”isdefinedby H(x,1) =A(t)(x), showthatHiscontinuous, sothatHoandHjarehomotopic asmaps from (R",R” ~{0}) to(R”,R"—{0}).Concludethatanon-singular linear transformation T:(R”,R” —{0}) >(R",R” ~{0}) with detT>0is homotopic totheidentity map. (b)Suppose f:R”>R”isC®andf(0) =0,while f(R"—{0})CcR"~{0}. IfDf(0) isnon-singular, show that f:(R",R" —{0}) >(R",R* —{0}) is The Tangent Bundle 105 homotopic toDf(0): (R",R" —{0})>(R",R" —{0}). Hint: Define H(x,t) = S(tx) for0<¢<1and H(x,0) =Df(0)(x). Toprove continuity atpoints (x,0), useLemma 2. (0)LetUbeaneighborhood of0€R"and f:U->R"ahomeomorphism with f(0) =0.LetB,CVbetheopen ballwith center 0andradius r,andlet A:R"—B,bethehomeomorphism hex)==arctanii)x;Pf then foh: (R",R" —{0})>(R",R" ~{0}). Wewillsaythat fisorientation preserving at0iffohishomotopic tothe identity map 1:(R",R" —{0}) >(R",R" ~{0}). Check that thisdoes not depend onthechoice ofB,CV. (d)Forp©R*,letTp:R®*>R"be7,9)=p+q.Iff:U>Visa homeomorphism, where U,V CR"areopen, wewillsaythat fisorientation preserving atpifT_sp)°f07,isorientation preserving at0.Show thatifM isorientable, then there isacollection Cofcharts whose domains cover Msuch thatforevery (x,U) and(y,V)in@,themapyox~ isorientation preserving atx(p) forall peUNV. (e)Notice thatthecondition onyox7!inpart(d)makes sense even ifyox! isnotdifferentiable. Thus, ifMisany(not necessarily differentiable) manifold, wecandefine Mtobeorientable ifthereisacollection Cofhomeomorphisms x:U—R"whose domains cover M, such that Csatisfies the condition in part(d),Toprovethatthisdefinition agreeswiththeoldoneweneedafact from algebraic topology: Iff:R”>R"isahomeomorphism with f(0) =0 andT:R">R®isT(x!,...,x”) =(x!,...,x"7!,x"),thenpreciselyone offandTofisorientation preserving at0.Assuming thisresult,showthat ifMhassuch acollection @ofhomeomorphisms, then foranyC®structure onMthetangent bundle TMisorientable. 33.LetM"CR beaC®n-dimensional submanifold. Byachord ofMwe meanapointofR¥oftheformp—qforp,g€M. (a)Prove thatifN>2n+1,then there isavector v€S%~! such that (i)nochord ofMisparallel tov, (ii)notangent plane M,contains v. Hint: Consider certain maps from appropriate open subsets ofMxMand TM toSN-}. 106 Chapter 3 (b)LetRY! cRYbethesubspace perpendicular tov,andx:RNY>R71 thecorresponding projection. Show that x|M isaone-one immersion. In particular, ifMiscompact, then x|M isanimbedding. (©)Every compact C®n-dimensional manifold canbeimbedded inR?"+}. Note: This istheeasy case ofWhitney’s classical theorem, which gives the same result even fornon-compact manifolds (H.Whitney, Differentiable manifolds, Ann. ofMath. 37(1935), 645-680). Proofs may befound inAuslander and MacKenzie, Introduction toDifferentiable Manifolds andSternberg, Lectures onDif ferential Geometry. InMunkres, Elementary Differential Topology, there isadifferent sortofargument toprove thatanot-necessarily-compact n-manifold Mcanbe imbedded insome RN(infact,withN=(”+1)?).Then wemayshow thatM imbeds inR?"+! using essentially theargument above, together with theexis- tence ofapropermapf:M—R,givenbyProblem‘2-30 (compareGuillemin andPollack, Differential Topology). Amuch harder result ofWhitney shows that M"canactually beimbedded inR™"(H.Whitney, Theself-intersections ofasmooth u-manifold in2n-space, Ann.ofMath.45(1944),220-246). CHAPTER 4 TENSORS Atheconstructions onvectorbundlescarriedoutinthischapterhavea common feature. Ineach case, wereplace each fibre x~!(p) bysome other vector space, andthen fitallthese new vector spaces together toform a new vector bundle over thesame base space. The simplest case arises when wereplace each fibre Vbyitsdual space V*. Recal} that V*denotes thevector space ofalllinear functions 4:V>R.If Jf:V¥>Wisalinear transformation, thenthereisalinear transformation S*: W* >Y*defined by (S*A)@) =ACfv). Itisclearthatif1y:V>Vistheidentity, then1”istheidentity mapofV* and ifg:U—V,then (fog)* =g*o f*. These simple remarks already showthatf*isanisomorphism iff:V>Wis,for(f7!0f)*=ly*and (fo fo) =lw*. The dimension ofV*isthesame asthat ofV,forfinite dimensional V.In fact, ifv;,...,Un isabasis forV,then theelements v*;€V*,defined by v*i(¥j) =8), areeasily checked tobeabasis forV*.The linear function v*;depends onthe entire setv1,...,Un, NOtjustonv;alone, andtheisomorphism from VtoV* obtained bysending v;tov*;isnolindependent ofthechoice ofbasis (consider what happens ifv;isreplaced by2v1). Ontheother hand, ifv€V,wecandefine v**€V**=(V*)* unambigu- ously by v**(A) =A(v) forevery} €V*. Ifv**(A) =0forevery A€V*,then A(v) =0forallA€V*,which implies that 107 108 Chapter 4 v=0.Thusthemapv+>v**isanisomorphism fromVtoV**.Itiscalled thenatural isomorphism from VtoV**. (Problem 6givesaprecise meaning totheword“natural”, formulated only after theterm hadlong been inuse. Once themeaning ismade precise, wecan prove thatthere isnonatural isomorphism from VtoV*.) Now let§=7:E>Bbeanyvector bundle. Let B=Ueto, pe anddefine thefunction 2’:E’+Btotakeeach [x~'(p)J* top.IfUCcB, and¢:#7'(U) >UxR”isatrivialization, then wecandefine afunction tin’"(U) >Ux(R")* intheobvious way:sincethemap/restricted toafibre, tywp) >{p)xR", isanisomorphism, itgives usanisomorphism (py! fp)" >fp}x(RY. Wecanmakex’:E’+Bintoavector bundle, thedualbundle &*of&,by requiring that allsuch 2’belocal trivializations. (We firstpick anisomorphism from (R")* toR",once andforall.) Atfirstitmight appear that £*~&,since each x~!(p) isisomorphic to x'—'(p). However, thisistruemerely because thetwovector spaces have the same dimension, The lack ofanatural isomorphism from VtoV*prevents usfrom constructing anequivalence between &*and§.Actually, wewillsee later that in“most” cases £*isequivalent to&;forthepresent, readers may ponder this question forthemselves. Incontrast, thebundle £** =(E*)* is always equivalent to§.Weconstruct theequivalence bymapping thefibre V of&over ptothefibre V**of&*over pbythenatural isomorphism. Ifyou ‘Tensors 109 think about how&*isconstructed, itwillappear obvious thatthismap isindeed anequivalence. Even if&canbepictured geometrically (eg,, ifisTM), there isseldom a geometric picture for&*.Rather, &*operates on&:Ifsisasection of§ando isasection of&*,then wecandefine afunction from BtoRby s(p) ex "(p) Pr o(p)(s(p)) = =o(p) en’ p)=2"(p)*. This function willbedenoted simply byo(s). When thisconstruction isapplied tothetangent bundle TM ofM,there- sulting bundle, denoted by7*M, iscalled thecotangent bundle ofM;thefibre ofT*M over pis(Mp)*. Like TM, thecotangent bundle T*M isactually a C®@ vector bundle: since two trivializations x,and y,ofTM areC®-related, thesame isclearly trueforx4’andy,’{infact,yx’0(X4!)—! =yx0(x4)7!). Wecanthus define C®, aswell ascontinuous, sections ofT*M. IfwisaC® section ofT*M and XisaC™ vector field, then w(X) istheC™ function Pr w(p)(X(p)). Iff:M+Risa C® function, then aC® section dfofT*M can be defined by df(ph\X) =X(f) forX©My. The section dfiscalled thedifferential off.Suppose, inparticular, that Xis de/dt\1, where ¢(to) =p.Recall that de] _ (4 ath, *\ath)” This means that de d dy> =C|— |=<! (fee) :ato , aF(e(O))=(foc)'(to) oree 7to 110 Chapter 4 Adopting theelliptical notations de de dg(t)S fr & BM for g'(t);a ae a sO thisequation takes thenice form de)_af(cl®)) af|—|=——.. if(&) di If(x,U) isacoordinate system, then thedx!aresections ofT*M over U. Applying thedefinition, weseethat a ; a=si dx!(p)(21)=i. Thus dx'(p),...,dx"(p) isjustthebasis ofMp*dual tothebasis /8x'|p,..., 8/8x"|p ofMy. This means that every section wcanbeexpressed uniquely onUas a w(p)=Ywi(p) dx'(p), i=l forcertain functions w;onU.Thesection wiscontinuous orC™ifandonlyif thefunctions w;are. We can also write n o=\ordx', i=l ifwedefine sums ofsections and products offunctions and sections inthe obvious way (“pointwise” addition andmultiplication). The section dfmust have some such expression. Infact, weobtain aclassical formula: Tensors 111 1,THEOREM. If(x,U)isacoordinate system andfisaC®function, then onUwe have nofi df=>oatdx'. PROOF. IfXp€Myis na X=oa!mal, then a!=X,(x!) =dx'(p)(Xp). Thus n ;af Af(D\(Xp) =Xp)=DaTG) i=l na ;=>Lodx!(p)(Xp).&i=1 3 Classical differential geometers (and classical analysts) didnothesitate totalk about“infinitely small” changes dx‘ofthecoordinates x’,justasLeibnitzhad. Noone wanted toadmit that this was nonsense, 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 thatdfshould beafunction ontangent vectors. Thedx!themselves thenmetamorphosed intofunctions, anditbecame clear thatthey must bedistinguished from thetangent vectors 3/dx'. Once thisrealization came, itwasonly amatter ofmaking new definitions, 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, likede/dt. Looking back attheclassical 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 MODERN FORMULATION Letfbeafunction ofthex!,...,x”, |Letfbeafunction onM,andxa sayf=f(x!,...,x"). coordinate system (sothatf=fox forsome function fonR",namely f= fox), Letx!befunctions of1,sayx4= Let¢:R>Mbeacurve. Then x!(r).Thenfbecomes afunction foc: R=R,where of4,fl)=fONO,....x"). foelt) =f(x!cc(t),...,x" ec(d)). We now have We now have af_ypafdt (Foxat 5ax!dr” ae isi= =PDifxe) -OFoo) (Theclassical notation, which ist suppresses thecurve ¢,isstillused yay nebyphysicists, asweshallpointout =DLpre) Gey onceagain inChapter 7.) in or ase) _Ha dx!(c(t)aDare: dt Multiplying bydrgives Consequently, “af i; de “af i(de= SF axi. a (=)=-yV = edxi (& af>axt@* if(3)>axCO)-ax3) (This equation signifies thattrue Since every tangent vector at¢(¢)is results areobtained bydividing by oftheform de/dt, wehave dtagain, nomatter whatthefunctions , x(t) are.Itistheclosest approach yp niinclassicalanalysistotherealiza- Cyzat tionofdfasafunction ontangent ~ vectors.) Tensors 113 Inpreparation forourreading ofGauss and Riemann, wewillcontinually examine theclassical way ofexpressing allconcepts which weintroduce. After awhile, the“translation” ofclassical terminology becomes onlyalittlemore difficult than the translation ofthe German inwhich itwas written. Recall that iff:M—NisC®, then there isamap f,:TM >TN; foreach p€M,wehave amap fep: Mp>Nyy). Since fypisalinear transformation between twovector spaces, itgives risetoamap Nip >My". Strict notational propriety would dictate that thismap bedenoted by(f.p)*, buteveryone denotes itsimply by IpsNyipy* >My”. Notice thatwecannot putall/;*together toobtain abundle map from T*N toT*M; infact, thesame g€Nmay be/(p;) formore than one p;€M, andthere isnoreason why fxp, should equal fp). Ontheother hand, wecan dosomething with thecotangent bundle that wecould notdowith thetangent bundle. Suppose wisasection ofT*N. Then wecandefine asection ofT*M asfollows: n(p)=(Ff(P))©fap ie, n(p)(Xp) =o(f(P))SapXp) forXp€My. (The complex symbolism tends tohide thesimple idea: tooperate onavector, wepush itover toNby/,,and then operate onitbyw.) This section 7is denoted, naturally enough, byf*w. There isnocorresponding way ofwans- ferring avector field XYonMover toavector field onN. Despite thesedifferences, wecansay,roughly, thatamapf:M>Npro- duces amap fsgoing inthesame direction onthetangent bundle andamap {*going intheopposite direction onthecotangent bundle. Nowadays such situations arealways distinguished bycalling thethings which gointhesame direction “covariant” andthethings which gointheopposite direction “con- travariant”. Classical terminology used these same words, and itjust happens tohave reversed this: avector field iscalled acontravariant vector field, while asection of7*M iscalled acovariant vector field. And noone has had the gallorauthority toreverse terminology sosanctified byyears ofusage. So it’svery easy toremember which kind ofvector field iscovariant, and which 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;) =e,then x(a!vy ++++a"_)=(a',...,0"), sothexcoordinate system isjust an“oblique Cartesian coordinate system”. Bares Lecce ' bv,{-~ iQx(p)=(a,b) ’ v2 av on Ifx’isanother such coordinate system, thenx4=7/2, aijx! forcertain ajj. Clearly aij=8x44/8x!, so nax’ in i. )iD Deal isl thiscanbeseendirectly from thefactthatthematrix (9x///4x*) istheconstant inatrix D(x! x7!) =x’ox7!. Comparing («)with 5 ;; ax'i(4 dx)=\~°—dx", (wa) Ls tl from Theorem 1,weseethatthedifferentials dx!“change inthesame way” as thecoordinates x“,hence theyare“covariant”. Consequently, anycombination n w=Yodx! i=l isalso called “covariant”. Notice that ifwealso have n w=Yo'ax", i=l Tensors 115 then wecanexpress thew’;interms ofthe@;.Substituting 5 P ax! j dx!=\*——dx La jel into thefirstexpression for@andcomparing coefficients with thesecond, we find that “ax! ’ w= erga isl Ontheother hand, given twoexpressions n n a a De -Le" ao = Sat 7 ita ON Ox foravector field, thefunctions a’!must satisfy n ;; Ox!in ia=vaaxt tl ‘These expressions canalways beremembered bynoting that indices which are summed over always appear once “above” andonce “below”. (Coordinate func- tions x!,..., x"used tobedenoted byx1,...,%n- This suggested subscripts w; forcovariant vector fields andsuperscripts a!forcontravariant vector fields. Af terthiswasfirmly established, theindices onthex’swere shifted upstairs again tomake thesummation convention work out.) Covariant and contravariant vector fields, i.e. sections ofT*M and TM, respectively, arealso called covariant and contravariant tensors (ortensor fields) oforder 1,which isawarning that worse things aretocome, Webegin with some worse algebra. IfVj,...,Vn arevector spaces, afunction Tixx VneR ismultilinear if VES Ty.)M15VsVeet+++»Um) islinear foreach choice ofv1,...,0%-1,Uk415--->Um- The setofallsuch T isclearly avector space. Ifi,...,Vm =V,thisvector space willbedenoted 116 Chapter 4 by7"(V). Notice thatT'(V) =V*. Iff:V>Wisalinear transfor- mation, then there isalinear transformation f*: 7"(W) >T”(V), defined completely analogously tothecase m=1: SJ?TM,..., Um)=T(f(1),--+5fm). ForT¢T*(V), andS€T!(V) wecandefine the“tensor product” T@S € TEH(V) by TQS, ..-5VesVegts s+Vet) =T(U1,---5 Uk)S(Uegass+Vet): Ofcourse, T@S isnotS@T. Ontheother hand, (S@T)@U =S@(T@U), sowecandefine n-fold tensor products unambiguously; thistensor product operation isitself multilinear, ($1+S2)@T =S;@T +S. @T, etc. In particular, if4,...,0,isabasisforVandv*),...,v%y isthedualbasisfor v*=7'(V), then theelements V7 BO vy, Il<t,...,% <0 areeasily scentobeabasis for7*(V), which thushasdimension n*. Wecanusethisnew algebraic construction toobtain anew bundle from any vector bundle §=1:E>B.Welet E’= TKx"(p)), peB and let n':E'-> Btake T#(x7(p)) top. IfUC Band t:a7\(U) >UxR" isatrivialization, then theisomorphisms tpi7"(p) >{p}xR” yield isomorphisms (tp")?: TR "p)) >{p}xTER"). Ifwechoose anisomorphism 7*(R") +R™onceandforall,these maps can beputtogether togive amap tsn'\(U) >UxR”, Tensors 117 Wemake x':E’->Bintoavector bundle *(&)byrequiring thatallsuch7’ belocal trivializations, The bundle &*isthespecial case k=1. Forthecase ofTM, thebundle 7*(7M) iscalled thebundle ofcovariant tensors oforder k,and asection iscalled acovariant tensor field oforder k.If (x,U)isacoordinate system, sothat dx'(p),...,dx"(p) isabasis for(Mp,)*, then thek-fold tensor products ax"(p)@---@dx"*(p) ET*(Mp) 1Sit,...,i¢ Sn areabasis forT*(M,). Thus, onUevery covariant tensor fieldAoforder k can bewritten Alp)=>Ad. (p)ax"(p)@+@dx"(p), heals orsimply A=DOAnuuig dxB+ax, where dx"! @-..@dx's now denotes asection ofT*(TM). Ifwealsohave A=DDAtaigAx"B+@dx"'k, Fyyeeeste then Pefe ik Ae.ncty==Ainturay Dyantoonik (theproducts arejust ordinary products offunctions). Toderive thisequation, wejust useequation («*) onpage 114, and multilinearity of®.The section A iscontinuous orC®ifandonly ifthefunctions Aj,..i, are. Acovariant tensor field Aoforderkcanjustbethought ofasanoperation A onkvector fields X1,..., Xxwhich yields afunction: A(X1,0.Xe)(Pp)=ACPXP), Xe(P))- Notice that Aismultilinear ontheset‘VofC®vector fields: ACM00Xi4X Xp)=AMyoXt Xe)+AM XiXb) A(X, -0Xi,.6. Xe)=@A(K,00.Xe) 118 Chapter 4 Moreover, because Aisdefined “pointwise”, itisactually linear overtheC° functions F;i.e,iffisC, then A(X5-005 LXine Xe)=LAM XinKids forwe have A(X15-225 £XigesXaMP)=APY KD) 5-2 LP)XCD), XP) =S(P)A( PX (p)s- +2Xe(p)> +++ Xe(P)) =fp) A(%,...,Xty--+ Xe)(p)- Wearefinally ready foranother theorem, onethat isused over andover. 2.THEOREM. If AL VX xVor Raia Atimes: islinear overF,thenthere isaunique tensor fieldAwithA=A. PROOF. Note firstthat ifv€Mpisanytangent vector, then there isavector field X€Vwith X(p) =v.Infact, if(x,U)isacoordinate systemand n av=>aial; far OXI then wecan define “ainc roe{2°gaton 0 outside U, where each a’now denotes aconstant function and fisaC®function with S(p) =1and supportfCcU. Now ifvy,...,v% €Mpareextended tovector fields X1,...,X% €Vwe clearly must define A(p)(U1,-- +5Uk)=A(X1,.-., Xe)(P)- The problem istoprove that thisiswell-defined: IfX;(p) =Y;(p) foreach i, weclaim that A(X,«025Xe)(p)=AM.Yep) « ‘Tensors 119 (The map &“lives atpoints”, tousetheinterminology) Forsimplicity, take the casek=1(thegeneral caseisexactlyanalogous). TheproofthatA(X)(p) = A(¥)(p) when X(p) =Y(p) isintwosteps. (I)Suppose first that ¥=Yinaneighborhood Uofp.LetfbeaC® function with f(p) =1andsupport fCU.Then fX=fY,so LA(X) =ACFX) =ALY) =SAY); evaluating atpgives A(X)(p) =ALY)(p). (2)Toprove theresult, itobviously suffices toshow that A(X)(p) =0if X(p) =0.Let (x,U) beacoordinate system around p,sothat onUwe can write a9 i X=La whereallb'(p)=0. Ifgis1inaneighborhood Vofp,andsupport gCU,then rr) re) Y=adivag =)85y7ial ist isawell-defined C™ vector field onallofMwhich equals XonV,sothat A(X)(p) =A(Y)(p), by(1). Now * a AMY)(p)=SBP) (ex))ist =0, since b'(p) =0. Because ofTheorem 2,wewillnever distinguish between thetensor fieldA and theoperation A,norwillweusethesymbol 4anylonger. Note that Theorem 2applies, inparticular, tothecase k=1,where T*(7M) =T*M, thecotangent bundle: afunction from V—¥which islinear over ¥comes from acovariant vector field «w.Just aswith covariant vector fields, aC°?map f:M—Ngivesamapf*takingcovariant tensorfieldsAoforderkonN tocovariant tensor fields {*A oforder konM: LAP Xgs++Xig)=ALP) SeXp«+2faXkp)- 120 Chapter 4 Moreover, ifAand Barecovariant tensor fields oforders kand/,respectively, then wecan define anew covariant tensor field A®Boforder k+d: (A@ B)(p) =A(p) ®Bp) (operating onMyx++.xMpk+1times). Although covariant tensor fields willbeourmain concern, ifonly forthe sake ofcompleteness weshould define contravariant tensor fields. Recall that acontravariant vector field isasection XofTM. Soeach Xp€Mp. Now an element vofavector space Vcanbethought ofasalinear function v:V*>R; wejust define v(A) tobeA(v). Acontravariant tensor field oforder kisjust asection Aofthebundle 7*(T*M); thus, each A(p) isak-linear function onM,*.Wecouldalsousethenotation 7,(7M), ifweuseT,(V) todenote all k-linear functions on V*. Inlocal coordinates wecan write i a a Atp)=DoAten) aaeean|Jumade ? ? (remember thateach 3/9x/|p operates onMp"), orsimply ,4, 0 a = Jred QeA=YEAhoO OT Jowodk Ifwehaveanother suchexpression, inode 9. a = ti-in 9 @...@ 9AmDoA 88ae Sivondk then weeasily compute that 1B gx! Bib = FoneOX x" 4=ONT ah Srvoode Acontravariant tensor fieldAoforder kcanbeconsidered asanoperator A taking kcovariant vector fields w1,...,@ into afunction: A(wr,..-.@k)(P) =ACp)w1(p)- ++,@%(P))- Naturally, there isananalogue ofTheorem 2,proved exactly thesame way, that allows ustodispense with thenotation A,and toidentify contravariant tensor fields oforder kwith operators onkcovariant vector fields that arednear over theC®functions F. Tensors 121 Finally, weareready tointroduce “mixed” tensor fields. Tomake theintro- duction lesspainful, weconsider aspecial case first. IfVisavector space, let TV) denote allbilinear functions T:VxV*>R. Avector bundle §=2:E—Bgives risetoavector bundle J;!(&),obtained by replacing eachfibre7~!(p) byJ;(-"(p)). Inparticular, sections of7;!(7M) arecalled tensor fields, covariant oforder 1and contravariant oforder 1. ‘There areallsortsofalgebraic tricks onecanplaywith J;!(V); although they should bekept toaminimum, certain ones arequite important. LetEnd(V) denote thevector space ofalllinear transformations T:V>V(“endomor- phisms” ofV),Notice thateachS€End(V) givesrisetoabilinear 5€7;'(V), 5:vxv*>R, bytheformula (*) S(v,A) =(S(v)). Moreover, thecorrespondence S++SfromEnd(V) tox!(V)islinear andone- one,forS=0implies thatA(S(v)) =0forall4,which implies thatS(v)=0, forallv.Since both End(V) andJ;!(V) have dimension n?,thismapisan isomorphism. The inverse, however, isnotsoeasy todescribe. Given 5,for each vthevector S(v) €Vismerely determined bydescribing theaction ofa) onitaccording to(#).Itisnothard tocheck that thisisomorphism ofEnd(V) andJ;!(V) makes theidentity map 1:V>VinEnd(V) correspond tothe “evaluation” map e:VxV*>R ind!(V) given by e(v,A) =A(v). Generally speaking, ourisomorphism canbeused totransfer any operation from End(V) to7;(V). Inparticular, given abilinear T:VxV*>R, wecantake thetrace ofthecorresponding S:V—>V;thisnumber iscalled thecontraction of7.Ifvj,...,v, isabasis ofVand T=OTvi@y, Ai 122 Chapter 4 then wecanfind thematrix A=(aij) ofS,defined by n Seu)=Yoanr;, j=l interms ofthe7/;infact, ay=v*)(S(%)) =Tv") =Ti}. Thus n contraction ofT=7. ised (Theterm“contraction” comes fromthefactthatthenumber ofindices iscon- tracted from 2to0bysetting theupper and lower indices equal and summing.) These identifications andoperations canbecarried out,fibre byfibre, inany fibrebundle 7;!(€). Thus, asection Aof7;!(£) canjustaswellbeconsidered asasection ofthebundle End(§), obtained byreplacing each fibre x~!(p) by End(x~"(p)). Inthiscase, each A(p) isanendomorphism ofz~}(p). More- over, each section Agives risetoafunction (contraction ofA): B>R defined by p+ contraction ofA(p) . =trace A(p) ifweconsider A(p) €End(x~'(p)). Inparticular, given atensor field A,covariant oforder 1and contravariant oforder I,which isasection ofJ;!(7M), wecanconsider each A(p) asan endomorphism ofMp, andweobtain afunction “contraction ofA”.Ifina coordinate system i aA= Jdxi@——DAaea al then a ; (contraction ofA)=>Ai, isl The general notion ofamixed tensor field isastraightforward generalization. Define7;(V)tobethesetofall(k+/)-linear T: V@---@VxV*®@---@V* oR.er eK ONee ktimes times Tensors 123 Everybundle &givesrisetoabundle 7;(&).Sections of7;¥(TM) arecalled tensor fields, covariant oforder kand contravariant oforder /,orsimply oftype ({),anabbreviation thatalsosaveseverybody embarrassment about theuseof thewords “covariant” and“contravariant”. Locally, atensor fieldAoftype(4) canbeexpressed as A= Appldxt@-@dxte ew@-ent j teoodi axh axa’ iuontooh and if Ae DOAp} ax@.@dx@~~ee foodinh ox? bythe Stvondl then a Oxtk 9x'B ax!Br (BiB— JonOX aoe (*) AN ak=,xicinFyn”JarenOaOxi’Healt Classical differential geometry books arefilled with monstrosities likethis equation. Infact, theclassical definition ofatensorfieldis:anassignment ofn*+# functions toevery coordinate system sothat(*)holds between then*+! functions assigned toanytwocoordinate systems xandx’.(!)Oreven, “aset ofn*+ functions which changes according to(*)”. Consequently, inclassical differential geometry, allimportant tensors areactually defined bydefining the functions Aj,interms ofthecoordinate system x,andthen checking that(+) holds. Here isanimportant example. Inevery classical differential geometry book, onewillfindthefollowing assertion: “The Kronecker delta 8/isatensor.” In other words, itisasserted thatifonechooses thesame n?functions 4/foreach coordinate system, then (*)holds, i.c., jax!ax'P BSgf2 BaeLsxrOxi? bi thisiscertainly true, for aax!ax!®=>ax!ax’?_3Cy!Bx!Oxi +Ox"Oxi 124 Chapter 4 From ourpoint ofview, what thisequation shows isthat A=vedx!®<aa ax ij isacertain tensor field, independent ofthechoice ofthecoordinate system x Toidentify themysterious map A(p): MpxMy*>R, weconsider v€Mpand A€M,* with theexpressions n a n= — = Brn):v=oata»=Sobpxp);eal iP Bal then A(p)(v,A) =8)ax'(p)@ (v,A), if‘ oxy n 3 a n= i dx! oo B=Lee Do"ge,)a7],(Loewen) if mal iP. 1p\Beal =oafa’b ij 1 =doalhy iI =A(v). Thus A(p) isjust theevaluation map MpxMj* —R;considered asanendo- morphism ofMp, itisjusttheidentity map. Thecontraction ofatensorisdefined, classically, inasimilar manner. Given atensor, i.e,acollection offunctions Aj,oneforeach coordinate system, sat- isfying .ax!ax'B Bei ae=DaiOx'®QxJ? Tensors 125 wenote that a a a ylax!ax ta J a=l aati Si] nax! ax’= Z — ij ast =Dass bj n isl sothat this sum isawell-defined function. This calculation tends toobscure theonepart which isreally necessary—verification ofthefactthatthetrace of alinear transformation, defined asthesum ofthediagonal entries ofitsmatrix, isindependent ofthebasis with respect towhich thematrix iswritten. Incidentally, atensoroftype(4)canbecontracted withrespecttoanypair ofupper and lower indices. Forexample, thefunctions Bit= ashe axl “transform correctly” iftheA%®¥do.Ifweconsider eachA(p)€7;'(Mp), then wearetaking B(p) €7;(M,) tobe B(p)(v1,02,41,A2) =contraction of:(v,4) +A(p)(v; U1,V2,At,A2,A). While acontravariant vector field isclassically asetofnfunctions which “transforms inacertain way”, avector atasingle point pisclassically justan assigument of1numbers a',...,a” toeach coordinate system x,such thatthe numbers a’!,...,a’ assigned tox"satisfy nP jOxd a}=) a'——(p).>gat(P) This isprecisely thedefinition weadopted when wedefined tangent vectors as equivalence classes [x,4].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. Thetangent bundle itself was almost avictim oftheexcesses ofrevolutionary zeal. Foralong time, theparty lineheld that 7M must bedefined either asderivations, orasequivalence classes ofcurves; thereturn totheolddefinition was influenced bythe“functorial” pointofviewofTheorems 3-1and3-4. ‘The modern revolt against theclassical point ofview hasbeen socomplete incertain quarters thatsome mathematicians willgive athree 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” and contractions, 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 “onvector fields, which miraculously turns out tobelinear over theC® functions ¥,and hence must come from some A.Attheappropriate time wewilldiscuss whether ornotthisisallabig cheat. Tensors 127 PROBLEMS 1.Let f:M" +N", and suppose that (x,U) and (y,V) arecoordinate systems around pand /(p), respectively. (a)Ifg: NR, then ages), ag a(yiof)ae (P=pspiLP)(0). (Proposition 2-3isthespecial case f=identity.) (b)Show that a 7aysof) a +(gh) <E%20-44,, ax!|p»axt 8y?Irn) and,moregenerally, express fx(S-jix1 4/0/x'|,) interms ofthe8/8y/|p. (c)Show that a2.ays stayy(p) =ED) -ax"tp). ised (@)Express r(DSannindy@---@dy") intermsofthedx!. 2.Iffg: M>NareC®,showthat d(fg) =fdgt+egdf. 3.Letf:M>RbeC®. Forv€My, show that Salo) =Uf)(py€Ry): 4,(a)Show thatifthe ordered bases vj,...,Unandwy,..., WnforVareequally oriented, then thesame istrueofthebases v*,,..., v*,andw*},...,w*, forV*. (b)Showthatabundleéisorientable ifandonlyif£*isorientable. 128 Chapter 4 5.The following statements and problems arealltaken from Eisenhart’s clas- sical work Riemannian Geometry. Ineach case, check them, using theclassical methods, and then translate theproblem and solution into modern terms. An “invariant” isjust a(well-defined) function. Remember that thesummation convention isalways used, so4/4; means 7/2; \‘4;. Hints andanswers are given attheend, after (xiii). ()Ifthequantity 4/2; isaninvariant andeither /or4;arethecomponents ofanarbitrary [covariant orcontravariant] vector field, theother setsarecom- ponents ofavector field. (i)IfAgi!arethecomponents of7vector fields [inann-manifold], where i fori=1,...,n indicates thecomponent and@fora=|,...,” thevector, and these vectors areindependent, thatis,det(Ag\') #0,then anyvector-field A/is expressible intheform A=a%al!, where the a’sare invariants. (iii)If4;arethecomponents ofagiven vector-field, anyvector-field A!satisfy- ing\'4;=0isexpressible linearly interms ofn—1independent vector fields Aaj!for@=1,...,2 —1which satisfy theequation. (iv)Ifa/=a4"forthecomponents ofatensorfieldinonecoordinate system, then a’=a'/'forthecoordinates inanyother coordinate system. ()Ifa!andb4arecomponents ofatensorfield,soareai+4.Ifadand by,arecomponents ofatensorfield,soarea!/by. (vi)IfayjA‘A4 isaninvariant for\/anarbitrary vector, thenaij+aj;arethe components ofatensor; inparticular, ifajjA/A/ =0,thenajj+aj;=0. (vii)IfajjA'A/ =0forallvectors A/suchthatA’; =0,where y4;isagiven covariant vector, ifv!isdefined [c.f.(iii)]byaijAa\'v/ =0,@ =1,...,7—1and pivi #0, and bydefinition ajv'=0; vig=r, then(aij—44170;)§/£4 =0issatisfied byevery vector field£/,andconsequently aijtay==u)+Hj0%). (viii)Ifa,sarethecomponents ofatensor andband¢areinvariants, show that ifba,s +cas, =0,then either 6=—cand arsissymmetric, orb=¢and arsisskew-symmetric. (ix)Bydefinition therankofatensor ofthesecond order aijistherank of thematrix (az). Show that therank isinvariant under alltransformations of coordinates. Tensors 129 (x)Show that therank ofthetensor ofcomponents ajby, where a;and by arethecomponents oftwo vectors, isone; show that forthesymmetric tensor a;b; +a;b; therank istwo. (xi)Show thatthetensor equation a/;; =aj,where @isaninvariant, can bewritten intheform (a!;—«8/;)A; =0.Show alsothata!;=6a, ifthe equation istohold foranarbitrary vector Aj. (xii)Ifa!;A; =cA;holds forallvectors A;suchthatj1/A;=0,where px!isa given vector, then | ; ; aj=08; +ojh!. (xiii) IF 0ifja=jpforsomea#Borig=igforsomea#B gaein = orfthi... ip)Aki, stp} fvdp 1 ifj1,..., jpisanevenpermutation ofi1,...,ép 1 ifji,.--, jpisanoddpermutation ofi1,...,ip then6!” arethecomponents ofatensorinallcoordinate systems. HINTS AND ANSWERS. (i)»isdetermined ifw(X) isknown forallX,and viceversa. (ii)Given w[with w(p) ¥0forallp],there areeverywhere linearly indepen- dent vector fields X1,..., Xn—1 which span ker ateach point. (This istrue onlylocally. Forexample, onS?xRthere isanwsuch thatkerw(p,£)consists ofvectors tangent toS?x{r}.) (vi)ForT:VxV>R,let7’(v, w)=T(w, v).Then 7+7" isdetermined by S(v) =Tv, v).For, T(v+w,v+w) =T(v,v)+T(v,w)+7(w,v)+7(w,wv). Similarly, T(v, v)=0forallvimplies thatT+7’=0. (vii)Given w[with w(p) #0forallp],choose Ycomplementary tokerw at allpoints. Ifo(Z)=T(Y,Z),thenT(Z,Z)=w(Z)o(Z)/w(Y) forallvector fields Z. 7 _(ix)T:VxV=Rcorresponds to7:V>V*[where T(v)(w) =T(v,w)]. TherankofTmaybedefined astherankofF(consider thematrix of7with respect tobases vj,...,Unandv*},...,0%n). (xii)Let V=M,*. IfT:V>Vandp€V*andT(v) =avforall v€kerp, there isaycomplementary tokerjzsuchthat Tv) =av+plv)y forallv. (Begin bychoosing ypcomplementary toker4andwriting vuniquely asvp-+cyo forvp€kerp.) 130 Chapter 4 (xiii) Define 6: Vx: xVxV*x---xVioR Sa * ptimes ptimes by S(U1,...5UpyAt,-.-,Ap) =det(Ai(vj)). 6.(a)Letiy:V>V™ bethe“natural isomorphism” éy(v)(A) =A(v). Show that foranylinear transformation f:V>W,thefollowing diagram com- mutes: i :y—e iy Vr wi”. we ‘b)Show that there donotexist isomorphisms iy:V—V*such that thefol- lowing diagram always commutes. Vvtvve |e wi”. we Hint: There does noteven exist anisomorphism i:R>R*which makes the diagram commute foralllinear f:R>R. 7.Acovariant functor from (finite dimensional) vector spaces tovector spaces isa function Fwhich assigns toeveryvectorspaceVavector spaceF(V)andtoev- crylineartransformation f:V>Walinear transformation F({): F(V)> F(W), such that F(ly) =lpqvy and F(g0f)=F(g)oF(v). {a)The “identity functor”, F(V) =V,F(f) =fisafunctor. (b)The “double dual functor”, F(V) =V**, F({) =f** isafunctor. (c)The“Jfunctor”, F(V)=Hj(V)=T*(V*), F(L)(T) Qa,+5dk)=TOfi. deof) isafunctor. (d)IfFisanyfunctor andf:V>Wisanisomorphism, thenF(/)isan isomorphism. Acontravariant functor isdefined similarly, except thatF(f): F(W) >F(V)and F(gof)=F(f)0F(g). Functors ofmorethanoneargument, covariant in some and contravariant inothers, may also bedefined. (e)The “dual functor”, F(V) =V*,F(f) =f*isacontravariant functor. (f)The“7*functor”, F(V) =T*(V), F(f) =f*isacontravariant functor. Tensors 131 8.(a)LetHom(V, W)denote alllinear transformations from VtoW.Choos- ingabasis forVand W,wecanidentify Hom(V, W)with themxnmatrices, and consequently give itthemetric ofR””. Show that adifferent choice of bases leads toahomeomorphic metric onHom(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=2:E+Bisanyvectorbundle, andFjiscontinuous, thenthereis abundle F(é) =x‘:E’>Bforwhich x'~!(p) =F(x7}(p)), andsuch that toevery trivialization t:a7(U) >UxR" corresponds atrivialization Usw'"(U) >UxFR"). (©)Thefunctor %(V) =71(V*) =V*iscontinuous. (The bundle Jj(7M) is justacase oftheconstruction in(b).) (d)Define acontinuous contravariant functor F,and show how toconstruct a bundle F(é). (©)The functor F(V) =V*iscontinuous. (The bundle 7*M isaspecial case oftheconstruction in(d).) Generally, thesame construction canbeused when Fisafunctor ofseveral arguments. Thebundles J;(4)areallspecial cases. Seethenexttwoproblems forother examples, aswellasanexample ofafunctor which isnotcontinuous. 9.(a)LetFbeafunctor from V",theclass ofn-dimensional vector spaces, toV*.Given A€GL(1,R) wecanconsider itasamap A:R”>R”.Then F(A): F(R") >F(R"). Choose, once and forall,anisomorphism F(R") > R*.Then F(A) canbeconsidered asamap (A): R&—R*. Show that h:GL(n,R) >GL(k,R) isahomomorphism. (b)Howdoesthehomomorphism 4depend ontheinitialchoiceofthe isomor- phism F(R") —R*? (©)Letv=(v,...,0,) and W=(w;,...,W») beordered bases ofVand let e=(e,...,&n) bethestandard basis ofR”.Ife+ydenotes theisomorphism taking e;tov;,show that thefollowing diagram commutes Rp £>%y R" 132 Chapter 4 where A=(aiy) isdefined by n w;=>»ajiv;. j=l After identifying F(R") withR*,thismeans that REFle> ¥)F(V) h(A|Tas ) RK alsocommutes. This suggests away ofproving thefollowing THEOREM. Ifh:GL(n,R) >GL(k,R) isanyhomomorphism, there isafunctor F,:V">V*suchthatthehomomorphism defined inpart (a)isequal toA. (d)Forq,q'€R*,define (v9) ~(Ww,4’) ifq=h(A)q' where w;=D7_, aj;0;. Show that~isanequivalence relation, andthatevery equivalence class contains exactly oneelement (v,q) foragiven v. Wewilldenote theequivalence class of(v,q) by[¥,4]. (ec)Show that theoperations [v.91]+[v.92]=[v.91+92] a-[v,q] =[v,aq] 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 S01) =Dja1 aij, anddefine FalSly.4)=[w,A(A)()]- Show that this isawell-defined linear transformation, that Fjisafunctor, and thatF,(A) =4(A) when weidentify F;,(R”) with R*by[e,g] >g. (g)Leta: R>Rbeanon-continuous homomorphism (compare page 380), andlet4:GL(1,R) >GL(,R) =Rbeh(A) =a(det A).Then Fy:V">V! isanon-continuous functor. Tensors 133 10.Inclassical tensor analysis there are,inaddition tomixed tensor fields, other “quantities” which aredefined assetsoffunctions which transform according toyetother rules. These new rules areoftheform axe ‘—— A!=Aoperatedonbyh(35). Forexample, assignments ofasingle function atoeach coordinate system x such that thefunction a’assigned tox’satisfies i a=dee() 0 axtt arecalled (even) scalar densities; assignments forwhich ax! {= —}]- areCalled oddscalar densities. TheTheorem inProblem 9allows ustocon- struct abundle whose sections correspond tothese classical entities (later we willhave amore illuminating way): (a)Leth:GL(n,R) >GL(,R) take Ainto multiplication bydetA.LetFy bethefunctor given bytheTheorem, and consider theI-dimensional bundle F,(TM) obtained byreplacing each fibre Mp with F,(Mp). If(x,U) isa coordinate system, then a a ox(p)=(siga)|©FalMy) isnon-zero, soevery section onUcan beexpressed asa-ayforaunique function a.Ifx’isanother coordinate system and @-@y=a‘-@y', show that ax! a’=det(3)oa. ax (b)If,instead, #takes Ainto multiplication by|det A],show that thecorre- sponding equation is ; 0|(=)| a’=|det|—— ]]-a. ax (©)Forthis4,show thatanon-zero element ofF;,(V) determines anorientation for V.Conclude that the bundle ofodd scalar densities isnottrivial ifMisnot orientable. 134 Chapter 4 (d)Wecanidentify 7/(R") withR"**" bytaking 07OeBeyBCj,e+Wj>(dayeestks /tye+-s.J)™ basisvectorofR™ Recall thatiff:V>V,wedefine 7;'(f): T'(V) >T;*(V) by TEAMT, «VesMyM STLDs SUM fd S)- Given A€GL(n,R), wecan consider itasamap A:R”+R”. Then TE(A): TER") >F(R") determines anelement F(A) ofGL(xk*!,R). Letkh:GL(n,R) >GL(n**!, R)bedefined by h(A)=(detA)"7;*(A) waninteger. ThebundleF(7M) iscalledthebundleof(even)relative tensors oftype (/) andweight w.Fork=/=0weobtain thebundle of(even) relative scalars of weight w[the(even) scalar densities arethe(even) relative scalars ofweight 1]. If(detA)”isreplaced by|detA|”(wanyrealnumber), weobtain thebundle ofoddrelative tensors oftype(*)andweight w.Show thatthetransformation lawforthecomponents ofsections ofthese bundles is APr-b—[detax?\]”>Aitedtaxitaaxleax'Pi ax‘Pateaie axFJ) me ax axAxi Axi (orthesame formula withdet(8x!/8x"’) replaced by|det(3x!/8x’/))), (e)Define +1 if41,...,é, isanevenpermutation ofI,...,7 €i).cin=4—!ifit,...,én isanoddpermutation ofI,...,7 0iffe=igforsomea#B. Show thatthere isacovariant relative tensor ofweight —1with these compo- nents inevery coordinate system, Also show thate/-“” =ej); arethecom- ponents inevery coordinate system ofacertain contravariant relative tensor ofweight 1.(See Problem 7-12 forageometric interpretation ofthese relative tensors.) CHAPTER 5 VECTOR FIELDS AND DIFFERENTIAL EQUATIONS W: returntoamoredetailed studyofthetangentbundleTM,andits sections, i.e.,vector fields. LetXbeavector field defined inaneigh- borhood ofp€M.Wewould liketoknowifthereisacurvep:(—¢,¢) >M through pwhose tangent vectors coincide with X,that is,acurve pwith —_— —_— = — p(0)=p 40Sd qd r(|)=a],<x00 a t|,dt|, —, “NSSS ’ Since thisalocal question, wewish tointroduce acoordinate system (x,U) around pandtransfer thevector field Xtox(U) CR”. Recall that, ingeneral, aX does not make sense forC® functions a:M—N.However, ifaisa diffeomorphism, then wedefine (2.X)q=O4(Xy-1(9)) [ses =Oyq1(9)(Xa-14)) Itisnothard tocheck (Problem 1)thataX isC°ona(M). Inparticular, we have avector field xX onx(U) CR". There isafunction f:x(U) >R” with (2X)q=SQ)q €R"g, i.e.,(%¢X)g has“components” f7(g),..., £"(g). Consider thecurve ¢=xop. The condition dp a7XP) means that dPs(|)=X(e()); 135 136 Chapter 5 hence dc d|=aap(Sl)=xa(¥((0)) =OX)aceon =Vet. Ifweusec’(t) todenote theordinary derivative oftheR’-valued function c, then thisequation finally becomes simply elt) =f(e(). This isasimple example ofadifferential equationforafunctionc:R>R", which may also beconsidered asasystem of»differential equations forthe functions ¢?, lta) =fi(cl(t),...,e%) G=1..n We also want the “initial conditions” cf(0)=x!(p). Solving adifferential equation used tobedescribed as“integrating” the equation (the process isintegration when theequation hasthespecial form c() =f(© forf:R =R,aform towhich ourparticular equations never reduce); solutions were consequently called “integrals” oftheequation. Part of thisterminology isstillpreserved. Acurve p:(—¢,¢) >Mwith pO) =p dp a7XO) iscalled anintegral curve forXwith initial condition p(0) =p.Similar ter- minology isapplied, ofcourse, tothedifferential equations oneobtains upon introducing acoordinate system. Forquite some time, wewillwork entirely in Euclidean space, andforawhile x,y,etc., willdenote points ofR”.IfUCR" isopen and f:U>R’,then acurve c:(—8,¢) >Mwith c(0)=x xeU c(t) =F(elt) iscalled anintegral curve forfwith initial condition c(0)=x. Vector Fields andDifferential Equations 137 Before stating themain theorem about theexistence anduniqueness ofsuch integral curves, weconsider some special cases. ‘The equation foracurve ¢with range R, et) =-[eP, which would bewritten classically interms ofafunction y:R>Ras dy 2 ax ~y isthespecial casef(a)=—a?.Thestandard method ofsolving thisequation istowrite dy=e=dx dy[i-fe 1 —=x+C y _it error Thus the curves 1 1)= —ete) t+C aresupposed tobesolutions. Thiscanbechecked directly ifyoudon’tbelieve theabove manipulations. (They really domake sense; theequation inquestion asserts that y’=f0y,so (:»)ylsoy)-y'=1;f hence, ifF’=1/f, then (Foy) =1 F(x) =x +C forsome C.)Toobtain theinitial conditions c(0)=a,wemust take I (1)=——_-.“= T78 This works inallcases except a=0.Inthiscase, thecorrect solution is c(t) =0 forallt 138 Chapter 5 (which wemissed bydividing byy).Intermsofvector fields,thecurves ¢are theintegral curvesof d =-a—,X(a) aa ee 0 0 Notice that nointegral curve, except c(t) =0,canbedefined forall¢,even though Xisdefined onallofR.Itmight bethought thatthissomehow reflects thefactthat X(0)=0,butthishasnothing todowith thecase. Fora>0,the curve c(t)=1/(t+1/a) isdefined forallJarge 1,andast>ooitapproaches, butnever reaches, 0,Ontheother hand, as¢>—1/a thecurve escapes to infinity because thevector field gets bigtoofast. This willcontinue tobetrue evenifwemodify thevector fieldnear0sothatitisnever0, Another phenomenon isillustrated bytheequation e=ceyr?, written classically as a_apa ‘There aretwodifferent solutions with theinitial condition c(0) =0,namely () c(t) =0 forallt, (2) c(t)axe forall¢. Inthiscase, thefunction /f,given byf(a) =a”,isnotdifferentiable. Unique- nesswil]always beinsured when f:U>R"isC!,butitcanalsobeobtained witharather Jessstringent condition. Wesaythatthefunction /satisfies a Lipschitz condition onUifthere issome Ksuch that If) -SOs Kix—yl forallx,yeU. Notice thatf(a) =2isnotLipschitz; infact,there isnoKwith If)—fs Kia forxnear 0,since nea 1/3 ea =x7/3—=x7'F ++00asx>OF. f=x Vector Fields andDifferential Equations 139 ALipschitz function isclearly continuous, butnotnecessarily differentiable (for example, f(x) =|x|). Ontheother hand, aC!function islocally Lipschitz, thatis,itsatisfies aLipschitz condition inaneighborhood ofeach point—this follows from Lemma 2-5. ALipschitz function isalsoclearly bounded onany bounded set. The basic existence anduniqueness theorem fordifferential equations de- pends onasimple lemma about complete metric spaces. I.THEOREM (THE CONTRACTION LEMMA). Let(M,p) beanon- empty complete metric space, andJetf:M>Mbea“contraction”, thatis, suppose there issome C<1such that PL) SO)sCoy) —forallx,yeM. Then there isaunique x€Msuch thatf(x) =x(thefunction fhasaunique “fixed point”). PROOF. Notice thatfisclearly continuous. Letx9€Manddefine asequence {%p} inductively by Xna =£(%n), ie, Xn=f"(X0) =fofo+++0 f(Xo). pioeeee Se, ntimes Then aneasy induction argument shows that P(Xn,Xn4t) SC" p(X *1)- Thus (Xn Xn4k) SP(%ns Xnga) +o +PXnsk-1 Xnak) S(CP$6 OMEN (x9,x1). SinceC<1,thesumD729C”converges, soC”----4+C" =!+0asin>00. ‘Thus thesequence {x} isCauchy, sothere issome xwith x= kim X,. nt00 Continuity offthen shows that L(%)=fim,f%n)=Jimxing=X. 140 Chapter 5 Wearegoing toapply theContraction Lemma tocertain spaces offunctions, Recal} thatif(44,9) isametric space andXiscompact, then thesetofall continuous functions f:X—Misametric space ifwedefine themetric oby o(f,8) =supp(f(x),8(%))- xeX IfMisbounded, then wedonoteven need Xtobecompact. Moreover, ifM iscomplete, thenthenewmetric spaceisalsocomplete; thisisbasically justthe theorem that theuniform limit ofcontinuous functions iscontinuous, plus the factthateachim,In)existssinceMiscomplete. Inparticular, ifMisa compact subset ofR”,thenthesetofallcontinuous functions f:X¥>Mis complete with themetric o(f,8)=If~sll, where||f||=sup|/()I.xeX Our basic strategy insolving differential equations willbetoreplace differen- tiable functions andderivatives bycontinuous functions andintegrals. IfUc R"and f:U>R’iscontinuous, then acontinuous functionw:(—b,b)>U, defined onsome interval around 0,clearly satisfies 0) a)=f@@) (0) =x ifitsatisfies theintegral equation 1 ) at)=x4fSle(u))du, where theintegral ofanR”-valued function isdefined byintegrating each com- ponent function separately. Conversely, if@satisfies (J),then @isdifferentiable, hence continuous; thus a’=f0@iscontinuous, so t 1 a(t)—x=a(t)—@(0)=fal(u)du=fS(a(u))du. 0 0 Fortheproof ofthebasic theorem, weneed only onesimple estimate. Ifa continuous function f:[a,b] >R”satisfies |f|<K,then blfsee5K(b~a).a Toprove this, wenote that itistrue forconstant functions, hence forstep functions, andthusforcontinuous functions, which areuniform limits on[a,b] ofstep functions. Vector Fields andDifferential Equations 141 2.THEOREM. Letf:U—R”beanyfunction, where UCR"isopen. Letxo€UandJeta>0beanumber such thattheclosed ballBza(xo), of radius 2aand center x9,iscontained inU,Suppose that ()If]<LonBoa(xo) _ (2)If) —SO)| SKix—y|forx,y€Bra(x0). Choose b>0sothat (3)b<a/L 4)6<1/K. Then foreachx€Ba(x9)there isaunique ax:(~b,b) >Usuchthat ax’) =Sex) (0) =x. PROOF. Choose x€Ba(xo), which wil}befixed fortheremainder oftheproof. Let M={continuous @:(—b,b) >Bra(xo)}. Then Misacomplete metric space. Foreach a€M,define acurve Saon (—b,b)by , Sa(t)=x+fSlau))du 0 {theintegral exists since fiscontinuous onB2_(xo0)). Thecurve Saisclearly continuous. Moreover, forany¢€(—b,5)wehave 1 isaq)—x1=|f" acydy <bL by(l) <a by(3). Since |x—xo|<@,itfollows that |Sa(t) —x9]<2a,forallt€(—b,b), so (*) Sa(t) €Bra(Xo) CBoa(xo) fort€(—b,b). Thus S: M>M. Now suppose a,8€M.Then : 15a-Sp1=sup| "fawn)—sew)a| + Vo <bK sup |a(u) -B@)| by(2) —b<u<b =bK\ja —Bi. 142 Chapter 5 SincewechosebK<1(by(4)),thisshows thatS:M—Misacontraction. Hence Shasaunique fixed point: There isaunique a:(—b,b) >Bog(xo) with ' a(t)=x+fS(a(u))du. 0 This, alas, isnotquite what thetheorem states. Having used theelegant Con- traction Lemma, wepayforitbyfinishing offwithafinickydetail: The mapaistheunique6:(—b,b)>Usatisfying t piy=x+ fsoudu. Reason: Weclaim thatanysuch factually liesinBzq(xo), infact, inBoa(xo). Consider first numbers 1>0.Wehave already seen (statement (+))that for eachtwithO <1<b, a (e*)BW)=x+fS(B@)duisinBog(Xo)—_[theopenball] 0 provided that B(u) €Bra(xo) forallwwith O<u <t, socertainly if B(u) €Bra(xo) foralluwith O<u<1. Wecannow useasimple Jeast upper bound argument. Let A={t:0 <t<band B(u) €Bog(Xo) for0<u<2}. Let«=supA.Suppose @<b.WeclearlyhaveB(u)€Boa(xo) for0<u<a. SoB(a) €Bog(xo), by(#*). This clearly implies that B(a+s) €Baa(xo) for sufficiently smal] s>0,which contradicts thefactthat@=supA.Soitmust bethat supA=6.Asimilar argument works for—b<1<0. Tosum up,theunique fixed point ayofthemap Sistheunique curve with thedesired properties. Vector Fields andDifferential Equations 143 Notice thatsolutions ofthedifferential equation a) =fe) remain solutions under additive changes ofparameter; that is,if : BU)=allo+4), then BY) =a'(o +1) =f(@lo+1)=S(BO)- This remark allows ustoextend theuniqueness part ofTheorem 2. 3,THEOREM. Suppose f:U->R”islocally Lipschitz, thatis,around each pointthereisabal]onwhich /satisfies condition (2)ofTheorem 2forsomeK (and hence alsocondition (1)forsome L).Letx€UandJet@1,2 betwomaps onsome open interval Jwith a(),a2(I) CU and ai'() =f(a) . i=1,2. (0)=x Then oy=a onI. PROOF. Suppose «1(to)=@2(to) forsome fo€I.Ifwedefine Bilt) =o(0 +4), then thefunctions ;satisfy thesame differential equation, B;'(t) =f(Bi()), andhave thesame initial condition B;(0) =0(f0)=2(fo) €U.Hence f(t) = B2(t) forsufficiently small t,byTheorem ].Thus theset {ee Tsa(t) =a2(t)} isopen. Itisclearly alsoclosed andnon-empty, soitequals I. ‘Wenow revert tothesituation inTheorem 2.Wewill write ax(f)asa(t,x), sothat wehave amap a:(-b,b) xBg(x0) >U satisfying a(0,x) =x cGx)=fiatWee) =Le») fie, Dia(t,x) =f(a(t,x)), butwewillfrequently use3/82 ord/dt inthis 144 Chapter 5 discussion]. Thismapaiscalledalocalflowforfin(—b,b) xBa(%o). To picture this map a,thebest wecan doistodraw theimages oftheintegral :|Bra(xo) curves ay.Ify=ax(fo), then theintegral curve awith theinitial condition @x(0) =xdiffers from theintegral curve @ywith initia] condition ay(0) =y only byachange ofparameter, sothetwoimages overlap. Foreach fixed x,the mapt ++a(t,x) for—b<1<bJiesalongpartofthecurve through x.Onthe otherhand, ifwefixt,thenthemap x a(t,x) gives theresult ofpushing each xalong theintegral curve through it,foratime interval of¢.Tofocus attention onthismap, wedenote itby¢;: G(x) =at,x) [=ax()]. This map ¢;isalways continuous. Infact, thewhole flow @iscontinuous (asa function ofboth ¢andx): 4.THEOREM. If/:U—R"islocally Lipschitz, then theflow @:(—b,b) xBa(xo) >U given byTheorem 2iscontinuous, PROOF. Letusdenote themap Sdefined intheproof ofTheorem 2bySx, toindicate explicitly therole ofx.Then Ila—Syaxl] =|Sxox —Syax] =|x—yl Vector Fields andDifferential Equations 145 Recall that |Sa —SB|| <bK|ja —BI. IfS”denotes then-fold iterate ofSy,then lox—Shere|] <lex—Syoexl| +|Syax —Shay] +++ +SP —SHorxl) 1 seent- ——|x — S(1+bK+--++OK)" ix-ylsTook Ji RecallalsothatinTheorem |thefixedpointeyofSyisthelimitofStefor anya.Hence ay=Jim,Syax, soweobtain War~ay<b|x~yl OTS TIER Oh Since |jax—ay|] =sup|a(¢,x)—a(t, y)|,thiscertainly proves continuity ofa. t Ifadditional conditions areplaced upon themap /,then further smoothness conditions canbeproved fora.Infact, Iff:U+R"isC¥,thentheflowce:(—b,b) xBa(xo) >UisalsoCK. Unfortunately, thisisavery hard theorem. Aclean exposition oftheclas- sical proof isgiven inLang’s Introduction toDifferentiable Manifolds (2nd ed.), and arecently discovered proof canbefound inLang, Real andFunctional Analysis (3rded.), pp.371-379. Inorder toread thishigh-powered proof, youmust first Jearn theelements ofBanach spaces, including theHahn-Banach theorem, and then read about differential calculus inBanach spaces, including theinverse and implicit function theorems (RealandFunctional Analysis, 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). Wewilljust accept thisfact. Notice that themaps ¢;areconsequently C° iffisC*. Since themap a:(-b,b) xBa(xo) >U satisfies @(0,x) =x,wehave a:{0}xBaja(x0) >Baj2(%0) CBa(xo)- 146 Chapter 5 Continuity of«andcompactness of{0}xBz/2(xo) imply thatthere issome €>0such that @:(—€,£)XBaja(Xo)>Ba(xo)-5] So Bra(xo) Ba(xo) (Ifx€Baya(xo), then theintegral curve with initial condition xstays inBa(xo) for|t|<.] Soif|s|<e, andx€Ba/2(Xo), then thepoint a(s,x) €Ba(Xo), sowecan also define YQ) =aa(s,x)) [tl<e. This satisfies : YO=SYO) (0) =a(s,x). We have also noted that Bt) =a(s+1,x), defined for|s+¢| <e, satisfies Be) =FBO) BO) =a(s, x). Consequently, B(t) =a(t,c¢(s,x)) for|t|<e.Inother words, f|si,\tl|lst+t]<e, then a@(t,e(s,x)) =a(s +4,x). Ifwenowlet$;:Ba/a(xo) >R"be¢;(x)=a(t,x)forx€Bay2(xo), Wecan say: if|s|,|¢],|s +2] <€andx,¢(x) €Ba/2(Xo), then Ps(br(X)) =Ps44 (2). Roughly speaking, $74: =70¢s=¢so¢1. This shows, inparticular, that for Is]<€each ggisadiffeomorphism, with inverse $,~! =¢-5. Everything we have said, since itislocal, canberesaid, without requiring anymore proof, on amanifold. Vector Fields andDifferential Equations 147 5.THEOREM. Let¥bea C®vector field onM,andletp€M.Then there isanopen setVcontaining pandan€>0,such thatthere isaunique col-lectionofdiffeomorphisms $7:V>¢,(V) CMfor|t|<ewith thefollowing properties: (1)@:(-e,€) xV>M,defined by$(¢, p)=¢(p), isC?. (2)IfIsl,lel,Is+41 <2, andq,¢,(g) €V,then Gsar(q) =bs©(9). (3)Ifg€V,then Xzisthetangent vector at¢=0ofthe curve +¢7(9). The examples given previously show that wecannot expect ¢;tobedefined forallt,oronallofM.Inonecase however, thiscanbeattained. The support ofavector field Xisjust theclosure of{p€M: Xp#0}. 6.THEOREM. IfXhascompact support (inparticular, ifMiscompact), thentherearediffeomorphisms ¢;:M—Mforallt€Rwithproperties (1), (2),(3). PROOF. Cover support Xbyafinitenumber ofopensetsVj,..., Vngivenby Theorem 5with corresponding ¢1,...,& anddiffeomorphisms ¢/.Let¢= min(e1,...,€n). Notice thatbyuniqueness, ¢/(¢) =$7(¢)forg€ViNVj.So we can define ; by ifgeV; bg)={1)#gevs q ifg¢support X, Clearly :(—€,£) xM>MisC™, and gr45 =$1ods if|tI,|s1,|+5]<€, and each ¢;isadiffeomorphism. Todefine ¢,for|t|=e,write t=k(e/2)+r with &aninteger, and |r|<¢/2. Let b=ej.+++0bej20by[e/2iteratedktimes] fork>0 1| bnej2 00 ognc/2°r [P-e/2 iterated ~ktimes] fork<0. Itiseasy tocheck that thisisthedesired {¢;}. 148 Chapter 5 ‘The unique collection {¢;}given byTheorem 6,ormore precisely, themap 1+@;fromRtothegroupofalldiffeomorphisms ofM,iscalledal-parameter groupofdiffeomorphisms, andissaidtobegenerated byX.IntheJocalcaseof 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 ofX ona C® function f:M>R.Recall that de. _af(e(t))_ . GeaAFeoyo- Thus, tosaythat X,isthetangent vector at =0ofthecurve &¢(q) amounts tosaying that «S(ba(4)) —Sl DQ)=Hef=fimLOM=SO) ‘This equation willbeused 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) #0.Then there isacoordinate system (x,U) around psuch that a X=jet OPUz. PROOF. Itiseasy toseethatwecanassume M=R”(with thestandard coor- dinate system 1!,...,7", say), andp=0€R".Moreover, wecanassume that X(0)=4/80'|,. Theideaoftheproofisthatinaneighborhood of0thereisa unique integral curve through each point (0,a?, ...,a"); ifqliesontheintegral Vector Fields andDifferential L-quations 149 curve through thispoint, wewillusea?,...,.a” astheJast7—1coordinates ofg and thetime interval ittakes thecurve togettogasthefirst coordinate. To dothis,JetXgenerate ¢,andconsider themap xdefined onaneighborhood of0inR”by x(a',....4") =$10,a,...,a"). Wecompute thatfora=(a!,...,a"), a ax(gal,)=galvem _ =fimUta! +h,0?,..-,2") =S(x(@))] poh _ 1 =fimFUGa4n(0,07, 44") —SX) h>0 _ =Him{LS@n(x@)) —S@) =(XS)(x(a@)). Moreover, fori>1wecanatJeastcompute a aX«(z,)Nega,fon 1 ime 0,...,/1,...,0)) —BimFU XO,2,-5.0) ~£00)] _ 1=lim—[f(0,...,4,-..,0) —SO jim{YO h)—£(0)] =f aul,” Since X(0) =9/8t"|, byassumption, thisshows thatx.0=7isnon-singular. Hence x=x7!may beused asacoordinate system inaneighborhood of0. This isthedesired coordinate system, foritiseasy toseethat theequation Xx(9/dt') =X©x,whichwehavejustproved, isequivalent to¥=0/Ax'. The second useoftheequation oo (XP)(p)=limFALGulp)—SP) ismore comprehensive. The fact that Xfcanbedefined totally interms of thediffeomorphisms ¢j,suggests thatanaction ofXonother objects canbe 150 Chapter 5 obtained inasimilar way. Toemphasize thefundamental similarity ofthese notions, wefirst introduce the notation Lxf for Xf. WecallLyfthe(Lie) derivative offwith respect toX;itisanother function, whose value atpisdenoted variously by(Lxf)(p) =Lxf(p) =(Xf)(p) = Xp(f). Now ifwisaC™ covariant vector field, wedefine anew covariant vector field, theLiederivative ofwwith respect toX,by A (Lxw)(p)=him[a*e)(p) ~@(p))- This isthelimit ofcertain members ofM,*. Recall that ifXp,€Mp, then ($1°m)(P)(Xp) =@(b4(P)) (bieXp)- Afairlyeasydirectargument (Problem 8)shows thatthislimitalways exists, and that thenewly defined covariant vector field LywisC™,butwewillsoon compute thisvector field explicitly inacoordinate system, and these facts wil] then beobvious. IfYisanother vector field, wecandefine theLiederivative ofYwith respect tox, _1 (LxY)(p)=fim7%~Pis¥Dp). The vector field ¢j4¥ appearing here isaspecial case ofthevector field a.Y defined atthebeginning ofthechapter, for@:M->Nadiffeomorphism andY avector field onM.Thus (fixY)p=$i«(¥¢_,(p)) isobtained byevaluating Y at$47!(p)=$-n(p), andthen moving itback topby$j,. (PhaYIp J’ integral curve te bi(p) Yo_u(0) ofXthrough p $-n(P) ‘The definition ofLY canbemade toJook more closely analogous toLxf and Lywinthefollowing way,Ifa:M—>Nisadiffeomorphism andYisa Vector Fields andDifferential Equations 151 vector fieldontherangeN,thenavector field@*YonMcanbedefined by (*YJp=(a)(Yacp))- Ofcourse, a*(Y) isjust(@~'),¥. Now notice that AUAE =fim2GRY)eley ge figgl1D»—Yo)=imgTp=fiOr—8-0)1 =fim[Wo~(bea¥)p]=(Lx)(0). Nevertheless, wewillstick totheoriginal (equivalent) definition. Wenow wish tocompute LywandLyYinacoordinate system.Thecal- culationismadealoteasierbyfirstobserving 8.PROPOSITION. IfLy¥;andLyw;existfori=1,2,then ()Lx +¥2)=LyY +LyYo, (2)Lx(+@2) =Lywy+Lywp. IfLyY and Ly exist, then (3)LxfY=Xf-Y+f-Lyy, (4)Lyf-w=Xf-wot+f-Lyo. Finally, ifw(Y) denotes thefunction p++w(p)(Yp) andLywandLyYexist, then (6)Lx(@()) =(Lx @)(¥) +(Ly). PROOF. (})and(2)aretrivial, The remaining equations areallproved bythe same trick, theoneused infinding (/g)'(x). Wewilldonumber (3)here. 1 (LxJV)p=himSY)p~OiaSV)p] _ 1 =fimFUL p~PieFY4-n(0)] h—0 h _ 1 =fimL(Y» ~£(b-1(P)) PheYo-n(o)] a 1 =fim£00)5%—ha¥o_n(or] +jim(“2=oh)SinYonlp- 152 Chapter 5 Thefirstlimitisclearly{(p)-LxY(p). Inthesecondlimit,theterminbrackets approaches «£(P)~S(bx(P)) HimLAPPLRP SX. fm, =k S(P)s while aneasy argument shows that$i4Yg,(p) >Yp-% ‘Wearenow ready tocompute Lyinterms ofacoordinate system (x,U) onM.Suppose ¥=7, a/9/9x!. Wefirstcompute Lx(dx!). Recall (Prob- Jem4-1)thatiff:M—Nandyisacoordinate system onN,then i rave /) *(dy!)=\°~— dx.Say’) xgara Wecanapply thisto$,*, where yisx.Then Lx(dx')(p) =lim|[@n*)ax!(p) —ax"(p)] 11 Aa(xiogs) i F=jim5[Esme (p)~dx'(p) |. Now thecoefficient ofdx/(p) is 1 fxtoga) gr]_p1[8x4edn) a(x!0go) Ean[ad|Bala Pe) _92d i ()=al,Jim,GLE"obu)—O0Go)] {this step wil]bejustified inamoment} ai,_dai BSaa,*e)=57(P)- Tojustify (*)wenote that themap A(h,g) =x!($4(q)) isC®from RxM toR;thus97A/dhdx/ =8?A/9x/ dh,which iswhat theinterchange ofJimits amounts to. Itnow follows that p_wad | Lydx!=>Fy j=l Wecould nowuse(2)and(4)ofProposition 8tocompute Lywingeneral, but Vector Fields andDifferential Equations 153 wearereally interested incomputing LyY.Tocompute Ly(8/4x') wecould imitate thecalculations ofLydx!;butthere would beacomplication, because hxonvector fields involves onemore composition than $j"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: ; (a (a a— 7 7on = A — A —O=LySh=Ly[ax(=)|=(Lydx')(=)+dx’(ur35). ° a aaé :a dx! (Ly )=- *(xa) oxi? thus, a “aai a Using (3)weobtain a a a Ly(b/-—)=Lyd)-+bly( x(a)agigi+?x(a) re) aai 8 = —_——_ iA.deOx!SedxeSedxt Summing over jand then interchanging iandjinthesecond double sum we obtain “(064 dal)a a) a) LxY= fbos, X= f—, Ye i. *L(reaoatia Lege Y=LYge This somewhat complicated expression immediately leads toamuch simpler coordinate-free expression forLyY.Iff:M>NisaC® function, then ¥f isafunction, soXYf=X(Yf)makessense.Clearly 9 (,, oF abiaf gsOS X(Vfy=: i) = i +albi=. on)daax(dema)ueDatGxt+9FyTaxt The second partial derivatives which arise here cance] those intheexpression forY(Xf), and wefind that LxY =XY-YX, also denoted by[X,Y]. 154 Chapter 5 Often, [X,Y] (which iscalled the“bracket” ofXand Y)isjust defined as XY —YX; note that this means GYD) =XN) -—HAN- Astraightforward verification shows that [X,Y Sa)=S(PX Vp) +8(PX, Yo), sothat[X,Y]pisaderivation atp,andcantherefore beconsidered asamember ofMp. Wearenow inavery strange situation. Two vector fields Ly¥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. InChapter 3weproved aJemma which forthespecial case ofRsays thata C@function f:(-e,£) >Rwith f(0) =0canbewritten SO =tg) foraC®function g:(-e,e) >Rwith g(0) =f’(0), namely 1 a(t)=|Sst)ds. 0 This hasanimmediate generalization. 9,LEMMA. Iff:(~e,8) xM—RisC® and f(0,p)=0forallpeM, thenthereisaC®function g:(—e,e) xM>Rwith St, P)=18 P) af=. =g(0, p).ar(0P)=800,P) PROOF. Define ‘a att,p)={gel8hpds.& loOs Vector Fields andDifferential Equations 155 10.THEOREM. IfXand YareC™ vector fields, then LxY =[%,Y]. PROOF, Letf: M>Rbe C™. LetXgenerate ¢,,|t|<¢.ByLemma 9 there isafamily ofC°° functions g;onMsuch that Sod =f+18 go= Xf. Then (bheY oD) =bra(Yon oD) =Yona S©$1) =Yo_ntoyS+hn), so : _ . 1 JimFO—GiaYoD=fimFLAP) —CNG-1(P))) ~fimen)(@-n(P)) =(LxYf)(p) ~(Ygo)(P) =X,(¥f) —Vy(Xf). The equality LyY=[X,Y]=XY—YXreveals certain factsaboutLyY which areby10means obvious from thedefinition. Clearly [X,Y]=-¥,X], so [X,X]=0. Consequendy, LyY =-LyX, so LyX =0. Sinceweobviously haveLy(a¥;+bY2)=aLyY; +bLyYo,itfollows imme- diately thatZisalsolinearwithrespect toX: Lax,+bxX2¥ =aLly,Y +bLyY. Finally, astraightforward calculation proves the“Jacobi identity”: [X.1Y, 2)+[Z,X,Y] +01Z,XJ]=0. This equation iscapable oftwointerpretations interms ofLiederivatives: (a)Lx[¥,Z] =[LxY,Z]+[¥,LyxZ], (b)asoperators onC®functions, wehave Lux,y| =LxoLy—Ly0Ly(which might bewritten as[Lx, Ly]. 156 Chapter 5 Finally, note thatLyYislinearoverconstants only,notovertheC®functions F. Infact, Proposition 8,orasimple calculation using thedefinition of[X,Y], shows that [fX,g¥] =fglX,Y] +f(Xg)¥ —g(Vs)X. Thus, thebracket operation [,]isnotatensor—that is,[X,¥]p does notde- pend only onXpand Yp(which isnotsurprising—what can one dototwo vectors inavector space except take linear combinations ofthem?), butonthe vector fields Xand Y.Inparticular, even ifXp=0,itdoes notnecessarily follow that [X,Y] =0—in theformula [X,Yo SP)=Xp(VP) —Yo(XS) thefirstterm Xp(¥/f) iszero, butthesecond may notbe,forXfmay have a non-zero derivative intheYpdirection even though (X/)(p) =0. The bracket [X,Y], although notatensor, pops upinthedefinition ofprac- tically allother tensors, forreasons thatwil}become more andmore apparent, Before procecding toexamine itsgeometric interpretation, wewillendeavor tobecome more atease with theLiederivative bytaking time outtoprove directly from thedefinition ofLyYtwofactswhichareobviousfromthedefi- nition of[X,Y]. ()LyX =0. IfXgenerates gy,itcertainly suffices toshow that ($iX)p =Xpforallh. Recall that ($i+X)p =GiaX$_4(p). Now Xo_,,p) isjust thetangent vector at Xp P ent) g-n(p) time 1=—/tothecurve ¢+>¢;(p), and thus thetangent vector, attime ¢=0, tothe curve VO) =rut P)- Thus $iXg_,,(p) isthetangent vector, attime f=0,tothecurve n° V(t) =bu(Gr—n(P)) =$r(P). Butthistangent vector isjust Xp. Vector Fields andDifferential Equations 157 (2)IfX,andY,areboth 0,then LyY(p) =0. Since Xp=0,theunique integral curve ¢with c(0) =pandde/dt =X(c(t) issimply c(t)=p(anintegral curve starting atpcannever getaway; conversely, ofcourse, anintegral curve starting atsome other point cannever gettop). Then Y,=0and (PraY)p=bieYg_n(p) =PheYp=bix0=0, soLyY(p) =0. Todevelop aninterpretation of[X,Y]wefirstprove twolemmas. Il.LEMMA. Leta: M>Nbeadiffeomorphism andXavector fieldonM which generates {g;}. Then aX generates {a0,a7}. PROOF. We have (eX )q(S) =[OeXorg) (J) =Xan(fa) ~ | = = =fimFICS©a)(dula“"(@))) ~(F2@)(0"(@))) h>oh =JimHfodsoa™"a)) ~Sg)& 12.COROLLARY. Ifa: M— M,thena,X =Xifand only if¢,0w =aog, for allt. 13,LEMMA. LetXgenerate {¢,} and Ygenerate {y;}. Then [X,Y] =0if and only if$;0Ws=Ws0foralls,¢. PROOF. If$;o%s=Ws0%foralls,then$;4Y=YbyCorollary }2.Ifthis istrue forall1,then clearly LyY =0. Conversely, suppose that [X,Y] =0,sothat .| (*)O=fim7[Ya—(bie)q] forallg. Given p€M,consider thecurve c:(—€,¢) >Mygiven by C(t)=reY )p- 158 Chapter 5 Forthederivative, c’(t), ofthismap intothevector space Mpwehave ,. 1 =jim[c(t - CO=fimFlee+4)~cO] _ =fim[Germ )p~Ore¥)p] ~ 1 =fim7[dra(breYorp)~Pre¥ou(p)] _ 1=bre{yimF10b4YDet-Yoo} =¢14(0) using (*)with q=$-1(p) =0. Consequently c(t)=c(0), so¢ra¥ =Y.ByCorollary 12,¢¢0Ws=Ysog;for alls,t. Wehave already shown thatifX(p) #0,then there isacoordinate system x with X=9/dx'. IfYisanother vector field, everywhere linearly independent ofX,then wemight expect tofind acoordinate system with a a (*) Xagy Yao However, ashort calculation immediately gives theresult aoaLaa~ sothere isnohope offindingacoordinate systemsatisfying(*)unless[X,Y]=0. Theremarkable factisthatthecondition [X,Y]=0issufficient, aswellas necessary, fortheexistence ofthedesired coordinate system. 14.THEOREM. If%1,..., Xxarelinearly independent C®vector fieldsin aneighborhood ofp,and [Xa,Xg] =0for1<o,f <k,then there isa coordinate system (x,U) around psuch that a Xa=FaonU,a=l,...,k, PROOF. Asintheproof ofTheorem 7,wecan assume that M=R”,that p=0,and,byalinear change ofcoordinates, that a Xq(0)=al,@=],...,k. Vector Fields andDifferential Equations 159 IfXqgenerates {¢%}, define xby H(A") =bau(Baal(BGR(s+0,044, 50"). -))s Asintheproof ofTheorem 7,wecancompute that Xq(0)=9]gat..k (2) a0)=ae]eaTees X(aa|J= Ot"Io, aFh @=ktl...yn Thus x=x7!canbeused asacoordinate system inaneighborhood ofp=0. Moreover, justasbefore weseethat a Maa Nothing saidsofaruses thehypothesis [Xx,Xg]=0.Tomake useofit,we appeal toLemma 13;itshows that foreach abetween |andk,themap xcan also bewritten XC,5")=ha(hn(=Oy--0,08, 50")...)), andourprevious argument then shows that a =~—.%Xa=5 Wethus seethat thebracket [X,Y] measures, insome sense, theextent to which theintegral curves ofXand Ycanbeused toform the“coordinate lines” ofacoordinate system. There isamorecomplicated, moredifficult to prove, andlessimportant result, which makes thisassertion much more precise. IfXand Yaretwovector fields inaneighborhood ofp,then forsufficiently small fhwe can {I)follow theintegral curve ofX Y through pfortimeh; (2)starting from that point, follow the integral curve ofYfortime A; (3)then follow theintegral curve of¥ x backwards fortime/; ? 5 (4)then follow theintegral curve ofY backwards for time A. Y 160 Chapler 5 Ifthere happens tobeacoordinate system xwith x(p) =0and a a A Yoqr Yap then these steps take ustopoints with coordinates () (4,0,0,...,0) (2) (h,h,0,...,0) (3) (0,4,0,...,0) (4) (0,0,0,...,0), sothatthis“parallelogram” isalwaysclosed.EvenwhenXand¥are(linearly independent) vector fields with [X,Y] #0,theparallelogram is“closed up tofirstorder”. The meaning ofthisphrase [anextension oftheterminology “c= yuptofirst order at0”,which means that c/(0) =y/(0)] isthefollowing. Letc(/) bethepoint which step (4)ends upat, Ne©(h)=V-n(b-nVan(O)))-Sy Thenthecurve¢istheconstant curvepuptofirstorder, thatis, 15.PROPOSITION. c/(0) =0. PROOF. Ifwedefine a(t,h) =Ve(ba(P)) a(t, 4)=b-1(Walbn(P))) 13(0,h)=¥-1($—n (abn (P)))), then c(t) =03(1). Moreover, a) 2(0,1) =ay(t,t) () 3(0,1)=aa(t,t) Vector Fields andDifferential Equations 161 and forany C®function f:M>R, 5 2m) yyoa aC2 @) Bee=-Xfou ©) HF03)|_yroasat while afoa 0) HL220)0,1)=Xf(e(0,h)). Consequently, repeated useofthechain rulegives (f2¢)'(0) =Dif 0@3)(0,0) +Da(f©a3)(0,0) =Di(f 0a3)(0,0) +[Dif 0a2)(0,0) +Da(fea2)(0,0)] using (b) =Di(f 003)(0,0) +Di(fa2)(0,0) +[Dif o01)(0,0) +Da(f 2a1)(0,0)] using (a). Thus, (c),(d),(e),and(f)give (Ffec) =~YS(p) ~Xf(p) +YS(p) +Xf(p) =0.& Whenever wehave acurve c:(~€,£) >Mwith c(0) =pand c'(0) =0€ Mp,wecandefine anewvector c'"(0) ord?c/dt?|, by e"O)(S) =(f 2c)"(0). Asimple calculation shows, using theassumption c'(0) =0,thatthisoperator c”(0) isaderivation, c”(0) €Mp. (Amore general construction ispresented inProb- Jem17.)Itturns outthatforthecurve ¢defined previously, thebracket [X,Y]p isrelated tothis“second order” derivation, Until wegettoLiegroups itwill notbeclear how anyone ever thought ofthenext theorem. The proof, 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 Jineasly independent vector fields indimension 2. 162 Chapter 5 16.THEOREM. ¢”(0) =2[X,Y]p. PROOF. Usingthenotation oftheprevious proof,since(foc)(t)=(fos) (t,t) we have (4)(£06)"() =Dis(f°.@3)(0, 0)+2D2,1(f 003)(0,0) +D2,a(foa3)(0,0). Now () Dialf0a3)(0,0) =Di(-Yf oa3)(0,0) by(e) =YYf(P) by(@). We also have (2)2D2,1(f ©.@3)(0,0) =2D1(-Yf 003) by(e) =Di (Yf oa2)(0,0) +Da(¥f ©«2)(0,0)] by(b)andthechain rule =2XYf(p) —2D2(¥f ©a2)(0,0) by(d) =2XYf(p) —2[Di (YFo.a1)(0,0) +D2(¥f oa1)(0,0)] by(a)andthechain rule =2XYf(p)—2YYf(p)-2XYf(p) _by(c)and(f). Since (b)gives Dalf0.03)(0,8) =Di(f0@2)(S,8) +Dalf22)(5,5), we have (3)Daa(f 003)(0,0) =D1i(f ©@2)(0,0) +2D2,1(f ©a2)(0,0) +D2,2(f 2.2)(0,0) =Di(~Xf 0.a2)(0,0) +2D2(~Xf ©a2)(0,0) +D2,2(f 002)(0,0) by(d) =XXf(p) ~AD (Xf 001)(0,0) +Da(Xf 0.a1)(0,0)] +D2,2(f 0@2)(0,0) by(d)andthechainrule =XXf(p) ~2YXf(p) ~2XXf (p) +D2,2(f 0a@2)(0,0) by(c)and(f). Vector Fields andDifferential Equations 163 Finally, from Da(f 0@2)(0,5) =Di(f 001)(s,5) +Da(f o@2)(s,5) [from {a)] we have (4) Dzalfoa2)(0,0) =Di(foa)(0,0) +2D2,i(f 001)(0,0) +D2,2(f 001) (0,0) =YYf(p)+2XYf(p) +XXf(p) by(0)and(). Substituting (J)(4) in(#)yields thetheorem. ¢ 164 Chapter 5 ADDENDUM 1 DIFFERENTIAL EQUATIONS Although wehave always solved differential equations a pptoe)=Fleet,x)1 with the initial condition a(0,x) =x, wecould justaswell have required, forsome fo,that a(to,x) =x. Toprove this, onecanreplace 0byfoeverywhere intheproofofTheorem 2, orelsejustreplace @byf+ a(t—f9,x). Another omission inourtreatment ofdifferential equations ismoreglaring: thedifferential equations a’(1)=f(a(t)) donoteven include simple equations oftheform a(t) =g(t), letalone equations likea(t) =ta(¢). Ingeneral, we would Jiketosolve equations fates)=f(Le,x)) a(0,x) =x, where f:(—¢,c) xU>R”. One way todothisistoreplace f(a(t, x))by J(t,a(t, x))wherever itoccurs intheproof. There isalsoaclevertrick.Define Sf:(-¢,e-) xU>RM by 7 f(5,x) =(1,£(5,x))- Then there isaflow(@',@) =@:(-b,b) xW>RxR"with a A py88) =FAUX) (0,5, x)=(5,X). Forthefirstcomponent function &!thismeans that 951G EigeGx) = @'(0,5,x) =5; Vector Fields andDifferential Equations 165 thus @'(t,5,x) =s+, Forthesecond component &wehave C) ah 5x)=SEW,») =SE65,2),H5.x) =f(s+1,@ (1,5,x). Then B(x) =@7(,0,x) isthe desired flow with C gb) =SEB) B(O,x) =x. Ofcourse, wecould alsohave arranged forB(to,x) =x(byfirstfinding & with&(fo,5,x)=(5,x),notbyconsidering thecurve t+B(t—t0,x)). Finally, consider thespecial caseofadineardifferential equation a’) =g(t) a), where gisann Xnmatrix-valued function on(a,b). Inthiscase SEX) =BO +x. If¢isanynxn(constant) matrix, then (c-@)'(t) =c-a'(t) =g(t) c-a(t) 80¢+@isalsoasolution ofthe same differential equation. This remark allows us toprove animportant property ofJinear differential equations, distinguishing them from general differential equations a‘(f) =f(t,a(¢)), which may have solutions defined only onasmall time interval, even iff:(a,b) xR”>R” isCO. 17.PROPOSITION. Ifgisacontinuous 7xnmatrix-valued function on (a,), then thesolutions oftheequation al) =g(t) at) can allbedefined on(a,b). 166 Chapter 5 PROOF. Notice that continuity ofgimplies that f(¢,x) =g(t) -xislocally Lipschitz. Soforanyfo€(a,b) wecansolve theequation, with anygiven initial condition, inaneighborhood offo.Extend itasfaraspossible. Iftheextended solution @isnot defined for alltwith9<¢<5,Jet4;betheJeastupperbound ofthesetof2’sforwhich itisdefined. Pick 8with BO) =a(t): Bt) fort near 4 Bia) #0. Then B(¢*) #0fort*<1close enough tot).Hence there is¢with (c-B)@*) =ace"). Byuniqueness, ¢-8coincides with aontheinterval where they aredefined. {tras @may beextended past fyasc-8,acontradiction. Similarly, @must be defined foralltwitha <1 <t. Vector Fields andDifferential Equations 167 ADDENDUM 2 PARAMETER CURVES IN TWO DIMENSIONS Iff:U—Misanimmersion from anopen setUCR" into ann-dimen- sional manifold M,thecurve f+>f(@1,.-.,@7-1,4,@i41,.++,@n) iscalled a parameter curve intheidirection. Given 7vector fields X1,...,Xqdefined inaneighborhood ofp€Mandlinearly independent atp,weknow thatthere isusually noimmersion f:U—->Mwith p€f(U), whose parameter curves intheidirection aretheintegral curves oftheX;—for wemight nothave [X;,Xj]=0.However, wemighthopetofindanimmersion fforwhichthe parameter curves intheitdirection liealong theintegral curves oftheX;,but have different parameterizations. Asimple example (Problem 20)shows that even thismodest hope cannot befulfilled indimension 3. Ontheother hand, inthespecial case ofdimension 2,such animbedding can befound: 18.PROPOSITION. Let X1,X2belinearlyindependent vectorfieldsina neighborhood ofapoint pina2-dimensional manifold M. Then there is animbedding f:U>M,where UCR?isopen andp€f(U), whose i* parameter lines Jiealong theintegral curves ofXj. PROOF. Wecanassume that p=0€R?,and that X;(0) =(ei)o. Every point qinasufficiently smal} neighborhood of0isonaunique integral curve ofX;through apoint (0,x?(q))—we proved precisely thisfactinTheorem 7. Similarly, gisonaunique integral curve ofX2through apoint (x!(g),0). q °@) alg) Themapq+>(x!'(q),x?(q)) isC®,withJacobian equalto7at0(thesefacts also follow from theproof ofTheorem 7).Itsinverse, inasufficiently small neighborhood of0,istherequired diffeomorphism. ¢ 168 Chapter 5 Wecan always compose fwith amap oftheform (x,y) >(@(x), B(»)) fordiffeomorphisms aand BofR,which gives usconsiderable flexibility. If forexample, CCR?isthegraph ofamonotone function g,thenthemap ;c (x,y)FH0%,g()))takesthediagonal {(x,x)}toC.Moreover, foranyparticular parameterization ¢=(c},¢2): R+R?ofC,wecanfurther arrange thatc(t) maps (0(c(t), c(t), bycomposing with (x,y) +(c)~!(x), y).Consequently, we can state 19,PROPOSITION. Let¥;,X2 belinearly independent vector fields ina neighborhood ofapointpina2-dimensional manifold M,andlet¢bea curve inMwith c(0) =pand¢'(t) never amultiple ofX;orX2.Then there isanimbedding f:U>M,where UCR?isopen andp€f(U), whose i" parameter lines liealong theintegral curves ofX;,andforwhich f(t,1) =¢(¢). Vector Fields andDifferential Equations 169 PROBLEMS 1.@)Ifa: MN isC™, thena,: TM >TNisC™. (b)Ifw:M>Nisadiffeomorphism, andXisaC®vectorfieldonM,then aX isaC® vector field onN. (c)fa: R=Risa(t)=2°,thenthere isaC®vector fieldXYonRsuchthat aX isnotaC™ vector field. 2.Findanowhere 0vector fieldonRsuchthatallintegral curves canbedefined only onsome interval around 0. 3,Find anexample ofacomplete metric space (M,p)anda function f: M> Msuchthatp({(x), f(¥))<p(x,y)forallx,y€M,butfhasnofixedpoint. 4.Letf:(c,¢) xUxV>R"beC®, where U,V CR”areopen, and let (x0,.0) €UxV.Provethatthereisaneighborhood Wof(xo,yo)andanum- berb>Osuch thatforeach (x,y)€Wthere isaunique a=ayz,y): (-b,b) > Uwitha'(1)€Vfort€(—b,b) and a(t) =f(,att),e"() (0) =x (0) =y. Moreover, ifwewriteo(x,y)(t)=a(t,x,9),thena:(~b,b)xW>UisC®. Hint: Consider thesystem ofequations a(t) =Bt) B') =S,2(1), BO)- 5.Wesometimes have tosolve equations “depending onparameters”, a (*)Hie PX)=LbYoh YX) (0, y,x) =x, where f:(~c,¢) xVxU=R",foropen UCR” and VCR”, andweare solving foray,x): (~b,b) >Uforeach initial condition xand“parameter” y. Forexample, theequation a'(t) =pat) (0) =x, 170 Chapter 5 with solution a(t) =xe", issuch acase. (a)Define fii(-cc)xV xU+R"xR" by >SY,X)=0,f(6y,%). If(@,@) =&:(-b,b) xW>R™xR"isaflowforfinaneighborhood of (yo, 0),sothat a. a HeoYo)=LORY) &(0, y,x)=(YX), show that we can write E(t, YX) =(Vat) forsomea,andconclude that@satisfies (x). (b)Showthatequations ofthe form a (*) qth) =Sx.) @(0,x) =x canbereduced toequations oftheform (#)(and thus toequations a 97th) =Fa»), ultimately). [When oneproves thataC¥function f:U>R"hasaC¥flow a:(-b,b) xW—JU,thehardpartistoprovethatiffisC!,thenais differentiable with respect tothearguments inW,andthatifthederivative with respect tothese arguments isdenoted byD2a, then (44%) DyDya(t,x)=D2f(a(t,x))-Dra(t,x) (aresult which follows directly from theoriginal equation Dya(t,x) =f(a(t,x)) iffisC?,since DjDz=D2Dy). Since (##*) isanequation forDaofthe form (##), itfollows that Dae isdifferentiable ifD2fisC?,ie,iffisC?. Differentiability ofclassC*isthenproved similarly, byinduction] Vector Fields andDifferential Equations 171 6.(a)Consider alinear differential equation a(t) =g(t)a(t), where g:R->R,sothatwearesolving forareal-valued function «.Show that allsolutions aremultiples of a(t)=efear where fg(t)dtdenotes some function Gwith G’(t) =g(onecanobtain all positive multiples simply bychanging G).The remainder ofthisproblem inves- tigates theextent towhichsimilar results holdforasystem oflinear differential equations, (b)LetA=(aj)beannxnmatrix, andlet[A]denote themaximum ofall |aiz|. Show that 14+ BI<1A1 +141 {ABl salAl- 181. (c)Conclude that theinfinite series ofmxmmatrices Aa AB At aetna2 yfyf. expAsetal+A+s+ +a converges absolutely [inthesense thatthe(i,j) entry ofthepartial sums converge absolutely foreach (i,j)]anduniformly inanybounded set. (d)Show that exp(TAT~!) =T(expA)T7!. ()IfAB=BA,then exp(A +B)=(exp A)(exp B). Hin: Write y " (A+B)?AP BP >ae ai >ar)+Rw p=0 ‘p=0 ‘pao Po and show that [Rw] >0asN—>00. (()(exp A)(exp —A)=J,soexpAisalways invertible. (@)Themapexp,considered asamapexp:R”—R?”,isclearly differentiable (itiseven analytic). Show that exp'(0)(B) =B(=exp(0)- B). (Notice thatfor|A|,theusual norm ofA€R”,wehave|A]<|Al<71A.) 172 Chapter 5 (h)Use thelimit established inpart (g)toshow that exp’(A)(B) =exp(A) -B ifAB= BA. (i)LetA:R>R™bedifferentiable, andlet B(t) =exp(A()). IfB'(t) denotes thematrix whose entries arethederivatives oftheentries ofB, show that Bit) =AO) exp(A()), provided thatA(t)A'(t) =A'(t) A(t). (This isclearly ueifA(s)A(t) =A(t)A(s) forall5,1.) (j)Show thatthelinear differential equation a(t) =g(t) ale) has the solution , a(t)=exp(fstsyds)0 provided thatg(s)g(t) =g(t)g(s) foralls,. (This certainly happens when g(t) isaconstant matrix A,soevery system oflinear equations with constant co- efficients canbesolved explicitly—the exponential offjg(s)ds=1Acanbe found byputting AinJordan canonical form.) 7.Check thatifthecoordinate system xisx=x~!, forx:R"+M,then X=0/8x" isequivalent tox,(8/dt') =Xox. 8.(a)LetMand NbeC® manifolds. ForaC® function f: MxNR andq€N,let{(-,g) denote thefunction from MtoRdefined by pre f(p.9)- If(x,U)isacoordinate systemonM,showthatthefunction4f/4x',defined by af —9FC9)) gy(PD=aa Ps isaC™ function onMxN. (b)If6:(-€,£) xM—MisaI-parameter groupofdiffeomorphisms, show that forevery C® function {: M—R,thelimit 1lim> = fimFUL@n(P)—£7) Vector Fields andDifferential Equations 173 exists, and defines aC® function onM. (©)If$a:(-e,e) xTM—TM isdefined by Galt, v)=bee(), show that$4isC°, andconclude thatforevery C®vector field Xandcovari- antvector field wonM,thelimit —1es -ol, fim7GUn"o)(Xp) ~of‘2)) exists and defines aC® function onM. (d)Treat LyYsimilarly. 9.Give theargument toshow that $ix¥_,(p) >YpintheproofofProposi- tion 8, 10.(a)Prove that Ly(f-o) =Xf-wt f-Lryo Ly{o(¥)] =(Ly oY) +o(LxY). (b)HowwouldProposition 8havetobechanged ifwehaddefined (Ly¥)(p) as 1 im — =? fimG1@ieY po—Yo)? 11.(a)Show that o*(df(Y) =YF09). (b)Using (a),show directly from thedefinition ofLythatforY€Mp, [Lxdf(Pp) =Yo(Lx S), and conclude that Lydf=d(Lyf). The formula forLydx’,derived inthetext, isjustaspecial casederived inan unnecessarily clumsy way. Inthenext part wegetamuch simpler proof that LxY =[X,Y], using thetechnique which appeared intheproof ofProposi- tion 15. (c)Let¥and¥bevector fieldsonM,andf:M—RaC®function. If¥ generates {¢;}, define a(th)=Yo_o(f oh). 174 Chapler 5 Show that Dya(0,0) =—Xp(¥f) Daax(0, 0)=¥p(Xf). Conclude that forc(h) =a(h,h)wehave -c'(0) =Lx¥(p)(J) =1%YIN). 12.Check theJacobi identity. 13.OnR?letX,Y,Zbethevector fields aa Kary 95 a a ¥s-sp tay a a Z=y—-x—.Yon “By (a)Show that themap aX+bY+cZ+(a,b,c) €RB? isanisomorphism (from acertain setofvector fields toR?)andthat[U,V]= thecross-product oftheimages ofUand V. (b)Show thattheflowofaX+bY+cZisarotation ofR?about some axis through 0. 14. IfAisatensorfieldoftype(*)onNand¢:M+Nisadiffeomorphism, wedefine $*A onMasfollows. Ifuj,...,v% €Mp,andA1,...,Ay €Mp", then 16"A(PYQ1,- VesAds) =AOD) bed, «Pade, (GAL (GD). (a)Check that under theidentification ofavector field [orcovariant vector field)withatensor fieldoftype(?)[ortype(5)]thisagrees withourold$*Y. (b)Ifthevector field¥onMgenerates {¢,},andAisatensor fieldoftype(*) onM,wedefine ~1 (Lx A)(p) =him=[($4,"A)(p) —A(P)].ho>oh Vector Fields andDifferential Equations 175 Show that Ly(A+ B)=LyA+LyB Lx(A@ B)=(LyA)@B+A@LYB (sothat Ly(fA) =X(f)A+ fLxA), inparticular). (©)Show that LyaxA =Ly, A+Ly,A. Hint;Wealready know thatitistrueforAoftype(9),(7),(4): (a)Let c:ov)>TEV) beanycontraction (CT (01, --.5 K-15 AL Ara) =contraction of (VA) >TVs eyVrms UsVols eeVRAD ADs Apa AsABate seesArad). Show that Ly(CA) =C(LxA). (e)Noting that A(X1,..., Xg.@1,-..,«@7) canbeobtained byapplying contrac- tions repeatedly toA@X,@--- @Xp@@ @+++@a,use(d)toshowthat Ly(ACM, 0015Xk,15+++,01)) =(Lx A(X.XeOty) k PDEA yyLaXioeMeO11) i=l i FOAM KesOreyLeWis501)- ist ha) n (£)IfAbascomponents Aj'""/! inacoordinate system xand¥=)>a'd/dx', isl show that thecoordinates ofLyAaregivenby n gAiwdekm aie (Lxaypinpt=oa!ale=apfetinae ist asl j=l ton i fied +>>Aiviiergmetye’ al ist 176 Chapter 5 15.LetDbeanoperator taking theC™functions FtoF,andtheC™vector fields Vto‘V,such that D: F+¥and D: V—Varelinear over Rand DY) =f+DY+Df-Y. (a)Show that Dhasaunique extension toanoperator taking tensor fields of type(#)tothemselves, suchthat (I)Dislinear over R (2)D(A®@B)=DA@B+A@ DB (3)foranycontraction C,DC =CD. IfwetakeDf=XfandDY=LxY,thenthisunique extension isLy.(b)LetAbeatensorfieldoftype (}),sothatwecanconsider A(p) €End(Mp); thenA(X)isavectorficldforeachvectorfieldX.Showthatifwedefine Daf =0,DaX =A(X), then Dyhasaunique extension satisfying (1),(2), and (3). (c)Show that (Dse)(P) =—A(p)*(o(p)- (@)Show that Lyx ={Lx —Dyeas. Hint: Check this for functions and vector fields first. (©)IfTisoftype(2),show that B n n (DaTib=>7EA+DT)AL—OTHAG. a= a=l a=t Generalize totensorsoftype (). 16.(a)Letf:R>Rsatisfy f’(0) =0.Defineg(t)=(V1)for1>0.Show thattheright-hand derivative -A)—(0)_f"(0) /(0)=|g(t)=8(0)=. 840) poor h 2 (Use Taylor’s Theorem.) (b)Given c: R-+Mwithc/(0) =0€My,define y(t)=¢(Vi) fort>0. Show that thetangent vector ¢”(0) defined by¢”(0)(/) =(f0¢)""(0) canalso bedescribed by¢”(0) =2y'(0). Vector Fields andDifferential Equations 177 17.(a)Letf:M—Rhavepasacritical point,sothatfzp=0.Given vectors Xp,¥p€Mp,choosevectorfields¥,¥with¥,=Xpand¥,=Yp. Define 7 San(Xp, Yp)=Xp(¥P). Usingthefactthat[¥,¥]p(f) =0,showthatf.s(Xp, Yp)issymmetric, and conclude that itiswell-defined. (b)Show that n 2 n ,a a vri iS=tps—— f(Dml,uext|)DoSag ix Ip jan Ib? jet ()Therank of(8?//8x“x4 (p))isindependent ofthecoordinate system. (d)Letf:M+Nhave pasacritical point. ForXp,¥p€Mandg: N>R define _—Sae(X,Y)(g) =Xp(¥(g 0f))- Show that Sasi MpXMp>Nyipy isawell-defined bilinear map. ()Ife: R>Mhas0asacritical point, show that Cxx(0): RoxRo>Meco) takes (19,1p)tothetangent vector c”(0) defined by¢”(0)() =(fo¢)’"(0). 18.Letcbethecurve ofTheorems 15and 16.Ifxisacoordinate system around pwith x(p) =0,and “8 WY)=oa!mal; i=l axtPp show that ; ; x!(c(t)) =a‘t? +o(t), where o(t?) denotes afunction such that a 2) 72Timo(@?)/t?=0. 19.(a)IfMiscompact and0isaregular value off:M—R,then there is aneighborhood Uof0€Rsuch thatf~!(U) isdiffeomorphic tof~'(0) xU, 178 Chapter 5 byadifleomorphism ¢:f—(0) xU+f7!(U) with f(@(p,t)) =t.Hint: UseTheorem 7andapartition ofunitytoconstruct avector field¥ona neighborhood off—'(0) suchthatf.X=d/dt. (b)Moregenerally, ifMiscompact andq€Nisaregular valueoff:M— N,then there isaneighborhood Uofqanda diffeomorphism¢:f~!(q)xU> L(Y) with f(6(7,9')) =4". (€)Itfollows from (b)thatifallpoints ofNareregular values, then f~'(q1) andf~1(gz) arediffeomorphic for91,92sufficiently close.IffisontoN,does itfollow thatMisdifleomorphic tof~'(q) xN? 20.InR3,letYandZbeunitvector fields always pointing along thep-and z-axes, respectively, andletXwillbeavector field oneofwhose integral curves isthex-axis, while certain other integral curves areparabolas intheplanes y=constant, asshown inthefirstpart ofthefigure below. Using thesecond part ofthefigure, show that Proposition 18does nothold indimension 3. AP 0 CHAPTER 6 INTEGRAL MANIFOLDS PROLOGUE Amathematician’s reputation rests on Beauty isthefirsttest: there isno thenumber ofbadproofs hehasgiven. permanent place intheworld for [Pioneer work isclumsy] ugly mathematics. A.S. Besicovitch, quoted inJ.E.Littlewood, G.H.Hardy, AMathematician’s Miscellany AMathematician’s Apology ikthepreviouschapter,wehaveseenthattheintegralcurvesofavectorfieldonamanifold Mmay bedefinable only forsome small time interval, even though thevector field isC® onallofM. Wewill now vary ourquestion alittle, sothat global results canbeobtained. Instead ofavector field, sup- pose that foreach p€Mwehave a|-dimensional subspace ApCMp. The function Aiscalled a1-dimensional distribution (thiskindofdistribution has nothing whatsoever toclowith thedistributions ofanalysis, which include such thingsasthe“S-function”), ThenAisspanned byavectorfield/ocally;thatis, wecanchoose (inmany possilsle ways) avector field Xsuch that0-4Xg€Ag forallgimsome open setaround p.Wecall AaC® distribution ifsuch a vectorfieldXcanbechosentobeC®inaneighborhood ofeachpoint. Fora1-dimensional distribution thenotion ofanintegral curve makes no sense, butwedefine a(I-dimensional) submanifold NofMtobeanintegral manifold ofAifforevery p€Nwehave ix(Np) =Ap where i: N—>M_ istheinclusion map. Foragiven p€M,wecanalways find anintegral manifold NofaC® distribution Awith p€N;wejustchoose avector field Xwith 0#Yq€Ag forqinaneighborhood ofp,findanintegral curve ¢ofXwith initial condition c(0) =p,andthen forget about theparameterization of¢,bydefining Ntobe {e(t)}. This argument actually shows thatforevery p€Mthere isacoordinate system (x,U)such thatforeach fixed setofnumbers a?,...,a”, theset fg€U:x7(q) =a?,...,x"(q) =a} 179 180 Chapter 6 isanintegral manifold ofAonU,andthatthese aretheonly integral manifolds inU. This isstillalocal result, but because 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 (rather than like ||| orsomething even more complicated). Forexample, there isadistribution onthetorus whose integral manifolds alllook likethedense 1-dimensional submanifold pictured inChapter 2.Ontheother hancl, there isadistribution onthetorus which hasonecompact connected integral manifold, andallother integral manifolds non-compact. Ithappens thattheintegral manifolds ofthese twodistributions arealsotheintegral curves forcertain vector fields, butonthe Integral Manifolds 181 Mobius strip there isadistribution which isspanned byavector field only locally. enEy Cmat Weareleaving outthedetails involved infitting together these local integral manifolds because wewilleventually dothisover again inthehigher dimen- sional case. Forthemoment wewillinvestigate higher dimensional cases only locally. Ak-dimensional distribution onMisafunction p++ Ap,where ApCMp isak-dimensional subspace ofMp. Foranyp€Mthere isaneighborhood U andkvector fields X;,...,X, such that ¥1(q),...,X¢(q) areabasis forAg, foreach g€U.WecallAaC® distribution ifitispossible tochoose C® vector fields X},...,%q with thisproperty, inaneighborhood ofeach point p. A(k-dimensional) submanifold NofMiscalledanintegral manifold ofAif forevery p€Nwehave is(Np) =Ap where i: N—M_ istheinclusion map. Although thedefinitions given sofaralllook thesame asthe1-cimensional case, theresults willlook very different. Ingeneral, integral manifolds donot exisi, even locally. Asthesimplest example, consider the2-dimensional distribution AinR?for which Ap=Aq,b,) isspanned by a a aof here ax|p az|, ay|, Thus a ax|, ay|p az\, If'we identify TR? with R?xR?,then Apconsists ofall(7,5,6r)p. Thus Ap may bepictured astheplane with theequation z-c=b(x—-a). 182 Chapter 6 Thefigure below shows Apforpoints p=(a,b, 0).Theplane A(a,s,<) through (a,b,¢)isjustparallel totheonethrough (a,b,0). LELE LELELI LO ODL Ifyou canpicture thisdistribution, youcanprobably seethat ithasnointegral manifolds; aproof canbegiven asfollows. Suppose there were anintegral manifold NofAwith 0€N.The intersection ofNand {(0,y,2)} would bea curve yinthe(y,2)-plane through 0whose tangent vectors would have tole intheintersection ofA@,»,2) andthe(j’,z)-plane. The only such vectors have third component 0,soymust bethey-axis. Now consider, foreach fixed yo, theintersection NM {(x, y0,=)}. This willbeacurve intheplane {(x,yo,z)} through (0,yo,0), with alltangent vectors having slope yo,soitmust bethe line{(x,yo,¥ox)}. Our integral manifold would have tolook likethefollowing picture. Butthissubmanifold does notwork. Forexample, itstangent space at (1,0, 0)contains vectors with third component non-zero. Integral Manifolds 183 Toseeingreater detail what ishappening here, consider thesomewhat more general case where A(a,b,c) =Apis a C) a Ap=yrx-|+5z-] +[rf(a,b)+sg(a,5)]ienseR}; axl, orl, Z\, geometrically, Apistheplane with theequation ce =f(a,b)(x —a)+8(a,b)(y ~6). Asinthefirst example, theplane Aa,s,<) through (a,6,¢) will beparallel to theonethrough (a,6,0),sincefandgdepend onlyonaandb. Wenow askwhen thedistribution Ahasanintegral manifold Nthrough each point. Since Apisnever perpendicular tothe(x,»)-plane, thesubmanifold is given locally asthegraph ofafunction: N={(x,y,2) 12=a(x,y)}.SS Now thetangent space atp=(a,b, (a,b))isspanned by a a aPal+(a,b)zl; dx|, ax” az\, a da as-|+7)z|5 ys|ay”del, These tangent vectors areinApifandonly if da S(a,b)=FyoO da g(a,b)=By)- Soweneed tofind afunction @:R* +Rwith da da (*)eh ye 184 Chapter 6 Itiswell-known thatthisisnotalwayspossible. Byusingtheequality ofmixed partial derivatives, wefindanecessary condition onfandg: af _og ()ay ax” Inourprevious example, S(a,b) =b,of1,ay ag g(a,b)=0,9=, sothisnecessary condition isnotsatisfied. Itisalsowell-known that theneces- sarycondition (#*)issufficient fortheexistence ofthe function asatisfying («)in aneighborhood ofanypoint. 0.PROPOSITION. Iff/,g:R?>Rsatisfy ) af_ag (we ay ax inaneighborhood of0,and zo€R,then there isafunction @,defined ina neighborhood of0€R?,suchthat @(0,0) =zo da (x) bxoS da Fen PROOF. Wefirst define @(x,0) sothat «(0,0) ==oand () Pa) =f(x.0); a(x,0) ax ‘ > 0 namely, wedefine x a(x,0)=zo+fS(t,0)dt. 0 Integral Manifolds 185 Then, foreach x,wedefine @(x, y)sothat (2) Fyed)=BI d. iii namely, wedefine y atsy)=a(.0+ [”ex,nat 0 x - -a+f seoars fgtx.dr. 0 0 This construction does notuse(##), and always provide uswith an@satisfy- ing(2),0a/Ay =g.Weclaim thatif(«#)holds, then also8a/4x =f.Toprove this, consider, foreach fixed x,thefunction da >FeOny)—O59).x This is0fory=0by(I).Toprove that itequals 0forally,wejust have to show that itsderivative is0.But itsderivative atypis Fa af a(aa arDyan2)—ay)=(¥)(x,9)—ayy) ag, af|.=FO) ~FO” by) =0_ by(x4). & Wearenow ready tolook atessentially themost general case ofa2-dimen- sional distribution inR?: Ap=pe+3+[f(p)+ wz inseRP=ae|tSdy,PSP)+aC?)xl,°” , where f,g: R?>R.Suppose that N={(x,yz)12=a(x,y)} 186 Chapter 6 isanintegral manifold ofA,The tangent space ofNatp=(a,b,a(a,b)) is spanned, once again, by z|+ZemZl.ax|,|ox” zl, a da asr)+5-ab Z|5 ay|ay3Ip These tangent vectors areinApifandonly if F(a,b,(@,6))=(0,0), (*) g(a,b,a(a,b)) =Sa). Inorder toobtain necessary conditions fortheexistence ofsuch afunction a, weagain usetheequality ofmixed partial derivatives. Thus (*)and thechain rule imply that oa ar ar ao© (a,b)=+(a,b,0(a,b)) +2(a,b,a(a,b))»“(a,b dyin” )ay(4,b,a(a,b)) +Fo(a,b,aa.))ay) il oa ag ag aoSe (a,b)= ned ors 7day )=Gy(a,b,a(4,b))+5(a,b,a(a,b)) dy(a,6) This condition isnotvery useful, since itstillinvolves theunknown function a, butwecansubstitute from (+)toobtain a a.Leab,a(a,b)) +a(a,b,a(a,b))+g(a,b,a(a, b)) a, a, =Fela,b,a(a,b)) +E(a,b,a(a,b)) -fla,b,a(a,6)). Now wearelooking forconditions which willbesatisfied byfandgwhen there isanintegral manifold ofAshrough everypoint, which means thatforcach pair (a,b) these equations must hold nomatter what @(a, 6)is.Thus weobtain finally thenecessary condition afaf|_ag,ag . (*)dyta28oxoef Integral Manifolds 187 Jnthismore general case, thenecessary condition again turns outtobesuf- ficient. Infact, there isnoneed torestrict ourselves toequations forasingle function defined onR?;wecantreat asystem ofpartial differential equations fornfunctions onR™(i.e.,apartial differential equation forafunction from R”™ toR”). Inthefollowing theorem, wewilluse¢todenote points inR™and x forpoints inR";soforafunction f:R™xR">R*weuse ofFvforDif, a,ceforDm+if. 1,THEOREM. LetUxVCR™xR”beopen, where Uisaneighborhood of 0€R™, and letfj: UxV>R"beC™ functions, fori=1,...,. Then forevery x€V,there isatmost one function a:W-YV, defined inaneighborhood Wof0inR™,satisfying a(0) =x (*) aor a =fj(t.a(t)) forall teW. (More precisely, anytwosuch functions 7and@2,defined onW;and Wo,agree onthecomponent ofW,NW; which contains 0.)Moreover, such afunction exists(andisautomatically C®)insomeneighborhood Wifandonlyifthere isaneighborhood of(0,x) €UxVonwhich af—afi“Of pk Oipx aR ()orag+Dp -Les =0 i,f=ly..sm. kal kal PROOF. Uniqueness willbeobvious from theproof ofexistence. Necessity of theconditions (##)islefttothereader asasimple exercise, andwewillconcern ourselves withproving existence iftheseconditions dohold.Theproofwillbe likethatofProposition 0,with adifferent twist attheend. Wefirst want todefine a(1,0,...,0) sothat @(0,0,...,0) =x U a® Fp05-040)=fillsOy-50,0605..-50)). 188 Chapter 6 Todothis,weconsider theordinary differential equation By(0) =x By'(t)=fil0,...,0, Br). This equation hasaunique solution, defined for|}<1.Define @(t,0,...,0) =y(t) ith<e1. Then (I)holds forjt}<¢. Now foreach fixed f!with jf'}<e1,consider theequation B2(0) =a(t',0,...,0) Ba!(t)=falt,,0,... 40,B2(0)- ‘This hasaunique solution forsufficiently small t.Atthispoint thereader must refer back toTheorem 5-2,andverify thefollowing assertion: Ifwechoose e, sufficiently small, then for|t}<e,thesolutions oftheequations forB2with theinitial conditions £2(0) =a(t',0,...,0) willeach bedefined for|t]<e2for some £2>0,We then define a(t!,t,0,...,0) =Bolt) I)<a,It<e2. Then (0,0,0,...,0) =x 8a ' ' (2)pat31,0,...,0)= fo(t',1,0,...,0,a(¢',14,0,...,0)) i]<a, itl<e2. Weclaim thatforeach fixed ¢!with |t']<2;wealsohave, forall¢with [1]<e2, 8)O=g()= 01,05.--50)— Fillt.0,.055050009.10,.--50)). Note first that (4) g(0) =0 by(I). Wenow derive anequation forg’(f). Inthefollowing, allexpressions invoh~ ing@aretobeevaluated at(t', 1,0,...,0) andallexpressions involving f;are tobeevaluated at(f7,1,0,...,0,a(¢!,1,0,...,0)). Wehave Pa af dfdak D j=—-=- ——.80=Fran~98>axka7?” Integral Manifolds 189 and thus a(da)Af,Ch pe ,0)80>Fi(38)a?>axe? by@) apOoafaak AfCah x . = saa = 2)ort+haxkar!ar?»axe? by(2)again ah ahyie k=ntrae sos] af af a:=aaa=p>oakSk bydefinition,(3) =1 "af,=o by(#4). kal 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(t) =0forall1.So(3)istrue. Itisasimple exercise tocontinue thedefinition of@until itiseventually defined on(—€, £1)x+++x(En, €n)andsatisfies (4). Theorem |essentially solvesforustheproblem ofdeciding whichdistributions have integral manifolds. Our investigation oftheproblem sofarillustrates one basic fact about theorems indifferential geometry: Many ofthefundamental theorems ofdifferential geometry fallinto one oftwo classes. The first kind oftheorem says that ifone hasa 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, and arecalled “integrability conditions”. The second kind oftheorem justifies this terminology, byshowing that the“integrability conditions” aresuffi- cient forrecovering thenice situation, The remaining parts ofourinvestigation, inwhich wewillessentially begin anew, illustrates aneven more important factabout thetheorems ofdifferential geometry: There arealways incredibly concise andelegant ways tostate thein- tegrability conditions, and prove their sufficiency, without ever even mentioning partial derivatives. 190 Chapter 6 LOCAL THEORY Iff:M—NisaC® function, and X¥and YareC® vector fields onM and N,respectively, wesaythat ¥and Yaref-related iffap(Xp) =Yp) for each p€M.Ifg:N>RisaC™ function, then Yyy(8) =SapXo(8) =X,(ge Sf), * (Yayof=X(fe8). Conversely, ifthisistrueforallC®functions g:N—R,thenXandYare f-related. Ofcourse, agiven vector field ¥may 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. IfYisa C™ vector field onNwith Yep) €Spx(Mp), then there isaunique C® vector field ¥onMwhich isf-related toY. PROOF. Clearly wemust define Xptobetheunique element ofM,with Yy(p) =foxXp. Toprove that XisC®, weuseTheorem 2-10(2): there are coordinate systems (x,U)around p€Mand (y,V) around f(p) €Nsuch that yofox(al,...,a") =(a',...,a7,0,...,0). This iseasily seen toimply that a a Thus if ng = —ayeay where@!areC®functions, then, Fx=) ph where a!of=B'.This implies thatthefunctions f/areC®(Problem 3). The most important property off-relatedness forusisthefollowing: 3,PROPOSITION. IfX;and ¥;aref-related, fori=1,2,then[X;,X2]and 1%, Yo]aref-related. Integral Manifolds 191 PROOF. Ifg:N>Ris C®, then () Wighof=Xilgof) i=1,2. So {M, Yelgho f=(M(hrg}of —MMNghof =X(glo f)-—X2(INglo /) by(1),with greplaced byYzgand Yig, respectively .=XM(X2lg0f))—X2(M(g 0f))_by(Il) =1%, Heo f). Now consider ak-dimensional distribution A. Wewill saythat avector field Xbelongs toAifXp€Apforallp.Suppose that Nisanintegral manifold ofA,andé:N Mistheinclusion map.IfXandYaretwovector fields which belong toA,thenforallp€Nthere areunique Xp,¥p€Npsuch that Xp=inXp, Yp=leYp- Inother words, XandXarei-related, andYand¥arei-related. Proposition 2 shows that¥and¥areC®vector fields onN,andProposition 3thenshows that[¥,¥]and[X,Y] arei-related. Thus iY, ¥]p=14,YIp. Here [¥,¥], €Np;thistherefore shows that[X,¥]p €Ap.Consequently, if there isanintegral manifold ofAthrough every point p,then [X,Y]alsobelongs wd. Foramoment lookbackatthedistribution AinR?given by p= fre] +2] +ovtseoe| insereVa, I, aid earal : Thevector fields a a X=mt foix+S9 a a Yastes belong toA.Using theformula onpage 156,weseethat _(agaf,,gaf)a aria(-o+ seg he This belongs toAonlywhen theexpression inparentheses is0,which isprecisely thecondition forAtohave anintegral manifold through every point. 192 Chapter 6 Ingeneral, Aiscalled integrable if[¥,Y]belongs toAwhenever X¥andY belong toA.This condition canbechecked fairly easily: 4.PROPOSITION. If%},...,X% spanAinaneighborhood Uofp,thenAisintegrable onUifandonlyifeach[X;,Xj]isalinearcombination k 1%,Xj]=2ChXa a= forC®functions Cj}. PROOF. Suchfunctions clearlyexistifAisintegrable, since[X;,Xj]y€Ag, shich isspanned bytheXe(q). Conversely, suppose such functions exist. If¥ and Ybelong toAwecanclearly write k X= HX iz k Y=>oeiXi. ia Toprove [X,Y] belongs toA,itobviously suffices totreateach[f;Xj,gjXj] separately. Since wehave (fX,8¥] =felX.Y1 +S(Xg)Y —8(VS)X, clearly [f¥,gY] belongs toAifX,Y and [X,Y] do. Wearenow ready forthemain theorem. Itisequivalent toTheorem 1;in fact, Theorem |canbederived from it(Problem 7).Buttheproof isquite different. 5.THEOREM (THE FROBENIUS INTEGRABILITY THEOREM; FIRST VERSION). LetAbeaC® integrable k-dimensional distribution onM.Forevery p€Mthere isacoordinate system (x,U)with x(p) =0 x(U) =(-€,€) x+++x(~é,€), such thatforeach a**!,,.., a”withall[a!|<e,theset {qeU:x**N(q) =ak, .,x(q) =0"} isanintegral manifold ofA. Anyconnected integral manifold ofArestricted toUiscontained inoneof these sets. Integral Manifolds 193 PROOF. Wecanclearly assume that weareinR",with p=0.Moreover, we canassume that AgCR"oisspanned by mal a7a aL Let7:R”>R*beprojection ontothefirstkfactors. Then 4:Ag>RXis anisomorphism. Bycontinuity, x,isone-one onAgforgnear 0.Sonear 0, wecanchoose unique X19), +-+>XK(g) €Ag sothat F XiQ=—> EWhoeort.eMD=oile‘ Then thevector fields X;(onaneighborhood of0€R")and4/87’ (onR*)are m-related. ByProposition 3, Ca malXj,Xj]q=Eedees =0. But,[Xj,Xj]q¢€Agbyassumption, andz,isone-one onAg.So[X;,Xj]=0. ByTheorem 5-14, there isacoordinate system xsuch that a A Magy fsheok. Thesets(g€Usxt1(g) =ak4),..,,x"(g) =a}areclearly integral man- ifolds ofA,since their tangent spaces arespanned bythe4/x/ =X;for f=..k. IfNisaconnected integral manifold ofArestrictcd toU,with inclusion map i:N—U,consider d(x” 0/)fork+1<m <n. Foranytangent vector Xq ofNzwehave d(x" 0i)(Xq) =Xq(x™ 0)=inXy(x™) =0, since isX_ €Ag,which isspanned bythe0/Ax/|q forj=1,...,k. Thus d(x" oi)=0,which implies that x”0/isconstant ontheconnected mani- fold N.% 194 Chapter 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,andifaround every point p€Mthere isacoordinate system (x,U), with x(U) =(6,6) x+++x(—8,€), such that thecomponents ofNOU arethesetsoftheform {qeU:Fg) =a,x") =a") ail<e. Eachcomponent ofNiscalled afolium orleafofthefoliation N.Notice that twodistinct components ofNU might belong tothesame leafofthefoliation. 6.THEOREM. LetAbeaC®k-dimensional integrable distribution onM. Then Misfoliated byanintegral manifold ofA(each component iscalled a maximal integral manifold ofA). PROOF. Using Theorem 1-2,weseethat wecancover Mbyasequence of coordinate systems (x;, U;)satisfying theconditions ofTheorem 5.Forsuch a coordinate system (x,U),letuscalleachset qeUsxk) =ak¥, x(q) =a"} aslice ofU. Itispossible forasingle slice SofU;tointersect U;inmore than oneslice ofUj,asshown below. ButS9Ujhasatmost countably many components, Integral Manifolds 195 andeach component iscontained inasingle sliceofU;byTheorem 5,soSAU; iscontained inatmost countably many slices ofU;. Givenp€M,chooseacoordinate system(x9,Uo)withp€Uo,andletSo bethesliceofUpcontaining p.AsliceSofsomeU;willbecalledjoinedtop ifthere isasequence O=io,f1,.-..4=8 and corresponding slices So=SigsSits. Si,=S with Sia Sigg #9 @=0,...,0-1. Since there areatmost countably many such sequences ofslices foreach se- quence io,...,i,andonlycountablymanysuchsequences, thereareatmost countably many slices joined top.Using Problem 3-1,weseethat theunion ofallsuch slices isasubmanifold ofM.Forg#p,thecorresponding union iseither equal to,ortotally disjoint from, thefirst union. Consequently, M isfoliated bythedisjoint union ofallsuch submanifolds; thisdisjoint union is clearly anintegral manifold ofA.¢ [Ifweareallowing non-metrizable manifolds, theproof iseven easier, since wedonothave tofindacountable number ofcoordinate systems foreach leaf, andcanmerelydescribe thetopology ofthe foliation asthe smallest one which makes each slice anopen set. Inthiscase, however, thediscussion tofollow willnotbevalid—in fact, Appendix Adescribes anon-paracompact manifold which isfoliated byalower-dimensional connected submanifold.) 196 Chapter 6 Noticethatif(x,U)isacoordinate systemofthesortconsidered intheproof ofthetheorem, then infinitely many slices ofUmay belong tothesame folium. aCreeSAAS However, almost countably many slices can belong tothesame folium; otherwise thisfolium would contain anuncountable disjoint family ofopen sets. This allows ustoapply aproposition from Chapter 2. 7.THEOREM. LetMbeaC®manifold, andM,afolium ofthefolia- tiondetermined bysome distribution A.LetPbeanother C®manifold and ff:P— MaC® function with /(P) CMi. Then fisC® considered asa map into M,. PROOF. According toProposition 2-1], itsuffices toshow that fiscontinuous asamap into M). Given p€P,choose acoordinate system (x,U) around S(p) such that theslices 4g€U:xk(g)=ak, ...,x(q) =0"} areintegral manifolds ofA.Now /iscontinuous asamap into M,softakes S(P) M Mm Integral Manifolds 197 some neighborhood Wofpinto U;wecanchoose Wtobeconnected. For k+1<i <n,ifwe badxi(f(p’)) #alforanyp'€W,thenx!fwould take onaltvalues between a!andx!(f(p’)), bycontinuity. This would mean that J(W) contained points ofuncountably many slices, contradicting thefactthat SW) cM. Consequently, x/(/(p’)) =a!forallp’€W.Inother words, /(W) is contained inthesingle slice ofUwhich contains p.This makes itclear thatf iscontinuous asamap into My. % 198 Chapter 6 PROBLEMS . 1.(@)Let€=2:E>Bbeann-plane bundle, andé!=x’:E’>Ba k-plane bundle suchthatE’CE.Ifi:E!>Eistheinclusion map,andIg:B>Btheidentity map,wesaythat€’isasubbundle of€if(i,12)isa bundle map. Show that ak-dimensional distribution onMisjustasubbundle ofTM. (b)Forthecase ofC® bundles €and&’overaC®manifold M,defineaC° subbundle, andshow thatak-dimensional distribution isC®ifandonly ifitis aC® subbundle. 2.(a)Intheproof ofTheorem 1,check theassertion about choosing ¢;suffi- ciently small. (b)Supply theproofoftheuniqueness partofthetheorem. 3.(a)IntheproofofProposition 2,showthat a a(4\)©ay!Len’ (b)Complete theproofofProposition 2byshowing thatif na Y=) a'— sothat X=yBi—saxi? i= with a!of=Bi,then thefunctions BareC®. 4.Intheproof ofProposition 4,show thatthefunctions Cfactually areC*. 5.LetAi,..., Agbeintegrable distributions onM,ofdimensions dj,...,d4. Suppose that foreach p€M, My=(Ai)p ®--- ®(An)p- Show thatthere isacoordinate system (x,U)around each point, such thatAy isspanned by8/8x!,...,0/x%, etc. Integral Manifolds 199 6.Prove Theorem |from Theorem 5,byconsidering thedistribution Ain R”xR”(with coordinates #,x),defined by m9 nm a= Yi ik 5 i= a1 Sisl Ip Notice thateven when thefjdonotdepend onx,sothat theequations areof the form Ajor qiO=LO, with theintegrability conditions fh_hi or ari* wenevertheless work inR™xR”,rather than R”. This isconnected with the classical technique of“introducing new independent variables”. 7.This problem outlines another method ofproving Theorem 1,byreducing thepartial differential equations toordinary equations along lines through the origin. Asimilar technique willbevery important inChapter II.7. (a)Ifwewant a(ut) =B(u,t) forsome function B:[0,e) xW—V,show that Bmust satisfy theequation 3 mFawn=OV-Glut,Pn) i B(0,t) =x. Weknow that wecan solve such equations (we need Problem 5-5, since the equation depends onthe“parameter” ¢€R”). One hastocheck that one& canbepicked which works forallteW. (b)Show that B(u, vt)=Blu, t). (Show that both functions satisfy thesame differential equation asfunctions ofw,with thesame initial condition.) Byshrinking W,wecan consequently assume that ¢=1. (c)Conclude that ap 3pye) =u:gy(hee): 200 Chapter 6 (d)Usetheintegrability condition onftoshow that Foun andv-fy(vt,B(v,1)) satisfy thesame differential equation, asfunctions ofv.Use(c)toconclude that thetwofunctions areequal. (e)Define a(r)=B(1,1). Noting thata(t)=B(v,1), showthat@satisfies the desired equation. 8.This problem isforthose who know something about complex analysis. Let f:€xC =Cbecomplex analytic. Ifwedenote thecoordinate functions in CxCby21,22 =¥1,91.2, 2,then f=wtivsatisfies theCauchy-Riemann equations du au ax; Oy, ye i=1,2. au av Oy ax; UseTheorem |toprovethatwecansolvetheequations 30_wee,ysa'w,9),24,9)=oe FeMOY a(xY)OP(a,Y))=Dy 2 1ue=u(x,a(x,y),07(x,y)=== inaneighborhood of0€C(orofanypointzo€C),andconclude thatthe differential equation $2) =Se, 0(2)) (inwhich ‘denotes thecomplex derivative) hasasolution inaneighborhood ofzo,with anygiven initial condition $(zo) =wo. CHAPTER 7 DIFFERENTIAL FORMS W:turnourattention oncemoretotensorfields,butwewillbeconcerned with aspecial kind oftensor field, thediscussion ofwhich requires some more algebraic preliminaries. LetVbeann-dimensional vector space overR,Anelement T€T*(V) is called alternating if Tne esUjyevesUfyenerUk)=O ifw=y (iFJ). IfTisalternating, thenforanyw,..., Ux,wehave O=T(my,...,U; +Uj,...,U; FUys..., UR) STU eeVine reVigne eyUh)AT(V1s 0ViseeeyUfone esUk) FTV 2Ufyeee Vigeee Ue)HT(UayoeUpseens YseeesUe) SORT Us6Mise UseeeUR)AT(UssoeUser esUseesUk)+0, Therefore, Tisskew-symmetric: TU yee Viney UpseeUe)=HT(Uy eyUseeesVisooUh) Ofcourse, ifTisskew-symmetric, thenTisalsoalternating. [Thisisnottrue inthespecial case ofavector space over afield where I+1=0;inthiscase, skew-symmetry isthesame assymmetry, andthecondition ofbeing alternating isthestronger one.] Wewilldenote by2*(V) thesetofallalternating T¢T*(V). Itisclear that2*(V) cT*(V) isasubspaceofT*(V).Moreover, iff:V>Wisa linear transformation, then {*:T*(W) —T*(V) preserves these subspaces— 7:OF(W) >QK(V). Notice that2'(V) =T'(V) =V*,so2(V) has dimension n.Itisalsoconvenient toset2°(V) =T°(V) =R.Atthemoment itisnotclear what thedimension of2*(V) equals fork>1,butonecaseis well-known. The most familiar example ofanalternating Tisthedeterminant function det €7"(R”), considered asafunction ofthe rows ofamatrix— weshail soon seethat this function is,inacertain sense, themost general alternating function. Most discussions ofthedeterminant begin byshowing that ofanytwoalternating n-linear functions onR”,oneisamultiple ofthe 201 202 Chapter 7 other; inother words, dim2"(R”) <1.Then one proves dimQ"(R") =1 byactually constructing thenon-zero function det(itfollows, ofcourse, that dim2"(V) =1ifVisanyn-dimensional vector space). The construction ofdetisusuallybyamessy,explicitformula, whichisaspecialcaseofthe definition tofollow. LetS,denote thesetofallpermutations of{1,...,}; anelement o€Syis afunction i++o(i). If(y,..., Ug)isak-tuple (ofanyobjects) weset 0+(Ury+++sUk)=(Vorays«++Yocky)+ This definition hasabuilt-in confusion. Ontheright side, thefirstelement, forexample, istheo(1)* ofthev’sontheleftside; ifthese u’shave indices running insomeorderotherthan1,...,k, thenthefirstelement ontherightisnor necessarily that vwhose index is(1). The simplest way tofigure outsomething like o-(v3, v2.11,-..)istorename things:v3=wi,v2=w2,v)=ws,.... Thus warned, wecompute O-(D> (Vis++5UK))=F+(Uptays---s Yotky) bysetting Up) =Wi, -+-5 Up(k) =Wk, sothat O(P+ (U,.-.5UK)) =O+(Wi,..-, We) =(Wo(ayr-- +5Wotky) =(Yp(oay)s- ++»Vototky)) SinceWa=Vora. Thus (*) ©+(P+(U1,--+5Uk)=(PO)+(U1... UR). Now foranyT€7*(V) wedefine the“alternation ofT” 1 ALT=SYsna-Too, oeSy ie., 1 AlT1258) =DYsgne-T(veays---s York), oeS, where sgno is+1ifoisaneven permutation and —1ifoisodd. Differential Forms 203 1.PROPOSITION. (I)IfT€TeV), thenAl(T) €2*(V). (2)Ifw€Q*(V), thenAltw =o. (3)IfT€T*(V), thenAle(Al(T)) =Alt(T). PROOF. Lefttothereader (orseepp.78-79 ofCalculus onManifolds). 4 Wenow define, forw€2*(V) andn€2!(V), anelement wAn€QkH(V), thewedge product ofwand n,by k+0! wAn=a Alt(w@n). Thefunny coefficient isnotessential, butitmakes some things work outmore nicely, asweshall soon see. Itisclear that ()Aisbilinear: yg PEC oA(m +m) =O@AmM+OAM awAn=w Aan=a(wAn) (2)f*@An)= ftwfn. Moreover, itiseasy toseethat (3)Ais“anti-commutative”: @An=(-1)"n Aw. Inparticular, ifkisodd then wA\w=0. Finally, associativity ofAisproved inthefollowing way. 2.THEOREM. (I)IfSeT*(V) andT€T!(V) andAlt(S) =0,then Alt(S @T)=Alt(T @S)=0. 2)Al(Al(@ @n)@8)=Alt(w @7@8)=Alt(w @Alt(n @8). (3)IfweQ*(V), nEMV), OEN™(V), then _(k+l+m)! (WAN)AB=HA(NAO)=rm AM@1@8). 204 Chapter 7 PROOF. (I)Wehave (k+1)IAh(S @T)(v5,.-., Ue47) =YPsgno-(S@T)- (0+(ur... 441) OES =YEsgna-Seqys..-s Yocky) *TYoteettys«+sVoter)» 6S1 Now letGCS41 consist ofallawhich leave k+1,...,4 +/fixed. Then DYsang+S(voayy+sYo)+Totetnys+++Yoeto) oe =]YEsano’: S(vorys.York)|«Tego ets) oeSy =0. Suppose now that09¢G.Let0G ={a90': o'€G}.Then YEsgno-(S@T)o-(r,..-, 442) can =sgnoo- D>sgno’-(S @T)(o" +(9-(U,---5 Yess) by(a). oe Wehave just shown that thisis0(since go-(¥j,..., ¥44) isjust some other (k+J)-tuple ofvectors). Notice that GN.a0G =G,forif¢€GNa0G, then o@=090" forsome a’€G,so09=o(0')~! €G,acontradiction. Wecanthen continue inthisway, breaking S,4; upinto disjoint subsets, thesum over each being 0.The relation Alt(T @S)=0isproved similarly. (2)Clearly Al(AN(] @8)—7@8)=Alt(n@8)~Alt(7@8)=0, so(I)implies that 0=Alt(w@[Al(7@8)—7@6) =Alt(w @Alt(n @8)) —Alt(w @n @8); theother equality isproved similarly. (3)Wehave (k+l4+m)! (wAnAb=“ERaDimt Aleam@8) _k+ltmyi (kt)! =ead! mayAle @1@8). The other equality isproved similarly. ¢ Differentiul Forms 205 Notice that (2)juststates that Aisassociative even ifwehad omitted the factor (k+1)!/K!U! inthedefinition. Ontheother hand, thefactor 1/k! inthe definition ofAltisessential—without it,wewould nothave Alt(Alt 7)=Alt7, and thefirst equation intheproofof(2)wouldfail.[IfwehaddefinedAltjust likeAlt, butwithout thefactor 1/k!, then Acould bedefined by ee orAn= ge @n). Thismakessense,evenoverafieldoffinitecharacteristic, because eachterminthe sum Alt(@@n)(1,..., Ves)occurs k1/!times(sincewand»arealternating), and 1/k!/! canbeinterpreted asmeaning that these k!/! terms arereplaced by justone.] The factor (k+/)!/k!/! hasbeen inserted into thedefinition ofA forthefollowing reason. Ifvy,..., U,isabasis ofV,and¢),...,@n isthedual basis, then 14-40)! PIAAon=CeO ang@---@dn) =Vosgno-(h1@-+-@gn)oo. oeSn Inparticular, (Pr Av+-A bn)(V1, +++Un)=1. (Soifv),...,¥, isthestandard basis forR”,then gA--+ A@n=det.) Abasis for2*(V) cannow bedescribed. 3.THEOREM. The setofall iyNo Abig l<ii<-<ipsn isabasis for2*(V), which therefore hasdimension i)_nt k}~ ki@-ky (Inparticular, 2*(V) ={0}fork>7.) PROOF. IfweQ*(V) CT*(YV), wecanwrite o=>in...ig DiyO°@Hig. ftyoatk 206 Chapter 7 So @=Al(w) =>ai. Alt(Gi, @+@dig). Aseathe Each Alt(¢;, @«+@i)iseither0or=£(1/K!) $j,A-+-AGj,forsome jy<+++<jg,80theelements $j,A«+Agy,forjr<--><jespan2*(V). If 0= aig birAoA Gigs hhe~siy then applying both sides to(uj,5.++5 Viz) ives Giy..ig =O. 4.COROLLARY. Ifa1,...,0% €21(V), then a,...,@% arelinearly inde- pendent ifand only if OrAv Na #0. PROOF. Ifwn,...,wx arelinearly independent, thereisabasisU,...,Ug,---.Un ofVsuch that thedual basis vectors gi,...,¢ks--+.dn Satisfy ¢;=@;for 1<i<k. ThenaA--+A ayisabasis element of2*(V), soitisnot0. On theother hand, if @) =ag, ++++ aOR, then @)AW. A+ AWK=(an@2 +++ +UKWK) AW2 A+++ Nw =0. Toabbreviate formulas, itisconvenient tolet/denote atypical “multi-index” (a... sig),andlet$ydenote $j, A+++A$i,.Then every element of2*(V) is uniquely expressible as La or. i Notice thatTheorem 3implies thatevery w€2*(R") isalinear combination ofthe functions Uy (U1,...,U%) >determinant ofa kxk minor of |:]. Uk One more simple theorem isinorder, before weproceed toapply ourcon- struction tomanifolds. Differential Forms 207 5.THEOREM. Let v,..., UnbeabasisforV,letw€2"(V), andlet n wesay i=1.040 j=l Then w(wWi,..., Wn)=det(aij) -@(Ur,..-, Un). PROOF. Define n€T"(R") by (arr vs@na)y very(dts+++@nn))=o(ranessDam):jal m= Then clearly n€2"(R"), son=¢-detforsome c€R,and ©=M(Ciy--r€n) =O(U1s-+4Un). 6.COROLLARY. IfVisn-dimensional and 0#@€2"(V), then there isa unique orientation #forVsuch that [ur,....tn) = ifandonly ifw(u,...,Un) >0. With our new algebraic construction athand, weareready toapply itto vector bundles. If€=: E>Bisavector bundle, weobtain anew bundle 2(€)byreplacing eachfibre7~(p) with2*(2~1(p)). Asection wof2*(é) isafunction withw(p) €2*(x~"(p)) foreach p€B.If7isasection ofQ/(é), thenwecandefine asection wAnof2k+(E) by(wAn)(p) =@(p) An(p)€ Qkn-"(p)).Inparticular, sections of2*(7M), which arejustalternating covariant tensor fields oforder k,arecalled k-forms onM.A1-form isjust acovariant vector field. Since 2*(7M) canobviously bemade intoaC®vector bundle, wecan speak ofC™ forms; allforms will beunderstood tobeC® forms unless the contrary isexplicitly stated. Remember that covariant tensors actually map contravariantly: Iff:M—NisC®, and isak-form onN,then f*w isa k-form onM.Wecanalso define @;+@2and wAn. The following properties ofk-forms areobvious fromthecorresponding properties for2*(V): (a1+a2)An=aAN+02AN oA(m+m)=oAM+OAMm fonn=on fn=flwrn) wrn=(-Ik naw Swan =fon f'n. 208 Chapter 7 If(x,U)isacoordinate system, thenthedx!(p) areabasis forMp*, sothe dx''(p) A+++ adxtk(p) (iy<-++<ig)areabasis for2*(p). Thus every k-form wcanbewritten uniquely as = DLWiig dxAoAaxis iyensig or,ifwedenote dx! A+++Adx'*bydx!forthemulti-index J=(i1,..., 4%), w= oydx!. i The problem offinding therelationship between thew;andthefunctions w'; when 0=o dx!=So' a! I I islefttothereader (Problem 16),butwewilldoonespecial case here. 7.THEOREM. Iff:M>NisaC®function between n-manifolds, (x,U) isacoordinate system around p€M,and (y,V)acoordinate system around q=S(p) EN, then i L(gdyA.Ady")=eop-ae(*P) dx!+.dx".x PROOF. Itsuffices toshow that iSi(dyn+dy")=det(257) dx!A.»Adx".7" Now, byProblem 4-1, a a (dy) Aven dy" aon eon S*(dy!A+Ady"\(p)(3||) =dy ay" u dg =dy'(q)A---Ady"(g)Fea ial, Savio fa =dy) vee Ly" —_* — wannao LPorgy.» “a(yiof), \a|) seDaa (ag Ayiof=det(2570) ,byTheorem5, Differential Forms 209 8.COROLLARY. If(x,U) and(y,V)aretwocoordinate systems onMand gdy' A+--Ady"=hdx' A+Adx", then . ay?=g-det(—}.h=g-det (x) PROOF. Apply thetheorem with f=identity map. ¢ [This corollary shows that n-forms arethegeometric objects corresponding to the“even scalar densities” defined inProblem 4-10.] If =: E>Bisann-plane bundle, then anowhere zerosection wof2"(&) hasaspecial significance: Foreach p€B,thenon-zero w(p) €2"(x-"(p)) determines anorientation ppofz~!(p) byCorollary 6.Itiseasytoseethat thecollection oforientations {4p}satisfythe“compatability condition” setforth inChapter 3,sothat 4={fp} isanorientation of&.Inparticular, ifthere is anowhere zero n-form wonann-manifold M,then Misorientable (i.e., the bundle 7M isorientable). The converse also holds: 9.THEOREM. IfaC®manifold Misorientable, thenthereisann-form @ onMwhich isnowhere 0. PROOF. ByTheorem 2-13 and2-15, wecanchoose acover ©ofMbyacol- lection ofcoordinate systems {(x,U)},andapartition ofunity{¢y}subordinate [email protected] pzbeanorientation ofM. For each (x,U)choose ann-form wy onUsuchthatforu,...,07 €Mp,p€Uwehave wy(Uy...5Un) >0ifand only if[u,...,Un] =Hp. Now let w=¥due. UcO Then wisaC® n-form. Moreover, forevery p,ifv1,...,Un €Mpsatisfy [u1,.--5 Un]=Mp,theneach (guwy)(p)(U1,-.-,Un) 20, andstrict inequality holds foratleast oneU.Thus w(p) #0. 4 210 Chapter 7 Notice that thebundle 2"(7M) isI-dimensional. Wehave shown that ifM isorientable, then 2"(7M) hasanowhere 0section, which implies that itis trivial. Conversely, ofcourse, ifthebundle 2”(7M) istrivial, then itcertainly hasanowhere 0section, soMisorientable. (Generally, if€isak-plane bundle, then2(€)istrivialifandonlyif€isorientable, provided thatthebasespaceB is“paracompact” (every open cover hasalocally-finite refinement).] Justas2°(V) hasbeen introduced asanother name forR,a0-form onM willjust mean afunction fonM(and fAq willjust mean f-o), For every 0-form fwehave theI-form df(recall thatdf(X) =X(/)), which ina coordinate system (x,U)isgiven by "of df=>sax.if»axl“ Ifwisak-form w=>w,dx’, I then each dw, isaI-form, andwecandefine a(k+1)-form dw,thedifferential ofw,by du=Yd dx! 7 n =pMatna’. 7a=1 * Itturns outthatthisdefinition does notdepend onthecoordinate system. This canbeproved inseveral ways. The firstwayistouseabrute-force computation, comparing thecoefficients ’;intheexpression w=Yo'rdx! I with the w;. The second method isalotsneakier, Webegin byfinding some properties of dw(stilldefined with respect tothisparticular coordinate system). 10. PROPOSITION. (I)d@ +a2)=day +day. (2)Ifwisak-form, then d(wyA@2)=dayAw+(—1)kay Adon. (3)d(dw) =0.Briefly, a?=0. Differential Forms 211 PROOF. (\)isclear. Toprove (2)wefirstnote that because of(I)itsuffices to consider only wo=fax! @,=gdx’. Then aAw =fedx!Adx?and d(@ Aa)=d(fg)Adx!Adx? =gdf ndx! adx!+fdgndx'ndx! =dwAw,+(—1)* fdx!adgadx? =dwAa) +(~1)kay Ador. (3)Itclearlysuffices toconsider onlyk-forms oftheform w=fax’, Then ,af I do=Leo" Adx a=l so (SN OPS baer gpl d(dw)=X(zsotOxfdx"Ade) Inthis sum, theterms ar ITaedxpdxndx and Pyf Brdxtoetax8 dx®ndx?ndx cancel inpairs. ¢ Wenext note that these properties characterize donU. 11,PROPOSITION. Suppose a’takesk-forms onUto(k+1)-forms onU, forallk,and satisfies ()d'(@ +2) =d'w, +d'or. (2)d'(w Awp)=d'w,Aw +(Ikan Ad'or. (3)a'(d'f) =0. (4)d'f=(theold)df. Then d!=d onU. 212 Chapter 7 PROOF, Itisclearly enough toshow thatd’w=dwwhen w=fdx!. Now by(2), a'(f dx!) =d'f ndx'+fnd(dx') =dfndx'+fnad'(dx') by(4). Soitsuffices toshow that d/(dx!) =0,where dx! =dx'"n--- 0ax’ =d'x!'n.--na’x"* by(4). Wewilluseinduction onk.Assuming itfork—1wehave d'(dx") =d'(d'x" n..-n d'x'k) =d'(d'x") Ad’x!?W--Nd’xik —d'xit xd'(d'x" n-..Ad'x'*) by(2) =0-0, _by(3)andtheinductive hypothesis. 12.COROLLARY. There isaunique operator dfrom thek-forms onMto the(k+1)-forms onM,forallk,satisfying d(@ +a) =da, +dw, d(wA@)=do,Aw.+(-1)kan Ader d=0, and agreeing with theolddonfunctions. PROOF, For each coordinate system (x,U) wehave aunique dydefined. Given theform w,andp€M,pick anyUwith p€Uanddefine do(p) =dy(wlU)(p). The third way ofproving that thedefinition ofd@does notdepend onthe coordinate system istogive aninvariant definition. Differential Forms 213 13,THEOREM. Ifwisak-form onM,thenthereisaunique (k+1)-form dwonMsuchthatforeverysetofvector fields %1,..., Xe41 wehave (#)dw(%,.... X41) A+r . =EDM YON, FisKea)) i= SDDEot)aEL1(0. 0.010. HEELSCREIED.CIEEEED.C2) Isi<jsk (=B)+Ea,say) where~ over X;indicates that itisomitted. This (k+1)-form agrees with dw asdefined previously. PROOF. Theoperator whichtakes(%1,..., X41)toD1+Zpisclearlylinear over R.Moreover, itisactually linear overtheC™functions #.Infact, ifXiqis replaced byfXjp, then E)becomes LE+DED OGNoh.KissKea), iia andusing theformulas (X.Y) =f1X,Y]-Ys-X LXSY] =SIX, Y14 XS-Y, itiseasily seen that D2becomes LEa+DAIMYfoKinsMyo.RinsesRiggsoesXen) isin =Se NoigsX15-06sKins+KjXean)s io} abrief inspection then shows that 5)+£2becomes f¥i +fZ2. Theorem 4-2shows that there isaunique covariant tensor field dwsatisfy- ing(x).Itiseasy tocheck that dwisalternating, sothatitisa(k+1)-form. Tocompute dwinacoordinate system (x,U)itclearly suffices tocompute d(fdx"), Moreover, byrenumbering, wemight aswellassume w=fdx'n.ndxk, 214 Chapter 7 Fordw,asforanyform, wehave da= > dex(/Ax™,...,/Ax#+") dxAoAdxth+1, ey<< Itisclear from (#)that dw(0/0x',...,8/8x**+!) =0 unless some (0),...,@,...,@%41) isapermutation of(1,...,%). Since the@’sareincreasing, thishappens only if (H.-HkJ) F>k, inwhich case Fl a.dw(@/0x%,...,0/8x%, a/axt)=(kL, ax! so af j5 ke 1 k dw=Yenak AeAax”Adxt ek a, . =yeaxiAdx!rn.ndxk —axijek na,=vax Adx!n--Adxk,—axi ja whichisjusttheolddefinition. This isourfirstrealexample ofaninvariant definition ofanimportant tensor, and ourfirst useofTheorem 4-2. Wedonotfind dw(p)(v},..., ¥41) directly, butfirst find dw(X1,...,X%41), where X;arevector fields extending v;,and then evaluate this function atp.Bysome sort ofmagic, thisturns outtobe independent oftheextensions X,,...,X%41. Thismaynotseemtobemuchof animprovement over using acoordinate system andchecking that thedefinition isindependent ofthecoordinate system. Butwecanhardly hope foranything better. After all,although dw(X},..., ¥x41)(p) does notdepend onthevalues ofX;except atp,itdoes depend onthevalues ofwatpoints other than p— this must enter into our formula someliow. One other feature ofour definition iscommon tomost invariant definitions oftensors—the presence ofaterm involving brackets ofvarious vector fields. This term iswhat makes theoperator Differential Forms 215 linear over theC®functions, butitdisappears incomputations inacoordinate system, Intheparticular casewherewisaI-form,Theorem 13givesthefollowing formula, dw(X,Y) =X(W(¥)) —¥(w(X)) -o([X, Y]) This enables ustostate asecond version ofTheorem 6-5(The Frobenius Inte- grability Theorem) intermsofdifferential forms. Define thering2(M) tobe thedirect sum oftherings of/-forms onM,foralt/.IfAisak-dimensional distribution onM,then £(A) C&(M) willdenote thesubring generated by thesetofallforms wwith theproperty that(if@hasdegree /) o(h,...,X) =0 whenever X),...,X; belong toA. Itisclear that w,+w2€£(A) ifw,w2 €J(A), and that 7Aw€(A) if w€f(A) [thus, £(A) isanideal inthering 2(M)]. Locally, theideal £(A) isgenerated byn—kindependent I-forms w**!,...,«”. Infact,around any point p€Mwecanchoose acoordinate system (x,U)sothat a a Dr] ooae Ap. ax|axk|ta Then dx!(p) A+» dx*(p) isnon-zero onAp. Bycontinuity, thesame istrue forqsufficiently close top,which byCorol- lary4implies thatdx'(q),...,ax*(q) arelinearly independent imAg.There- fore,thereareC®functions Sfsuchthat k dx%(q) =)~fgg)dxP(q) restricted 6Bg==k+1,...,0. p=) Wecantherefore let k wo=dx—>fpdx. b=) 14,PROPOSITION (THE FROBENIUS INTEGRABILITY THEOREM; SECOND VERSION). Adistribution AonMisintegrable ifandonlyif d(£(A)) cf(A). 216 Chapter 7 PROOF. Locally wecanchoose 1-forms w',...,@" which span M,* foreach q suchthatw*+1,...,«" generate £(A). LetXj,...,%_ bethevector fields with : w!(Xj) =8). ‘ThenX},..., XxspanA.SoAisintegrable ifandonlyifthere arefunctions Cf with k inX%l=CRXpfads. sk. fa Now des(Xj,Xj)=Xi(w%(Xj)—Xj(w*(Xi)-@%(Xs,X})). For|<i,j<kand a>k,thefirst two terms ontheright vanish. So dw%(X;, Xj)=0ifandonlyifw%([X7, Xj])=0.Buteachw%([Xi, Xj])=0 ifand only ifeach [X;,Xj]belongstoA(ic.,ifAisintegrable), whileeach dw*(X;, X;)=0ifandonly ifdw €L(A). + Notice thatsince thew!Aw/ (i<j)span 2?(M,) foreach g,wecanalways write dot=Vefw! nw! i<j =067 Aw! forcertain forms 6%. i If@>k,andfo,jo<karedistinct, wehave 0=dw(Xig,Xin)=Y(OFAw)(XigsXio) i =OF,Xia)s sowecan write thecondition d(f(A)) C£(A) as dot= 08nw?, Bok Once wehave introduced acoordinate system (x,U)such thattheslices {geUzx*4q) =ak, x"(g) =a"} areintegral submanifolds ofA,theforms dx*+!,...,dx” areabasis forL(A), sowk+,...,w" must belinear combinations ofthem. Wetherefore have the following Differential Forms 217 15.COROLLARY. Ifw*t!,...,«” arelinearly independent I-forms ina neighborhood ofp€M,thenthere are1-forms 6¢(a,B>k)with do*=>6gNw? D ifandonlyiftherearefunctions /f,g° (a,B>k)with wo=> fea’. B Although Theorem 13warms theheart ofmany aninvariant lover, thecases k>1willhardly ever beused (avery significant exception occurs inthelast chapter ofVolume V).Problem 18gives another invariant definition ofdw, using induction onthedegree ofw,which ismuch simpler. The reader may reflect onthedifficulties which would beinvolved inusing thedefinition of Theorem 13toprove thefollowing important property ofd: 16.PROPOSITION. Iff:M—NisC® and wisak-form onN,then I*(dw) =d(f*w). PROOF. Forp€M,let(x,U) beacoordinate system around f(p). Wecan assume : :w=gdx" yn...nrdxik, Wewilluseinduction onk.Fork=0wehave, tracing through some defini- tions, I*(dg)(X) =dg(f.X)=[feX1(8)=X(g0f) =d(go f)(X) (and, ofcourse, f*g istobeinterpreted asgof),Assuming theformula for k—1,wehave d(ftw) =d((ftg dx!n--- dx) af*tdxik) =d(f*(g dxn---ndxtt)) aftdx' 40 since df*dx'k =dd(x'* of)=0 =f*(d(g dx"n---ndx'k)) aftdxik bytheinductive hyposthesis =f*(dg ndx" n.--n dx) nftdx'k =f*(dg ndxi A---ndx'k- ndx'k) =f*(dw). & 218 Chapter 7 One property ofdqualifies, bythecriterion oftheprevious chapter, asa basic theorem ofdifferential geometry. Therelation d?=0isjustanelegant way ofstating thatmixed partial derivatives areequal. There isanother setof terminology forstating thesamething. Aformwiscalledclosed ifdw=0 andexact ifw=dnforsome form 9.(The terminology “exact” isclassical— differential forms used tobecalled simply “differentials”; adifferential wasthen called“exact” ifitactually wasthedifferential ofsomething. Theterm“closed” isbased onananalogy with chains, which willbediscussed inthenext chapter.) Since d?=0,every exact form isclosed. Inother words, dw=0isanecessary condition forsolving w=dy.IfwisaI-form 1 o=)ojdx', ist then thecondition dw=0,i.e., Bw; _dws ax! ~Axi isnecessary forsolving w=df,ic., af ag=e Now we know from Theorem 6-1 that these conditions are also sufficient. For 2-forms thesituation ismore complicated, however. Ifwisa2-form onR3, w=Ady Adz —Bdx ndz+Cdx dy, then w=d(Pdx +Qdy+Rdz) ifand only if aOR 90 4 ay ae oP aR a ox78 aQ ap te te The necessary condition, dw=0,is aArnoBmaC0dx dy Or Differential Forms 219 Ingeneral, wearedealing with arather strange collection ofpartial differential equations (carefully selected sothatwecangetintegrability conditions). Itturns outthat these necessary conditions arealso sufficient: ifwisclosed, 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 thecase ofaclosed 1-form wonR?: a,a. w=fdx+gdy,withuv=x Weknow howtofindafunction @onailofR?withw=de,namely x yate foAawars f°ena.xo yo Ontheotherhand,thesituation isverydifferent ifwisdefined onlyonR?—{0}. Recall thatifLCR?is[0,00) x{0},then 6:R-LSR,+ZL@ defined inChapter 2,isC™; infact, (7,0): R?—L>(r:r>0}x(0,22) istheinverse ofthemap (a,b) +(acosb,asinb), whose derivative at(a,6)hasdeterminant equal toa#0.Bydeleting adifferentrayL;wecandefineadifferent function 6).Then6;=8intheregionA;and 6,=8+2z intheregion Az.Consequently d@andd@,agree ontheir common 2’A Ot1 °Ap-=)19000 220 Chapter 7 domain, sothattogether theydefine aI-form wonR?—{0}.Acomputation (Problem 20)shows that -y x ora Otay dy. The 1-form wisusually denoted by4,butthisisanabuse ofnotation, since w=d6only onR?-L. Infact,wisnotdfforanyCfunction f:R?-{0} >R. Indeed, ifw=df,then df=d0 onR*-L, sod(f-8)=0onR?—L,which implies thatf/8x =90/ax andaf/ay = a/ay andhence f=6+constant onR?—L, which isimpossible, Nevertheless, dw=0[thetworelations d(d0)=0 onR?-L d(d@;)=0 onRP-Ly clearly imply thatthisisso]. Sowisclosed, butnotexact. (Itisstillexact ina neighborhood ofanypoint ofR?—{0}.) Clearly @isalsonotexact inanysmall region containing 0.This example shows that itistheshape oftheregion, rather than itssize, that determines whether ornotaclosed formisnecessarily exact. Amanifold Miscallcd (smoothly) contractible toapointpo€Mifthereis aC® function H:Mx(0,1)> M such that H(p,l) =(p= P forpeM. H(p,0) =po Forexample, R”issmoothly contractible to0€R”;wecandefine H:R" x(0,1)>R” by H(p,t) =1p. More generally, UCR"iscontractible topo€UifUhastheproperty that Differential Forms 22] p€Uimplies po+t(p —po)€Ufor0<1<1(sucharegion Uiscalled star-shaped with respect topo). Ofcourse, many other regions arealsocontractible toapoint. Ifwethink of[0,1] asrepresenting time, then foreach timefwehaveamapp++H(p,t) ofMintoitself;attime1thisisjusttheidentity map,andattime0itisthe constant map. Wewillshow that ifMissmoothly contractible toapoint, then every closed formonMisexact. (Bytheway,thisresultandourinvestigation oftheformd@ prove theintuitively obvious factthatR?~{0}iszofcontractible toapoint; the same result holds forR”~{0},butwewillnotbeinaposition toprove this until thenext chapter.) The trick inproving ourresult istoanalyze Mx[0,1] (foranymanifold M),andpayhardly anyattention atalltoH. Fort€[0,1]wedefine ip: M>Mx(0,1) by .i(p) =(pt). WeclaimthatifwisaformonMx[0,1]withdw=0,then i\*w —ig*w isexact; 222 Chapter 7 wewillseelater (and you may trytoconvince yourself right now) that the theorem follows trivially from this. Consider firsta1-formwonMx[0,1].Wewillbeginbyworking inacoordi- nate system onMx[0,1]. There isanobvious functiontonMx[0,1](namely, theprojection zonthesecond coordinate), andif(x,U)isacoordinate system onM,while zy istheprojection onM,then (x)onm,...,x" 07M,1) isacoordinate system onUx[0,1].Wewilldenote x!omybyx,forconve- nience. Itiseasy tocheck (orshould be)that 2 n ; iet(Soenas!+fat)=Vei(-,a)dx', i=l isi where w;(-,@) denotes thefunction p++ w;(p,a). Now for@=72, 0;dx!+fdtwehave 2 n :: a0; 2; OfiG dw =[t ving dt]—>—azndt sea!ndt. wo=[termsnotinvolving df]2aoAthe xt Sodw=0implies that da; _af ar ax! Consequently, 90; @i(p;1)—@;(p,0)=fHy(etat 0 it lar=| Sw.odfpg(Pt)at, so 2 n (Spar ; 1 i(p,I)dx!- is we ci i)alo,axLevn.0v4y X(fBata) ax, Ifwe define g:M—Rby 1 8(p)=fS(p,t)dt, Differential Forms 223 then ag ‘ar (2) =— = =S(p.t)dt. @) Ho=[Shino Equations (1)and(2)show that ii*w —igtw =dg. Now although weseem tobeusing acoordinate system, thefunction f,and hencegalso,isreallyindependent ofthe coordinate system. Notice that for thetangent space ofMx[0,1] wehave («) (Mx(0,Dip =kere @kere. kerzm» Mx(0,1)oe aker, Ie Ifavector space Visadirect sum V=V;@V2oftwosubspaces, then any w€2(V) canbewritten O=01 +02 where (0) +02)=w(0}) @2(v) +V2)=w(v2). Applying thistothedecomposition (#),wewritetheI-form@onMx(0,1]as @+@9;there isthen aunique fwith w2=fdt. Ingeneral, forak-forma,itiseasytosee(Problem 22)thatwecanwrite© uniquely as @=, +(dtan) where @1(v},...,0,) =0ifsome v;€kermyx, and 7isa(k—1)-form with the 224 Chapter 7 analogous property. Define a(k—1)-form JwonMasfollows: 1 Ha(pyory-esb =fMPNndiyoosrade-s)at0 Weclaim that dw=0implies that iw —io*w =d(Jw). Actually, itiseasier tofind aformula fori;*@ —ip*w that holds even when dw#0. 17,THEOREM. Foranyk-form wonMx[0,1]wehave iy*w —iptw =d(Iw) +I(dw). (Consequently, ij*w —iptw =d(Iw) ifdw=0.) PROOF, Since Iwisalready invariantly defined, wecanjust aswell work in acoordinate system(%!,...,5",1). Theoperator Jisclearlylinear,sowejust have toconsider two cases. (I)w=fdk"n---nax* =fdx!. Then a dw=__.+Lanas; itiseasy toseethat ba, Hdw)(p)=(fZoe) ax!(p) 0 =[f(r1) ~f(p,0)dx"(p) =i)*w(p) —io*w(p). Since Jw=0,thisproves theresult inthiscase. (2)@=fidt nds rn... adsik-! =fdt \d&!, Then is*w =ipto =0. Now H(dw)(p)=(-osLand&*nd"\(p) om)axe “re ar=- ai a tSD(fpale.dt) dx”ndxa=l and 1 dw)=d(ffp.nat) dx! 0 .a ; cs 7 -Le(ffo.nat) dx*ndx', Clearly I(dw) +d(Iw) =0. Differential Forms 225 18.COROLLARY. IfMissmoothly contractible toapoint po€M,then everyClosed formwonMisexact. PROOF, WearegivenH:Mx[0,1]>Mwith A(p,1) =(D=P alpeM.H(p,0) =po Thus Hoi,;: M—Mis theidentity Hoig: M—M_ isthe constant map po. So @=(H0i;)*(w) =i*(H*) 0=(A0ip)*(w) =ig"(H*w). But d(H*w) =H*(dw) =0, so. @—0=1)"(H*w) —ip”(H*w) =d((H*w)) bytheTheorem. Corollary 18iscalled thePoincaré Lemma bymost geometers, while d?=0 iscalled thePoincaré Lemma bysome (Idon’t even know whether Poincaré had anything todowith it.)Inthecase ofastar-shaped opensubsetUofR”,where wehave anexplicit formula forH,wecanfind(Problem 23)anexplicit formula forI(H*w), forevery form wonU.Since thenewform isgiven byanintegral, wecansolvethesystem ofpartial differential equations w=dyexplicitly intermsofintegrals, There areclassical theorems about vector fields inR?which canbederived from thePoincaré Lemma and itsconverse (Problem 27),and originally dwas introduced inorder toobtain auniform generalization ofall these results. Even though thePoincaré Lemma anditsconverse fitvery nicely into ourpattern forbasic theorems about differential geometry, ithasalways been something ofamystery tomejust why dturns outtobesoimportant. Ananswer tothisquestion isprovided byatheorem ofPalais, Natural Operations onDifferential Forms, Trans, Amer. Math. Soe. 92(1959), 125~141. Suppose we have anyoperator Dfrom k-forms to/-forms, such thatthefollowing diagram 226 Chapter 7 commutes foreveryC®mapf:M—N[itactually suffices toassume that thediagram commutes only fordiffeomorphisms /]. *k-forms onM£7 k-forms onN >| |e ft i-forms onM+——— /-forms onM Palais’ theorem says that, with fewexceptions, D=0.Roughly, these excep- tional cases arethefollowing. Ifk=/, then Dcanbeamultiple oftheidentity map, butnothing else. If/=k+1,then Dcanonly besome multiple ofd. (Asa corollary, d?=0,since d?makes theabove diagram commute!) There is onlyoneothercasewhere anon-zero Dexists—when kisthedimension ofM and/=0.Inthiscase, Dcanbeamultiple of“integration”, which wediscuss inthenext chapter. Differential Forms 227 PROBLEMS 1.Show that ifwedefine 4(Vy,0665Vk)=(Ugmt(1ys-- +5YomI(hy)s then 0+pe(dy,..., 0%)=Opo(d},-..,VK). 2.LetAltbeAltwithout thefactor 1/k!, anddefine wn =Alt(w @n).Show that7isnotassociative. (Tryw,n€21(V) and6€27(V).,) 3.LetS’CSgyy bethesubgroup ofallowhich leave both sets{1,...,k} and {k-+1,...,k 41}invariant, Acrosssection ofS’isasubset KCSky containing exactly oneelement from each leftcoset ofS’. (a)Show that foranycross section Kwehave WAN, -.-5URI) =>SENT+OBN(Vo(ys+++Vork4l))- oeK This definition may beused even inafield offinite characteristic. (b)Show fromthisdefinition thatwA7isalternating, andwAn=(—1)"’n aw. (Proving associativity isquite messy.) (c)Apermutation o€Sxyy iscalled ashuffle permutation ifo(1) <(2) <+++ < o(k) and o(k +1)<o(k +2) <+-- <o(k 4). Show that thesetofallshuffle permutations isacross section ofS’. 4.ForveVandw€2*(V), wedefine thecontraction vw €2*-!(V) by (VA@)(U1,. 66,Vent)=OY,V1,0,Vea). Thisissometime alsocalledtheinnerproduct andthenotation iywisalsoused. (a)Show that vi(wito) =—wi(via). (b)Show thatifvj,..., 0,isabasisofVwithdualbasis1,...,¢n, then 0 j#any igYjI(bi,A=Abig)= = pee 4bin i){(=I)!bi,AoAbigAoAGJStee (c)Show thatfor@€2*(V) andw2€2!(V) wehave vA(@) Aw) =(v1}) Aw?+(—I)kay A(v4.2). (Use (b)andFinearity ofeverything) 228 Chapter 7 (d)Formula (c)canbeused togive adefinition of@;Aw2 byinduction onk+/ (which works forvector spaces over anyfield): IfAisdefined forforms ofdegree adding upto<k+/, wedefine WA 02(Yj,...5 Vest) =[(¥1101) Awe)(v2,..65Ve4s) +(=IferA(v)1@2)](v2, Ve)» Show thatwith thisdefinition @Aw2isskew-symmetric (itisonly necessary to checkthatinterchanging v;andv2changes thesignofthe right side). (c)Prove byinduction thatAisbilinear andthatw)A@=(~1)w, A. (9If¥isavector field onMandwak-form onMwedefine a(k~1)-form X10 by (X¥4w)(p) =X(p)1o(p). Show that if@;isak-form, then XA(wA@2)=(X¥401)002+(—1koy A(X402). 5.Show that nfunctions fj,..., fn:4—Rformacoordinate system ina neighborhood ofp€MifandonlyifdfiA---Adfn(p) #0. 6.Anelement w€2*(V) iscalled decomposable ifw=¢A--+Agx forsome $iEV =QV), (a)IfdimV <3,then every w€2(V) isdecomposable. (b)If$j,¢=1,...,4 areindependent, then w=(giAd2)+(3Aga)isnot decomposable. Hint: Look atwAw. 7.Foranyw€2*(V), wedefine theannihilator ofwtobe Ann(w) ={6€V*: 6Aw =0}. (a)Show that dim Ann(w) <k, and that equality holds ifand only ifwisdecomposable. (b)Every subspace ofV*isAnn(w) forsome decomposable w,which isunique uptoamultiplicative constant. (c)If@and w2aredecomposable, then Ann(w) CAnn(w2) ifand only if @2=@Anforsome7 (d)Ifw;aredecomposable, then Ann(w1) QAnn(w2) ={0}ifand only if @;A@2#0.Inthiscase, Ann(0}) +Ann(w2) =Ann(@ Awr). (e)IfVhasdimension n,then anyw€2"—1(V) isdecomposable. (0)Since v;€Vcanberegarded aselements ofV**, wecanconsider vjA--+A vy,€2K(V*), Reformulate parts(a)-(d)intermsofthis Aproduct. Differential Forms 229 8.(a)Letw€27(V). Show thatthere isabasis ¢y,...,¢n ofV*such that @=($1Aba) +-+*+(harAdar). Hint: If w=Payhinvi, i=} choose ¢involving 1,¥3,...,¥n and¢2involving ¥2,..., UnSothat w= Agrto’, where w’does not involve ¥or¥2. (b)Show thatther-fold wedge product wA+++ Awisnon-zero anddecompos- able, andthatthe(r+1)-fold wedge product is0.Thus riswell-determined; itiscalled the rank ofw. (0If@=DjejaiiAVy,showthattherankofwistherankofthema- trix(aij). 9.Ifv1.2.5 UnisabasisforVandw;=D4 ajrvj, show that det(ajj)w*, A--- Aw,=UA Av n 10.LetA=(ajj) beannxn matrix. Let|<p<nbefixed, andletg=n—p. ForH=hy<-++<hyandK=ky<+++<kg,let Giyhy ees yh Aptiky +++ Aptijkg BYadel: :|,CK=det ; 5 : phy +++ Apyhip Gnky vee Anke (a)Ifv,...,UnisabasisofVand ” w=ajiny, j=l show that WA AWy=>BYvy H Wppi AvAWn=Vekox. K 230 Chapter 7 (b)LetH’={1,...,2}— H(arranged inincreasing order). Show that 0 K#¢H AUK =PHU VoywitAvAyK=H, where ey,#7isthesign ofthepermutation Ayshayeeshpskiseeeskg)™ (c)Prove “Laplace’s expansion” detA=)enn BUCH. a 11.(Cartan’s Lemma) Letg1,...4% €V*beindependent andsuppose that Visser Ve€V*satisly (brAWa)++++(keAVe)=0. Then k Vi=Dajigj, where aj=aij. jal 12.Inaddition toforms, wecanconsider sections ofbundles constructed from TM using Qandother operations. Forexample, if§=: E>Bisavector bundle, wecanconsider 2*(é*), thebundle whose fibreatpis2*([x~'(p)]*). Since wecanregard Beasanelementof(M,)™, anysection of2"(7*M) canbewritten locally as a a hagt 8aga (a)Show that if ugAoAughuNruBayt 8Gye=aT Se then , ay!V7 h=e[a(35)]. Differential Forms 231 This shows thatsections of2"(7*M) arethegeometric objects corresponding tothe(even) relative scalars ofweight —1inProblem 4-10. (b)Let7;*"1(V) denotethevectorspaceofall multilinear functions Vx XVxXVExxVEOMY).QCEAROOES LESAPESLAS ‘Atimes times Showthatsections of7;“"(7M) correspond to(even) relative tensors oftype (4)andweight 1.(Notice thatifvj,...,unisabasisforV,thenelements of 2"(V) canberepresented byrealnumbers [times theelement v*)A---Av*n].) (0)IfTky(V) isdefined similarly, except that2”(V) isreplaced by2”(V*), showthatsections ofJ;f\(TM) correspond to(even)relative tensors oftype(7) and weight —1. (4)Show thatthecovariant relative tensor oftype(°)andweight |defined in Problem 4-10, with components e!'-~", corresponds tothemap V*xeeexVESQV)RARRIOCORLE, ntimes. given by(¢1,..-,¢n) >o1A-+-A@n. Interpret therelative tensor with com- ponents &j,...;, similarly. (c)Suppose 2""(V) denotes allfunctions 9:Vx«-»xV—Rwhich areof the form N(1,--+5 Ux)=[W(V1,--.,¥n)]” waninteger forsomew€2"(V). LeeT4"™)(V) bedefined like7;”), except that2"(V) is replaced by2""”(V). Showthatsections of7;‘""}(7M) correspond to(even) relative tensors oftype(7)andweight w,Similarly forTif): (()Forthose who know about tensor products V@Wand exterior algebras A(V), theseresults canallberestated. Wecanidentify 7;{(V) with k U @VeQ@V=Vte--eV'eVe--eV.8 BUONO OY‘&times Ttimes Since 2"(V) =A™(V*) =[A™(V)]*, wecanidentify k U TV) with RQvre@ veay) k U Tiim(V) with &)V*@®@ Vea’. 232 Chapter 7 Consider, more generally, . k U woem) =@v*@@V@@A™YV) . k t wThaw”)=@V°@@Va@"anv. Noting thatA"(V)@---@A”(V) isalways 1-dimensional, show thatsections of THYTM)andTif,y(TM) correspond to(even)relativetensorsoftype(7)andweight wand —w, respectively. 13.(a)IfVhasdimension nandA:V-Visalineartransformation, then themap A*: 2"(V) +2"(V) must bemultiplication bysome constant c. Show that ¢=detA.(This may beused asadefinition ofdetA.) (b)Conclude that det AB=(det A)(det B). 14.Recall thatthecharacteristic polynomial ofA:V>Vis x(A) =det(Al —A) =N"—(trace AAT! +--+ 4(-1)" detA SAM A"m! 4cg? $+ (=On. (a)Show thatcz=trace ofA*:Qk(V) >QK(V). (b)Conclude thatcx(AB) =cx(BA). (©)Let8/'"7*beasdefinedinProblem 4-5(xiii).IfA:V+Vhasama- trix(a/)(withrespect tosomebasis), showthat 1 ;, COA=BLE aie a *dnessadpe Jivondk Thus, if5isasdefinedonpage130,andAisatensoroftype(1),thenthe function p+>ce(A(p)) canbedefined asa(2k)-fold contraction of A@---@ AS.CO Se&times 15.LetP(X;;) beapolynomial inn?variables, Forevery nxnmatrix A=(aij) wethen have anumber P(aj;). Call Pinvariant ifP(A) =P(BAB™?) for allAandallinvertible B.This problem outlines aproof thatanyinvariant P isapolynomial inthepolynomials c},...,¢n defined inProblem 14.Wewill Differential Forms 233 need thealgebraic result thatanysymmetric polynomial Q()1,...,)n) inthen variables y),...,, canbewritten asapolynomial inoj,...,0n, where 9;is thei"elementary symmetric polynomial ofyj,..., Yn.Recall thattheo;can bedefined bytheequation n T]0-90=9"-ory +--+ (-1)"on. ist Thus, they arethecoefficients, uptosign, ofthepolynomial with roots y1,..., Yn.Since theeigenvalues 41,...,4n ofamatrixAare,bydefinition,theroots ofthepolynomial x(A), itfollows that ci(A) =oj(Aas... An). ‘WewillfirstconsidermatricesAoverthecomplexnumbersC(thecoefficients ofPmay alsobecomplex). (a)Define Q()1,-.-,¥n) tobeP(A) where Aisthediagonal matrix Oy Then there isapolynomial &such that Qe Pn)=ROLVas Vande OnVisseesPn)» The polynomial &hasrealcoefficients ifPdoes. (b)P(A) =R(e1(A),...,€n(A)) foralldiagonalizable A. (c)Thediscriminant D(A) isdefined asJ],.j;(4:—4y)®, where A;arethe eigenvalues ofA.Show that D(A) canbewritten asapolynomial intheentries ofA. (d)Show that P(A) =R(¢,(A),...;¢n(A)) whenever D(A) #0.Conclude, by continuity, that theequation holds forallmatrices Aover C.(This lastconclu- sion follows even ifCisreplaced bysome other field, since thesetwhere D#0 isZariski-dense; thisis“the principal ofirrelevance ofalgebraic inequalities”, compare pg,V.375.) Now suppose thatthecoefficients ofParerealandthatP(A) =P(BAB™!) forallreal Aand reat invertible B. (c)Thesame equation holds forcomplex Aandcomplex invertible B.(Regard theequation asn?polynomial equations intheaj;andbj.) 234 Chapter 7 16.(a)Letvj,..., vpbeabasisforV,andletwi,..., wz€Vbegivenby 1 wy=Sayin; jal Forw€2*(V) show that @(Wi1,..., We)=>G7O(Vi,,+.5Vig), Imi <cig where aisthedeterminant ofthekxksubmatrix of(«;) obtained byselecting rows i},...5ik. (b)Generalize Theorem 7andCorollary 8tok-forms. (c)Check directly from (b)that thedefinition ofddoes notdepend onthe coordinate system. 17.Show thatd(Q)j<; aijdx!adx/) =0ifandonlyif Ooi;—Baik|DayK -oGak-a07+a=0foralli< j<k. 18. InProblem 5-14 wedefined LyAforanytensorfieldA. (a)Show thatifwisak-form, then soisLyw. (b) Show that Lx(wiAa) =LywyAw,+0ALywr. (0)Using 5-14(e), show that X(O(M,.--» Xe)=Lx(W(%,..-» Xk) =Lyo(%,..., Xe) k +PEDO, KA,Xs.KisXe) ist (d)Deduce thefollowing twoexpressions: do(X,..., Xe41) k+I . =Ven Ly0%... Ki.Kea) int +RI HX, Xp],XtoeKiseRyeeyMet) iy Differential Forms 235 dw(X,..., Xai) 1A+ =FCDM ON, KissMega) int $Lx,(May KinsMeta} (ce)Show that XAdw =Lyw —d(X 10), ie, do(M,...,Xksi) =(Lx, @)(X2, «++1X41) —d(Mi 1)(X2,..., Xe) (This may beused togive aninductive definition ofd.) (()Using(¢),showthatd(Lx#)=Ly(dw). 19.Letajjben?functions onR”withaij=aj.Show thatinorder forthere tobefunctions #1,...,¥, inaneighborhood ofany point inR”with _1(aur,au =3\aa*ax! itisnecessary and sufficient that aay Paix Pay Parea for alli, j,k1.axkaxl~Axlax!=Gxkaxt Oxfaxt OANAAka! Hint: Firstsetuppartial differential equations forthefunctions fj,=du,/Ox*— uz/Ax/, anduseTheorem 6-1. 20.Compute that «dy—yd. gor=X=Paxx+y (Atmost places 6=arctan y/x [+a constant]. 21.(a)Ifwisa1-form fdxon[0,1]with (0) =f(1), show that there isa unique number Asuchthatw—’dx=dgforsomefunction gwithg(0)=g(1). Hint: Integrate theequation w—Adx=dgon[0,1] tofind A. (b)Leti:S!—R?—{0} betheinclusion, andlet0=i*(d6). If¢:[0,1]>S! is c(x) =(cos2zx, sin2x), show that c*(o') =Qndx. (0)Ifwisaclosed 1-form onS!show that there isaunique number Asuch that w—Ao’ isexact. 236 Chapter 7 22.(a)Show thatevery w€2*(Yj @V2)canbewritten asasumofforms @;A@2where w;hasdegree @andw2hasdegree B=k—0and @1(0},...,Uz) =0ifsome v;€V2 @2(Y4,...,0g)=0 ifsome v;€Yi. (b)Ifdim¥=1,and0#4 €V4",then wcanbewritten uniquely as@;+ (wzAA), where @isak-form and w»isa(k—1)-form such that @(Uj,...,Uk) =0 ifsome v;€V2 @2(Yj,..-,Uk-1) =0ifsome y;€V2. 23,LetUc R"bean open setstar-shaped with respect to0,anddefine H:Ux [0,1] >UbyH(p,1) =tp.If = DYwhy dxAndxk ewnix onU,show that I(H*w) ze 1 .-— =vyeoe'(f Vaalts)a)x!@dxl'n...ndxlan...ndx'k, icp @=1 0 24.(a)LetUCR*beabounded open setsuch thatR?-U isconnected. Show thatUisdiffeomorphic toR?,andhence smoothly contractible toapoint. (The converse isproved inProblem 8-9.) Hint: Obtain Uasanincreasing union of sets,thek*setbeingafiniteunionofsquares containing thesetofpointsinU whose distance fromboundary Uis<1/k. \ COCEEC ECCCelecere eer EEREEERECEP asCHCOA Seagscnbessneyaceennneny NELOpETEBRTCEEEEEEEll BCESBAE anSSRNROyBal CN ee EESCEEECEES EE PCOSSC CeCe seeerCOCR (b)Find abounded open setUCR?such thatR?-Uisconnected, butUis notcontractible toapoint. Differential Forms 237 25.LetUcR"beanopen setstar-shaped with respect to0.IsUhomeo- morphic toR"? (Itwould certainly appear so,butthe“obvious” proof does notwork, sincethelength ofrays from 0totheboundary ofthesetcould vary discontinuously.) 26.Let{,)betheusual inner product onR”, (a,b)=Yate! ist (a)Ifv4,...,Un-1 €R”,show that there isaunique vector vjx+++XUn—1 €R” with w vy (uyX+++XUpna,W)-u(:forallweR”. Una (b)Show that x--.x€2"71(R"), andexpress itintermsofthe e*;,using the expansion ofamatrixbyminors. (©)ForR?show that vxw=(vw —vw, vw! —vw, ow? —vw). (First findalle;xe;,) 27.{a)Iff:R"—R,define avector field gradf,thegradientof/,onR” by n naf a a Introducing theformal symbolism “a v=»Di5e ia 238 Chapler 7 wecanwrite grad f=Vf.If(grad f)(p) =wp,show that Dyf(p) =(vw), where Dyf(p) denotes thedirectional derivative inthedirection vatp(or simply v,(f), ifweregard vp€R",). Conclude that V/(p) isthedirection in which fischanging fastest atp. (b)If¥=Dt, a!/ax! isavector field onR",wedefine thedivergence ofXas no. aa! dv¥= yo.ivX =>ial i=l (Symbolically, wecanwrite div¥=(V,X).) Wealsodefine, forn=3, curl ¥(=VxX) =(32_dat)a|(aa!_aa?)9|(aa?dal)a =Vani~and)axt7aad7ant)OtTat7ot) Define forms wy=a! dx+a"dy+a°dz my=aldyndz+a?dzadxt+a°dxady. Show that df=Wgrad (my) =Neux d(nx) =(div X)dxAdy Adz. (c)Conclude that curl gradf=0 div curl X=0. (@)IfXisavector field onastar-shaped open setUCR"and curl¥=0, then X=grad fforsome function f:U>R.Similarly, ifdiv¥=0,then X=curl¥forsomevectorfield¥onU. CHAPTER 8 INTEGRATION ahbasicconceptofthischaptergeneralizes lineandsurfaceintegrals, which first arose from very physical considerations. Suppose, forexample, that¢:[0,1] >R?isacurve andw=fdx+gdyisa1-form onR?(where tig:R?>R,andxandydenote thecoordinate functions onR?).Ifwe choose apartition 0=fo<-++<fq=1of[0,1], then wecandivide thecurve c intonpieces, thei"*piece going from c(tj-1) toc(t). When thedifferences t;~tj.aresmall, each such piece isapproximately astraight segment, with e() et) CE)A|<(4)—C41) (0)Ci) es!h(n) —eG) horizontal projection c!(#;) —¢!(4—-1) andvertical projection ¢?(4;) —¢?(t-1). Wecanchoose points ¢(&;) oneach piece bychoosing points &;€[f-1,4]- For each partition Pand each such choice &=(£1,...,), consider thesum " S(P,E) =>Sees) ~aN) +BCE) LOU) —71D). i= Ifthesesumsapproach alimitasthe“mesh” ||P]ofPapproaches 0,thatis, asthemaximum of4;~f;~; approaches 0,then thelimit isdenoted by [fdx+gdy.lc (This isacomplicated limit. Tobeprecise, if||P|=max(i; —4-1}, then the equation ‘ lim_S(P,&) = dx dy m5 >)fsx+gdy 239 240 Chapter8 means: forall¢>0,there isa5>0such thatforallpartitions Pwith ||P|]<6. we have sr~[pax+20]<é c forallchoices &forP.) Thelimitwhich wehavejustdefined iscalled a“lineintegral”; ithasanatural physical interpretation. Ifweconsider a“force field” onR?,described bythe vector field tut Ug™feo then S(P,)isthe“work”involvedinmovingaunitmassalongthecurve¢in thecasewherecisactually astraight linebetween s;_;and4andfandgare constant along these straight linesegments; thelimit isthenatural definition ofthework done inthegeneral case. (Inclassical terminology, thedifferential fdx+gdywould bedescriloed asthework done bytheforce field onan“in- finitely smal)” displacement with components dx,dy;theintegral isthe“sum” ofthese infinitely small displacements.) Before worrying about how tocompute thislimit, consider thespecial case where yo et) =@,yo). H Inthiscase, ¢'(t;) —c'(ti-1) =t—1, while €2(t;) ~c7(4-1) =0,so a SCPE=DoSi,YON~G1).isl Integration 241 These sums approach 1 [se+edy-[SPlx,yo)dx. F 0 Ontheother hand, if Yor #———* e(t) =b+ (1—1)a, yo), a b then el(ts)~cl(ti-1) =(@~a) —4-1), 80 1 S(P,) =(b~a)- S>Gib+(1~a,Yo)ti~ti-1). iat These sums approach 1 b o-ofS(xb+(1—x)a,yo)dx~fSe,yo)dx. 0 a Ingeneral, foranycurve c,wehave, bythemean value theorem, Ma) —Ga) =u -tr) ar€[r=4] (4) =ia) =(BMG 1) Bi€[fetta]. So n S(P.E)=SO{SleEyeaa) +aleGde"(B)} ~4-1). isl Asomewhat messy argument (Problem 1)shows thatthese sums approach what itlooks likethey should approach, namely 1freare"o +ecw" 4. cy Physicists’ notation (orabuse thereof) makes iteasy toremember thisresult. The components c!,c? of¢aredenoted simply byxandy[i.e., xdenotes 242 Chapter 8 xoe and ydenotes yo¢;thisisindicated classically bysaying “letx=x(¢), y=y()”]. The above integral isthen written idx dy [resvea= ['[son$ +eon$] a. Inpreference tothis physical interpretation of“line integrals”, wecan in- troduce amore geometrical interpretation. Recall that de/dt(g;) denotes the et) cide,Fe) e(4—1) tangent vector of¢attime ;.Then thesums 2 de @ Lown ($a) =aist n = Use"Es)+alee" EN)i—1) i=l clearly also approach 1 fete" +etenre"oyar Consider thespecial case where ¢goes with constant velocity oneach (t;—1, 4). Integration 243 Ifwechoose any&€(f;~1, 07),then lengthofa)=theconstantspeedon(¢;-1,41) __length ofthesegment from e(é~1) toc(t) 7 uta , so [leneofSe}+(tj—G1)=lengthofsegmentfromc(;-1)toet). Inthis case, “ de>[ienethofea]“(i-t-1) isl a isthelength of¢,and thelimit ofsuch sums, forageneral c,canbeused asa definition ofthelength ofc.The lineintegral [w=limitofthesums(*) ec canbethought ofasthe“length” of¢,when ourruler ischanging contin- uously inaway specified byw:Notice that therestriction ofw(c(¢)) tothe i-dimensional subspace ofR?,¢n spanned byde/dt isaconstant times “signed length”, The natural way tospecify acontinuously changing length along c istospecify alength onitstangent vectors; thisisthemodern counterpart of theclassical conception, whereby thecurve ¢isdivided into infinitely small parts, theinfinitely small piece atc(t), with components dx,dy,having length S(e()) dx+g(e(t)) dy. Before pushing thisgeometrical interpretation toofar,weshould note that there isnoi-form wonR?such that f{w=length of¢forallcurves.ec Itistruethatforagiven one-one curve ¢wecanproduce aform wwhich works for¢;wechoose w(c(¢)) €2'(R%() sothat x x d x oo($f)=1, . dt f ‘kernel ex(e(0)) x (choosing thekernel ofwarbitrarily), andthen extend wtoR*.Butif¢is 244 Chapter 8 notone-one thismaybeimpossible; forexample, inthesituation shown below, there isnoelement of&1(R?,,)) which hasthevalue 1onallthree vectors. Ingeneral, given anywonR?which iseverywhere non-zero, thesubspaces Ap=kerw(p)forma1-dimensional distribution onR?;anycurvecontained inanintegral submanifold ofAwillhave “length” 0.Later wewillseeaway ofcircumventing thisdifficulty, ifweareinterested inobtaining theordinary Jength ofa curve. Forthepresent, wenote that thesums («),used todefine this generalized “length”, makesenseevenifcisacurveinamanifold M(where there isnonotion of“Jength”), andwisa1-form onM,sowecandefine fo asthe limit ofthese sums. One property oflineintegrals should bementioned now, because itisob- vious with ouroriginal definition and merely true forournew definition. If p:[0,1] +[0,1] isaone-one increasing function from [0,1]onto[0,1], then the curve¢op:[0,1]>Miscalled areparameterization ofc—ithasexactly the same image asc,buttransverses itatadifferent rate. Every sum S(P,&) for¢ isclearly equal toasum S(P’,£") for¢©p,andconversely, soitisclear from ourfirstdefinition thatforacurve ¢:[0,1] +R?wehave feLeic [cop (“the integral ofwover ¢isindependent oftheparameterization”). This isno longer soclear when weconsider thesums («)foracurve ¢:[0,1] >M,noris itclear even foracurve c:[0,1]>R?,butinthiscasewecanproceed right to theintegral these sums approach, namely 1f[PeM)c"O +g(e@))e*) dt. Integration 245 The result then follows from acalculation: thesubstitution ¢=p(w) gives 1 [rarer +eemeonar py 5 =fseo,LEMME (PL)+8(PON) HCN)]pw)du A 1 =f[fle>pw)y(eep)"(u)+gle>pU))(e°p)*"u))du. 0 Foracurve inR",andaI-form »=7", @dx', there isasimilar calcula- tion; forageneral manifold M,wecanintroduce acoordinate system forour calculations if¢([0, 1])liesinonecoordinate system, orbreak ¢upintoseveral pieces otherwise. Wearebeing abitsloppy about allthisbecause weareabout tointroduce yetathird definition, which wil] eventually become our formal choice. Consider once again thecase ofa1-form onR?,where 1fw=f[FeM)e"@ +gee] at. ic0 Notice that if¢isthestandard coordinate system onR,then forthemap c:[0,1]>R?wehave c*(f dx+gdy)=(f oc)e*(dx) +(goc)c*(dy) =(fcc)d(xoc) +(g0c)d(yoc) =(foc)e’ dt+(g occ” dt, sothat formally wejust integrate c*(f dx+gdy); tobeprecise, wewrite e*(f dx+gdy) =hdt (intheunique possille way), and take theintegral of4on(0,1]. Everything wehave said forcurves c:[0,1] >R"could begeneralized to functions ¢:[0,1]?>R”. Ifxandyarethecoordinate functions onR?,let ac ac ae_ (2 y & ax Lox ae(a) _=~ =a(—}. ay ay Forapairofpartitions so<---<5mandf<---<tof[0,1],ifwechoose 246 Chapter 8 &j€[si-1,51]x[41,4] and@isa2-form onR",then AA SET Sint Si Ea ac wtetéy)(FE)xe)6G ~4-0) isa“generalized area” oftheparallelogram spanned by ac ac Pra wei ‘Thelimitofsumsofthese terms canbethought ofasa“generalized area”of¢. Tomake aJong story short, wenow proceed with theformal definitions. AC®function c:[0,1]>Miscalledasingular k-cube inM(theword “singular” indicates that¢isnotnecessarily one-one). WewillJet[0,1°=R°= 0€R,sothat asingular 0-cube ¢isdetermined bytheone point c(0) €M. Theinclusion mapof[0,1J*inR*willbedenoted by7*:[0,1]!>RF;itis called the standard k-cube. Ifwisak-form on(0,1}*,andx!,..., x*arethecoordinate functions, thenw canbewritten uniquely as w=fdx'a---rdx*. We define =fSO,....x*) dat... dxt 0. |5ms|f inclassicalnotation,whichmodern },foe font notation attempts tomimic asfaraslogic permits Ifwisak-form onM,andcisasingular k-cubeinM,wedefine [oefctw, ec (oye where theright hand sidehasjustbeen defined. Fork=0,wehave aspecial definition: a0-form isafunction f,andforasingular 0-cube ¢wedefine [f=P09. lc Antegration 247 1.PROPOSITION. Letc:[0,1]” >R"beaone-one singular n-cube with detc’>0on[0,1].Letwbethen-form w= fdx'r---rAdx". Then ¢ forfe [Le © eaten" PROOF. Bydefinition, [De|ctw) le fone =|(fec)(dete!)dx! A---Adx" byTheorem7-7 foray" =|(fec)|detel|dx! A...Adx" byassumption fo,1)" =ffbythechangeofvariableformula.¢ e({0,1)") 2.COROLLARY. Letp:[0,1]*>[0,1]}* beone-one onto with detp’>0, letcbeasingular k-cube inMand letwbeak-form onM.Then {[email protected] cop PROOF, We have fw=f(cop)to=fP*(c*w) cop (0,118 (0,1)* =/c*(w)_bytheProposition, sincepisonto fo" =[0.% lc 248 Chapter 8 Themapcop: [0,1] >Miscalled areparameterization ofcifp:[0,1]> [0,1}isaC&one-one onto map with detp’#0everywhere (sothatp~!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 that there would benosuch result ifwetried todefine theintegral over ¢ ofa C® function {: M—Rbytheformula |foe. {0,1 Forexample, ifc:[0,1] >Mthen 1 1 fSle(t))dt isgenerally #fSle(po) dt. 0 0 Fromaformal pointofview,differential forms arethethings weintegrate be- cause they transform correctly (i.¢., inaccordance with Theorem 7-7, sothat thechange ofvariable formula willpopup);functions onamanifold cannot be integrated (wecanintegrate afunction fonthemanifold R*onlybecause it gives usaform fdx'A+--Adx*), Our definition oftheintegral ofak-form wover asingular k-cube ¢can immediately begeneralized. Ak-chain issimply aformal (finite) sum ofsingular k-cubes multiplied byintegers, e.g.. ley—2¢2 +303. The k-chain 1c)=1-cywillalsobedenoted simply bycy.Weaddk-chains, andmultiply them byintegers, purely formally, e.g., (cy +34) +(—2)(cr +3 +¢2)=~2e2 —2e3+64. Morcover, wedefine theintegral ofwover ak-chain ¢=7;a;¢;intheobvious way: w=)a;|ow. ona Ee, 7 The reason forintroducing k-chains isthattoevery k-chain ¢(which may be just asingular k-cube) wewish toassociate a(k~1)-chain ac,which iscalled theboundary of¢,andwhichissupposed tobethesumofthe various singular Integration 249 (k=1)-cubes around theboundary ofeach singular k-cube inec.Inpractice, it isconvenient tomodify thisidea. Theboundary of/?,forexample, willnotbe thesumofthefoursingular 1-cubes indicated below ontheleft,butthesum, =I -1 +1 +1 with theindicated coefficients, ofthefour singular 1-cubes shown ontheright. (Notice that thiswillnotchange theintegral ofa1-formover0/7.)Foreachi with1<i<nwefirstdefine twosingular (n—1)-cubes Tio)andTay(the (i,0)-face and(i,1)-face of1”)asfollows: Ifx€(0,1]"~!, then Too) =11 10,240 x74) Hl. ox xt), Ty) =1", Bt) a(t ixt xt). ay ,Tio Ki Tho) Ta Te.) 250 Chapler 8 The(i,a)-face ofasingular n-cubecisdefined by Cay =6°Thay)» (1,0) etl) can c C@,0) (0) aD Now we define n c=>YH)! cG,0- i=1 a=0,1 Finally, theboundary ofann-chain 37,a;¢;isdefined by (Sar) =4/8).a i These definitions allmake sense only forn>1.Forthecase ofa0-cube c:[0,1]° >M,which wewillusually simply identify with thepoint P=c(0), wedefine actobethenumber 1€R,andfora0-chain 7;ajc;wedefine a(Saier) =Saja) =Yai.i a i Notice that foraI-cube ¢:[0,1] >Mwehave 8¢=e(1,1) -€41,0)> so (ac) =1-1=0. Wealsohave, forasingular 2-cube c:[0,1]? >M, qSs Be=41,1)=Ce,1)=€44,0)+€2,0)5 oo rom a(ac)=(R—Q)—(R-S) P —(S-P)+(Q-P)2.0) =0. R Q ea) Integration 251 From apicture itcanbechecked that thisalsohappens forasingular 3-cube, agood exercise because thisinvolves figuring outjustwhat theboundary ofa 3-cube looks like. Ingeneral, wehave: 3.PROPOSITION. Ifcisanyn-chain inM,then (dc) =0.Briefly, 3?=0. PROOF. Leti<j<n—1,andconsider(Jf.4y)y,g).Forx€(0,1]"-?,we have, from the definition Tay G,B)09=MuayLGpO) Ee(HNCane aSEEPS SS) =I tax! Bad), Similarly, Casey =1641.0) MayPD =Hanpyle ext 0x8...) HIM xa xt IT Bix x7). Thus (4as)¢,p) =UParp))gay rt<j<= 1,Iefollows easily forany singular n-cube ¢that(¢(7,«))¢j,8) =(¢y+1,6))(é,a) fori<j<n—1. Now a :ae=a(>DO(-0'*eu0)f=1a=0,1 n m~t 7 HV VY LVoiew* ease.s)- .i=1 @=0,1j=1 B=0,1 Inthissum, (ci,a))(j,A) and(¢(j-41,6)) (a)Occur with opposite signs. Therefore allterms cancel inpairs, and 8(c) =0.Since thetheorem istrue forsingular n-cubes, itisclearly also true forsingular n-chains. Notice that forsome n-chains ¢wehave notonly (9c) =0,buteven dc=0. Forexample, thisisthecase if¢=cy—¢2,where ¢;and czaretwo I-cubes 252 Chapter 8 with ¢)(0) =¢2(0) and ¢;(1) =¢2(1). If¢isjust asingular 1-cube itself, then a2CQ dc=0precisely when c(0)=e(1),i.e.,whencisa“closed” curve. Ingeneral, anyk-chain cjscalled closed ifd¢=0. Recall thatadifferential formwwithdw=0isalsocalled “closed”; this terminology hasbeen purposely chosen toparallel theterminology forchains {ontheother hand, achain oftheform deisnotdescribed, reciprocally, by theclassical term of“exact”, butissimply called “aboundary”). This parallel terminology wasnotchosen mcrely because oftheformal similarities between ¢ and@,expressed bytherelations d?=0and9?=0.Theconnection between forms andchains goes much deeper than that. Forexample, wehave seen that onR?—{0}there isa1-form “d” which isclosed butnotexact. There isalsoa I-chain ¢which isclosed butnotaboundary, namely, aclosed curve encircling Integration 253 thepoint 0once. Although itisintuitively clear that ¢isnottheboundary ofa 2-chain inR?—{0},thesimplest proofusesthetheorem whichestablishes the connection between forms, chains, d,and @. 4.THEOREM (STOKES’ THEOREM). Ifwisa(k—1)-form onMand is ak-chain inM,then fdw=fet c ac PROOF. Mostoftheproofinvolves thespecialcasewhere@isa(k—1)-formonR*andc=J*,Inthiscase,wisasumof(k—1)-forms ofthetype Sax!nondenendsk, anditsuffices toprove thetheorem foreach ofthese. Wenow compute. First, alittle notation translation shows that kx,1ai ke PoweTG) (fdx'A+--Adx!A+++Adx") 0 ny di =||Slax dxldx®itysi. fo. Therefore fSax!Avindxinsnak ark ui —=Dyc fThgtfdsA.AdaAondx*)jaland) foe =(-nH fLOscyeex®) dot...dat fo,ny* +(-1!fLO... 0...) dat.dak, fo.1y* 254 Chapter 8 Ontheother hand, ffafdx!n---nOxin--ndx*) qk =fDifdx!dx!N.ndxtnondx® f0,1)7* =(-p! |Dif. fo,1* ByFubini’s theorem and thefundamental theorem ofcalculus wehave fdas nnd ncnds) tk 1 1 _ =nf=|Difax) ds!)dx'...dsl...dx* 0 o 71 1 =n fff[P68ete) 0 0 F080 24)dx...dx...ax® RCV fF cetoAdstat toa +(-1|T0850, axl dk. fon ‘Thus ffde=f@. tk ark Foranarbitrary singular k-cube, chasing through thedefinitions shows that ae ark Therefore feo=f rtd)=faera)=fcoxfo. c tk ik ark ac The theorem clearly follows fork-chains also. + Integration 255 Notice that Stokes’ Theorem notonly uses thefundamental theorem ofcal- culus, butactually becomes thattheorem when ¢=J!andw=f. Asanapplication ofStokes’ Theorem, weshow that thecurve c:[0,1] > R?—{0}defined by cec(t)=(cos2x,sin2x1),rans although closed, isnot4c?forany2-chain c?.Ifwedidhave¢=4c?,thenwe wouldhavefo=fed=faas)=fo=0. c ‘cz 2 ic? Butastraightforward computation (which willbegood forthesoul) shows that -y xd=] Say =28. [@[weetae Ly [There isalso anon-computational argument, using thefactthat “d6” really isdOfor6:R?—(0,00) x{0})>R:Wehave f@=00-9-60, ellete} and@(1—£) —O(£) >2”ase 0] Although weused thiscalculation toshow that ¢isnotaboundary, wecould justaswell have used ittoshow that w=“d@” isnotexact. For, ifwehad w=dfforsome C®function f:R?—{0}+R,then wewould have a=fo=far=f sa[reo. cic ae 0 Wewere previously able togive asimpler argument toshow that “d6” isnot exact, but Stokes’ Theorem isthe tool which will enable ustodeal with forms onR"—{0}.Forexample, wewilleventually obtain a2-form wonR3—{0}, _XdyAdz—ydx Adz+zdx Ady Ck G+ y2+222 256 Chapter 8 which isclosed butnotexact. Forthemoment wearekeeping theorigin ofwa secret, butastraightforward calculation shows thatdw=0.Toprovethatwis notexact wewillwant tointegrate itover a2-chain which “fills up”the2-sphere S?CR? ~{0}.There arelotsofways ofdoing this,buttheyallturnouttogive thesame result. Infact, wefirst want todescribe away ofintegrating n-forms over n-mantifolds. This ispossible only when Misorientable; thereason will beclear from thenext result, which isbasic forourdefinition. 5.THEOREM. LetMbeann-manifold with anorientation yz,and letc1,c2: [0,1)"+Mhetwosingular n-cubes which canbeextended tobediffeomor- phisms inaneighlorhood of[0,1”.Assume that ¢,and czareboth orientation preserving (with respect totheorientation %onM,and theusual orientation onR"). If@isan n-form onMsuch that support @C¢([0, 1]”)Ne2([0, 1]"), then feo=fo.ct 2 PROOF. Wewant touseCorollary 2,and write fo-f o=fo.c2 je20(e2~1 0c) 1 Theonlyproblem isthat¢2!cyisnotdefined onallof[0,1]"(itdoessatisty det(c27' ec)’ >0,since ¢;and¢2areboth orientation preserving). However, a glance attheproof ofCorollary 2willshow thattheresult stillfollows, because ofthefactthatsupport @Ccy([0, 1]”)N¢2([0, 1]”). ‘Thecommonitumber[@,forsingularn-cubes¢:[0,1]>Mwithsup- lc port &C¢({0, IJ")and¢orientation preserving, willbedenoted by feM Ifwisanarbitrary n-form onM,then there isacover ©ofMbyopen setsU. eachcontained insome¢([0,1"),where ¢isasingular n-cube ofthis sort; if® isapartition ofunity subordinate tothiscover, then Lee'M Integration 257 isdefined foreach ¢€®,Wewish todefine w=)foo. I,gcaM Wewilladopt thisdefinition only when @hascompact support, inwhich case thesum isactually finite, since support @canintersect only finitely many ofthe sets{p:o(p)#0},which formalocally finitecollection. Ifwehaveanother partition ofunity W(subordinate toacover (’),then Lfeoe-Lf Lvl fvow: ea M ged yew ded yew these sums areallfinite, andthelastsum canclearly alsobewritten as LU fewe=y fwe, wedpedo M wey’ M sothat ourdefinition does notdepend onthepartition. (We really should denote thissum by (Mn) fortheorientation —.ofMweclearly have 1ied (M,-#) (Mu) However, weusually omit explicit mention of1.) With minor modifications wecandefine fy,@even ifMisann-manifold- with-boundary. IfMC R”isann-dimensional manifold-with-boundary and f:M—RBhascompact support, then ffax!Anat =ff. M M where theright hand sidedenotes theordinary integral. This isasimple conse- quence ofProposition }.Likewise, iff:M”—N”isadifleomorphism onto, and wisann-form with compact support onN,then f@ iffisorientation preserving 5N [fe= M-faiffisorientation reversing, IN 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 p€Mwehave op) =I|np| forsome np€2"(My), i.e,foranynvectors v1,..-.Un €Mpwehave (P)(V1y+++Un)=Mp(U1y+++Un)!20. Such afunction iscalled avolume element—on each vector space itdeter- mines away ofmeasuring n-dimensional volume (notsigned volume). If(x,U) isacoordinate system, then onUwecanwrite w=fldx'A---Adx"| forf=0; wecall»aC®volume element iffisC°°.Onewayofobtaining avolume element istobegin with ann-form nand then define w{p) =|n(p)|. However, notevery volume element arises inthisway—the form npmay notvary con- tinuously with p.Forexample, consider theMébius strip M,imbedded inR?. Since M,canbeconsidered asasubspace ofR3,,wecandefine (P)(Up, Wp)=area ofparallelogram spanned byvand w. Itisnothard toseethat isavolume element; locally, wisoftheform w=|n| forann-form n.But this cannot betrue onallofM,since there isnon-form 7 onMwhich iseverywhere non-zero. Theorem 7-7 has anobvious modification forvolume elements: 7-7,THEOREM. Iff:M>NisaC™function hetween n-manifolds, (x,U)isacoordinate system around p€M,and (y,V) acoordinate system around g=f(p) €N,then fornon-negative g:V+Rwehave acy! Igldy!AAdy") =(gof)-fac(5°) sds!Asnds"), PROOF, Gothrough theproof ofTheorem 7-7,putting inabsolute value signs intheright place. Jnlegration 259 7-8’, COROLLARY. If(x,U)and(y,V)aretwocoordinate systems onM and gidy!A---Ady"|=h|dx!a---Adx"| —g,h>0 then ay &=g-ldet|— ]]. «b(n [This corollary shows that volume elements arethegeometric objects corre- sponding tothe“odd scalar densities” defined inProblem 4-10.] Ttisnow aneasy matter tointegrate avolume element wover anymanifold. First wedefine fo=|ftforw=f|dx'a---adx"|, f20.be fo,1)" Then foran»-chain ¢:[0,1]" >Mwedefine fo=f[ ee.ic fo1y" Theorem 7-7’shows thatProposition |holds foravolume element w=f|dx!A -+-Adx"| even ifdete’ isnot >0.Thus Corollary 2holds forvolume elements even ifdetp’isnot >0,From this weconclude that Theorem 5holds for volume elements @onanymanifold M,without assuming ci, orientation preserving (oreven thatMisorientable). Consequently wecandefine fry© foranyvolume element with compact support Ofcourse, when Misorientable these considerations areunnecessary. For, thereisanowhere zeron-form7onM,andconsequently anyvolume element w can bewritten w=f\ln|, f20. Ifwechoose anorientation pufor Msuch that w(vj,...,Un) >0forv),...,Un positively oriented, then wecandefine [e=fiot™ M (Mn) 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 3-16 that ifMisamanifold-with-boundary, and p€ aM, then certain vectors v€M,canbedistinguished bythefactthat forany coordinate system x:U—H”around p,thevector x4(v) €H"¢cp) points “outwards”. Wecallsuch vectors v€My“outward pointing”. IfMhasan ca” < Lr orientation p1,wedefine theinduced orientation 9:for2Mbythecondition that [u1...-,Un—1] €(8), ifandonly if[w,v1,...,Un—1] €Mpforevery outward pointing w€My. Ifuistheusual orientation ofH",then forp=(a,0) «H” we have Mp=((er)ps-+++(€n)p) =(=1)""[en)ps (€1)ps+s(en=1)p] =(=1)"[(~en)ps (€1)ps---» (@n~1)p]- Since (~en)p isanoutward pointing vector, thisshows thattheinduced orien- tation onR"~! x{0}=9H” is(—1)” times theusual one. The reason forthis choice isthefollowing. Let¢beanorientation preserving singular n-cube in (M,p)suchthat8MNM¢([0,1}") =eG,0)([0, 1]"7!). Thenc(,0):[0,1]! > ;0) (8M, 0)isorientation preserving foreven n,and orientation reversing for oddn.Ifwisan(n—1)-form onMwhose support iscontained intheinterior Integration 261 oftheimageof¢(thisinterior contains pointsintheimageofc(n,0)),itfollows that [w=(-1)"[email protected]. ‘aM —support@ Butc(n,0) appears with coefficient (~1)" indc.So a fomf.o=cmf o=fo ae yen. cin aM Ifitwere notforthischoice of34wewould have some unpleasant minus signs inthefollowing theorem. 6.THEOREM (STOKES’ THEOREM). IfMisanoriented n-dimensional manifold-with-boundary, and4Misgiven theinduced orientation, and@isan (n—1)-form onMwith compact support, then [=fM ‘aM PROOF. Suppose firstthatthere isanorientation preserving singular n-cube ¢ inM—4Msuch thatsupport @Cinterior ofimage c.Then fdo=fdo=f®byTheorem4 M c ae =0 sincesupport wCinterior ofimage c, while weclearly have [eneaM Suppose next that there isanorientation preserving singular n-cube cinM such that2MNe([0, 1]")=¢(n,0)([0, 1]"~"), andsupportwCinteriorofimagec. Then once again fdu=faofo=fwoby).iM c ae aM 262 Chapter & Ingeneral, thereisanopencover©ofMandapartition ofunity®sub- ordinate to@such that foreach @€®theform ¢-aisone ofthetwo sorts already considered. Wehave 0=d(1)=a(Xe) =)4,ged geo so Vagnw=o. ged Since @hascompact support, thisisreally afinite sum, andweconclude that >fdpAw=0.geo lM Therefore fde=Lfo-dw=Lfdp\w+$-dwcs geoM oeaiM -rf ag-0)= >foxfwo.%eco! eco 10M Cot Oneofthesimplest applications ofStokes’ Theorem occurswhentheoriented n-manifold (M,4)iscompact(sothateveryformhascompactsupport)and aM=G.Inthiscase, ifisany(n—1)-form, then fdn=fn=0. M ‘3M Therefore wecanfind ann-form @onMwhich isnofexact (even though it must beclosed, because all(2+1)-forms onMare0),simply byfinding an with fwo#0. M Such aform @always exists. Indeed wehave seen that there isaform »such that forv),...,Un €Mpwehave (*) (1, --.,Un) >0if[r,..., Um]=wp. Tf¢:[0,1]” +(M,)isorientation preserving, thentheformc*won[0,1]”is clearly gdx' A+ Adx" forsome g>0on[0,1]”, Integration 263 sof,w>0.Itfollows thatfy,@>0.There is,moreover, noneed tochoose aform wwith (x)holding everywhere—we canallow the>sign tobereplaced by>.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, wecan obtain more information about theshape ofMbyanalyzing more closely theextent towhich closed forms arenotnecessarily exact. Inparticular, wewould now liketoaskjust how many non-exact n-forms there areonacompact oriented n-manifold M.Naturally, if»isnotexact, then thesame istrue forw+dnfor any(n—1)-form n,sowereally want toconsider @andw+dnasequivalent. There is,ofcourse, astandard wayofdoing this, byconsidering quotient spaces. Wewillapply thisconstruction notonly ton-forms, buttoforms ofanydegree. Foreach k,thecollection Z*(M) ofallclosed k-forms onMisavector space. Thespace B*(M) ofallexact k-forms isasubspace (since d?=0),so wecanform thequotient vector space HE(M) =Z*(M)/BK(M); thisvector space H*(M)iscalled thek-dimensional deRham cohomology vector space ofM. [deRham’s Theorem states that thisvector space isisomorphic to accrtain vector space defined purely interms ofthetopology ofM(forany space M),called the“k-dimensional cohomology group ofMwith real coef ficients”; thenotation Z*,B*ischosen tocorrespond tothenotation used in algebraic topology, where these groups aredefined.] Anelement ofH*(M) isanequivalence class [w]ofaclosed k-form w,two closed k-forms w,anda»being equivalent ifandonly iftheir difference isexact. Intermsofthesevectorspaces, thePoincaré Lemma saysthatH*(R") =0(the vector space containing only0)ifk>0,ormore generally, H*(M) =0ifM iscontractible and k>0. Tocompute H°(A4) wenote firstthat B°(M) =0(there arenonon-zero exact 0-forms, since there arenonon-zero (—1)-forms forthem tobethedif- ferential of).SoH°(M) isthesame asthevector space ofallC™functions f{:M >Rwith df=0.IfMisconnected, thecondition df=0implies thatfisconstant, soH°(M) ©R.(Ingeneral, thedimension ofH°(M) isthe number ofcomponents ofM.) Aside from these trivial remarks, wepresently know only oneother factabout H*(M)—if Miscompact andoriented, thenH”(M) hasdimension >1.The further study ofH*(M) requires acareful lookatspheres andEuclidean space. 264 Chapter 8 OnS"~) ¢R®—{0}there isanatural choice ofan(n—1)-form ofwith Sgn 6!>0:for(01)py.+65(Un-a)p €S”1p, wedefine P ’ Uv)'(p)((U1)ps--+5(Un-1)p) =det}. Uni Clearly thisis>0if(v1)p,...,(Un-1)p isapositively oriented basis. Infact, wedefined theorientation ofS”~) inprecisely thisway—this orientation isjust theinduced orientation when S”~! isconsidered astheboundary oftheunit ball{p€R”:|p|<1}with theusual orientation. Using theexpansion ofa dcterminant byminors along thetoprow weseethat o’istherestriction to S”~? oftheform ¢onR”defined by wu;: — 0=SH x!dxtAAdxt AAdx", ia The form o’onS”~? willnowbeused tofindan(n—1)-form onR”—{0} which isclosed butnotexact (thus showing thatH"~1(R" —{0}) +0).Consider themap r:R"—{0}+$*~! defined by P Pp r(pP)= =—. Ipl vp) Clearly r(p) =pifp€S"~); otherwise said, ifi:S"-! +R"—{0}isthe inclusion, then roi=identity ofS"7. (Ingeneral, ifACXandr: X—Asatisfies r(a) =afora€A,then ris called aretraction ofXonto A.) Clearly, r*o’ isclosed: d(r*o’) =r*do' =0. However, itisnotexact, forifr*o’ =dn,then o!=i*r*0' =di*n; but we know that o’isnot exact. Integration 265 Itisaworthwhile exercise tocompute bybrute force that forn=2, reo=2avdx_xdy—ydx _jg x24 ye ry] gay XdyAdz—ydx AdztzdxAdy forn=3,r*o’=e ] =yaltdy nde—ydxAdz+zdx Ady). Since wewillactually need toknow r*o’ ingeneral, weevaluate itinanother way: 7.LEMMA. IfaistheformonR"defined by n a=Anita! dxlandxt asndx", i= and 0’istherestriction i*oofotoS"~!, then o(p) (* r*o'(p) =——. (*)P=im So , rol==enix! axtaweAdxtAwAdx", ix PROOF. Atanypoint p€R”—{0},thetangent space R”, isspanned bypp andthevectors vpinthetangent space ofthesphere S"~1(| p|)ofradius|p|. Soitsuffices tocheck that both sides of(x)give thesame result when applied tom—Ivectors each ofwhich isoneofthese twosorts. Now ppisthetangent vector ofacurveylyingalongthestraightlinethrough0andp;thiscurveis taken tothesingle point r(p) byr,sor4(pp) =0.Ontheother hand, P P 9(P)(Pp,(Pi)ps+-+5(Un-2)p) =det} |=O. Una? Soitsuffices toapply both sides of(*)tovectors inthetangent space of S""(|p|). Thus (Problem 15),itsuffices toshow thatforsuch vectors vpwe have ! (Up) =—Ur(py-(Up. ae 266 Chapter 8 Butthisisalmost obvious, sincethevector vpisthetangent vector ofacircley lying inS"~"(|p|), andthecurve royliesinS*~? andgoes 1/|p| asfarinthe same time. 4% 8.COROLLARY (INTEGRATION IN“POLAR COORDINATES”). Let ff:B= R,where B={peR":|p| <j, anddefine g:S’~! >Rby 1 et)=fwFeu:pau.0 Then fr-f fastnunds=fgo’.B iB srmt PROOF. Consider S”~} x[0,1]andthetwoprojections pause m:8") x[0,1] >$7) gralmy:S™* x[0,1]=[0,]. in Letususetheabbreviation |xgmt a!Adt=m*o!Antal. >) If(y,U)isacoordinate system onS"~?, with acorresponding coordinate sys- tem(7,4) =(¥om,m2) onS”™ x[0,1], andof=ady' A--- Ady", then clearly aoAdt=odj!n---Adj"vdt. From thisitiseasy toseethatifwedefine h:S"~! x[0,1] +Rby h(p,u) =u" fu p), then fgo’=(-1)""! f ho’Adt. Isnt S15(0,1] Now wecandefine adifleomorphism $:B—{0}>S"~? x(0,1] by (Pp) =(r(p), v(p)) =(p/P Pl). Integration 267 Then o°(0Adt)=$*(m"0' Am2dt) =o'my"0' Ab*ny*dt =(m0¢)"0" A(m20)*dt =ro'avdt 1/< fei — Lg=a(e dx!a...ndxtA--ads") ayax!ial i=l =O xy?aeavende i= ey na ; =peroeAvAax", Hence o"(ho’ Adt) =(hog)¢*(o' Adt) yet dx! ax"=vSteTIx"Av Aax =(“If dxaeAdx". So, fSax!A.Adx"=curtfg*(ho!Adt)IB B~{0} 5cnf ho!Adt 51 (0,11 =fgo". Isnt (This last step requires some justification, which should besupplied bythe reader, since theforms involved donothave compact support onthemani- folds B—{0}audS"~! x(0,1] where theyaredefined.) ¢ Weareabout ready tocompute H*(A/) 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. 268 Chapter 8 ThedeRham cohomology vector spaces withcompact supports H*(M) are defined as HE(M) =Z5(M)/BE(M), where Z*(M) isthevector space ofclosed k-forms withcompact support, and BE(M) isthevector spaceofallk-forms dnwhere7isa(k—1)-form with compact support. Ofcourse, ifMiscompact, thenHk(M) =H*(M). Notice thatB‘(AM)isnotthesameasthesetofallexactk-forms withcompact support. Forexample, onR",iff>0isafunction with compact support, and f>0 atsome point, then w= fdx' rA-»-Adx" isexact (every closed form onR’is)andhascompact support, butwisnotdn foranyform nwith compact support. Indeed, if#=dywhere nhascompact support, then byStokes’ Theorem fo=[an=ffn=0. Re Re an" This example shows that H2(R") #0,and asimilar argument shows that ifMisanyorientable manifold, ten H?(M) #0.Wearenow going toshow that foranyconnected orientable manifold Mweactually have H?(M)®R. This means thatifwechoose afixed wwithfy@#0,thenforanyn-form w! with compact support there isareal number asucli that w’—aw isexact. The number acanbedescribed easily: if ow’—aw =dn, then [ef awe favo, mM: JM iM so M 'M theproblem, ofcourse, isshowing that 7exists. Notice that theassertion that H®(M) ®Risequivalent totheassertion that lo]>f5 iM isanisomorphism ofH7'(M) with R,i.¢.,totheassertion thataclosed form w with compact support isthedifferential ofanother form withcompact support iffyo=0. Integration 269 9.THEOREM. IfMisaconnected orientable n-manifold, thenH2(M) ~R. PROOF, Wewillestablish thetheorem inthree steps: (1)The theorem istrue for M=R. (2)Ifthetheorem istruefor(n—1)-manifolds, inparticular forS”~), then itistrue for R”. (3)Ifthetheorem istrue forR”,then itistrue foranyconnected oriented n-manifold. Step1.Let beaJ-form onRwithcompact support suchthatfy@=0.There issome function f(not necessarily with compact support) such that w=df. Since support wiscompact, df=0outside some interval [—N, N],sofisa f -N N constant ¢;on(—oo, —NV)andaconstant ¢2on(N,00).Moreover, o=fo-f a=[roa Fara. R R R Therefore ¢)=¢2=¢and wehave w=df-c) where f—¢hascompact support. Step2.Let@=fdx’A--» Adx"beann-form with compact support onR” suchthatfgn@=0.Forsimplicity assume thatsupportwC{p€R”:|p|<1}. Weknow that there isan(?—1)-form 7onR”such that w=dy.Infact, from Problem 7-23, wehave anexplicit formula forn, n 1 _np)=1)(frr)f(t.p)ar)xfdx’A.AdxtAAd", ist 0 270 Chapter 8 Using thesubstitution u=|p| thisbecomes tpt: Pp 1 (p)=fuw(«-5)du|— oe(I Ta))ioe ul5; —_ xPei xtdxtAvdlAvAdx" iat tpl P=fuf(«:md)du)-r'o'(p) byLemma7. 0 |p| Define g:S"~! >Rby 1 at0)=ffwf p)du. OnthesetA={p€R": |p|>1}wehave f=0,soonAwehave ' P ne=(f ws(ue)du)-2*0(D), ‘0lpi or n=(gor)-r*o’ =r*(go"). Moreover, byCorollary 8wehave forthe(1—1)-form go!onS*?, fgo'=fffds!nvndst . isnt IB .=fo=0. fan Thus, bythehypothesis forStef2, go’=dX forsome (n—2)-form )onS"~!. Hence n= rtd) =dr"). Let&:RB"—[0,1] beany C® function with 4=1onAand 4=Oina neighborhood of0.Then fr*A isaC®form onR"and w=dn=d(n—d(hr*d)): Integration 271 theform n—d(tr*)) hascompact support, since onAwehave n—d(hr*h) =n—d(r*d) =0. Step3.Choose ann-form wsuch thatfy@#0andwhascompact support contained inanopen setU¢M,with Udiffeomorphic toR". Ifw!isany other 7-form with compact support, wewant toshow that there isanumber ¢ andaformynwithcompact support suchthat wo!=cw+dn. Using apartition ofunity, wecanwrite a!=po!++++bo" where each ¢;! hascompact support contained insome open setU;CM with U;diffeomorphic toR".Itobviously suffices tofind ¢;and nrwith ¢ja’ = ¢j@+dnj, foreach 7.Inother words, wecanassume w’hassupport contained insome open VCMwhich isdiffeomorphic toR". Using theconnectedness ofM,itiseasy toseethat there isasequence of open sets U=N,....VY=¥ diffeomorphic toR”,with VM Vi+1 ¢9.Choose forms w;with support wyC paper CongSI ye VNVieandfy, #0.Since weareassuming thetheorem forR”wehave a)—qe@=dn @2—C01 =dag o!—¢p@,-1 =dnp, where allnj;have compact support (CcVj). From thisweclearly obtain the desired result. 272 Chapter 8 The method used inthelaststep canbeused toderive another result. 10.THEOREM. IfMisany connected non-orientable n-manifold, then H2(M) =0. PROOF, Choose ann-form wwith compact support contained inanopen setU diffeomorphic toR",suchthatfy,»#0(thisintegralmakessense,sinceUis orientable). Itobviously suffices toshow that w=dnforsome form nwith compact support. Consider asequence U=N,...,4-=V ofcoordinate systems (Vj,x;)whereeachx;0x;417? isorientation preserving, Choose theforms w;inStep3sothat, using theorientation ofV;which makes xi:V;>R"orientation preserving, wehavefy,a;>0;thenalsofy,7>0. Consequently, thenumbers G=fOj/f,@—)arepositive. 7 ¥; Itfollows that @=cw+dn where c> 0. Now ifMisunorientable, there issuch asequence where V,=V;butx,0x7? isorientation renersing. Taking w'=—w, wehave -w=cwo+dn fore>0 so : (-e-lw=dn for-c-1#0. & Wecanalso compute H"(M) fornon-compact M. 11.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=U;,U2,U3,Us,... Integration 273 ofsuch coordinate neighborhoods such that U;NU;4 #@,and such that the sequence iseventually inthecomplement ofanycompact set. supportaQ9S supportaco Now choose n-forms w;with compact support contained inU;NMUj41, such thatfy, #0.There areconstants ¢;andforms n;withcompact support ¢U;such that @=c0,+dm Oo;=G14 t+dnig, 121. Then w=dm +a, =dmt+eidnz +c1C2@2 =dm+eidn2 +crc2dns +1020303 Since anypoint p€Miseventually inthecomplement oftheU;’s, wehave w= dm+eidn2 t+crcadns +cye2cadng ++++, where theright side makes sense since theU;areeventually outside ofany compact set. Now itcanbeshown (Problem 20)that there isactually such asequenceUy,U2,Us,...whoseunionisallofM(repetitions areallowed, andU;may intersect several U;forj<i,butthesequence isstilleventually outside ofany compact set). The cover ={U}isthen locally finite. Let{¢y} beapartition ofunity subordinate [email protected] onM,then foreach U;wehave seen that gu,® =dn; where n;hassupport contained inU;UU4) UUi42U--- « Hence oo © oow=)due=dn=d(Sx)= i=l is i=) 274 Chapter 8 SUMMARY OF RESULTS (1)ForR”wehave weary {Pk=0 0 k>0. (2)IfMisaconnected n-manifold, then H°(M)*R Hn){[FitMisorientable0 ifMisnon-orientable HMM)={HeifMiscompact 0 ifMisnotcompact. ‘Wealsoknow that H"~!(R” —{0}) +0,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.Ifwisaclosed k-form onN,then f*w isalso closed (df*w =f*dw =0),sof*takes Z*(N) toZ*(M). Ontheother hand, J?alsotakes B‘(N) toBk(M), since f*(dn) =d(f*n). This shows thatf* induces amap Z*(N)/ BEN) -Z*(M)/B‘(M), also denoted byf*: f*:H*(N) >H*(M). Forexample, consider thecase k=0.IfNisconnected, then H°(N) isjust thecollection ofconstant functions c:N>R.Then f*(c) =cofisalso a constant function. IfMisconnected, then f*:H°(N) >H°(M) isjustthe identity map under thenatural identification ofH°(N) andH°(M) withR. IfMisdisconnected, with components Mg, @€A,then H°(M) isisomorphic tothe direct sum Ra, where eachRy*R; aed themap/*takesc€Rintotheelement of@Rewitha”component equal toc. IfNisalso disconnected, with components Ng,B€B,then r:@R >OR beB aed takes theelement {cg}ofges Rpto{cq},where c,=cpwhen f(Ma) CNp. Integration 275 Amore interesting case, andtheonly onewearepresently inaposition to look at,isthemap f*:H"(N) >H"(M) whenMandNarebothcompact connected oriented n-manifolds. There is nonatural waytomake H”(M) isomorphic toR,sowereally want tocompare ff*oandfwo M N for»ann-form onN,Choose onewowithfy@o#0.Thenthereissome number asuch that ffroy=a- fa. M N Since@+fy@isanisomorphism ofH"(M) andR(andsimilarly forN)it follows that forevery form wwehave [fora fio.‘M N The number a=degf,which depends only on/f,iscalled thedegree of/. IfMandNarenotcompact, butfisproper (theinverse image ofanycompact setiscompact), then wehave amap S*: HUN) >HEM) and anumber deg/,such that ffroatdeg ffoiM iN forallforms »onNwith compact support. Until one sees theproof ofthe next theorem, itisalmost unbelievable that thisnumber isalways aninteger. 12.THEOREM. Letf:M—Nbeapropermapbetween twoconnected oriented n-manifolds (M,) and(N,v). Letg€Nbearegular valueoff. Foreach p€f~1(q), let 1 Wffap: Mp>Neisorientation preserving sign,f= {usingtheorientations ypforM,andvgforNg) —1 iff<pisorientation reversing. 276 Chapter 8 Then degf= Sosign,f (=0if f-(p)=9). pef-'@) PROOF. Notice first that regular values exist, bySard’s Theorem. Moreover, 7(q)isfinite,sinceitiscompact andconsists ofisolated points, sothesum above isafinite sum. . Letf-'(g) ={pi,-++s Pk}. Choose coordinate systems (U;,x;) around p; such thatallpoints inU;areregular values off,andtheU;aredisjoint. We want tochoose acoordinate system (V,y)around gsuchthatf-1(V) =UU .-:UUg. Todothis, first choose acompact neighborhood Wofg,and let ee W'CcMbethecompact set W'=f(W)—(YU--- UY). Then {(W’) isaclosed setwhich does notcontain g.Wecantherefore choose V¢W—f(W’). This ensures thatf-"(V) CUU---UUg. Finally, redefine U; tobe UNf-(V). Now choose wonNtobew=gdy' A--- Ady"where g>0hascompact support contained inV.Then support f*@ CU;U++.UUg. So k tw=ftw. Since fisadiffeomorphism from each U;toVwehave ffto=f®iffisorientation preserving Up v =~f®iffisorientation reversing. Vv Since fisorientation preserving {orreversing] precisely when sign,f=}[or —1] thisproves thetheorem. Integration 277 Asanimmediate application ofthetheorem, wecompute thedegree ofthe “antipodal map” A:S”—S”defined byA(p) =—p.’ We have already seen that Aisorientation preserving orreversing atallpoints, depending on whether nisoddoreven. Since A~!(p) consists ofjustonepoint, weconclude that degA=(-1)"7!. Wecandraw aninteresting conclusion from thisresult, butweneed tointro- duceanother important concept first.Twofunctions f,g:M>Nbetween twoC®manifolds arecalled (smoothly) homotopic ifthere isasmooth function H:Mx([0,])>N with H(p,0) =emo rareleTeTieH(p,1) =g(p) themapHjscalled a(smooth) homotopy between fandg.Notice thatMis smoothly contractible toapoint po€Mifandonly iftheidentity map ofMis homotopic totheconstant map po.Recall thatforevery k-form wonMx[0,1] wedefined a(k—1)-form JwonMsuch that iy"@ —io*w =d(Iw) +I(dw). Weused thisfacttoshow that allclosed forms onasmoothly contractible man- ifold areexact. Wecannow prove amore general result. 13.THEOREM. Iff,g: M—Naresmoothly homotopic, then themaps f*:H*(N) >H*(M) g*:H*(N) >H*(M) areequal, f*=g*. PROOF. Byassumption, thereisasmooth mapH:Mx[0,1]>Nwith f=Hoig g=Hot. Anyelement ofH*(N) istheequivalence class[w]ofsome closed k-form w onN.Then gto —ftw =(Hoh)*w -(H 0ig)*o =i)*(H*@) —io*(H*@) =d(IH*w) +I(dH*w) =d(IH*w) +0. But this means that g*([w]) =f*([w]). 4% 278 Chapter 8 14,COROLLARY. IfMand Narecompact oriented n-manifolds and the mapsf,g:M—Narehomotopic, thendegf=degg. 15.COROLLARY. If2iseven, then there does not exist anowhere zero vector field onS”. PROOF. Wehave already seen thatthedegree oftheantipodal map A:S”> S"is(-1)""1. Since theidentity map hasdegree 1,Aisnothomotopic to theidentity for7even. Butifthere isanowhere zero vector field onS”,then wecanconstruct ahomotopy between Aand theidentity map asfollows. For each p,there isaunique great semi-circle ypfrom ptoA(p) =—pwhose tangent vector atpisamultiple ofX(p). Define H(P,1) =yo). & For1odd wecanexplicitly construct anowhere zero vector field onS". For p= (X1,..-,%n41) €8”wedefine X(p) =(—¥1, X0,—X3, X25+. Xt Xn) thisisperpendicular top=(m,%2,...,Xn41), andtherefore inS”p. (OnS! thisgives thestandard picture.) The vector field onS”canthen beused togive ahomotopy between Aand theidentity map. Foranother application ofTheorem 13,consider theretraction r:R"- {0}3s"? r(p)=P/\pl. Ifi:S"™™) —R"—{0}istheinclusion, then roi: S"~) +§"~) istheidentity 1ofS"). Integration 279 The map for:R"—{0}>R"—{0} ior(p) =p/Ip| is,ofcourse, nottheidentity, butitishomotopic totheidentity; wecandefine thehomotopy Hby rac) 7(9) a H(p,t) =tp+(1 —¢)r(p) €R”—{0}. (t=0) Aretraction with thisproperty iscalled adeformation retraction. Whenever r isadeformation retraction, themaps (r0/)* and(ior)* aretheidentity. Thus, forthecase ofS”~? ¢R”—{0},wehave r HE(S"™!) —>HER" —{(0}) HE(R" —{0})>HE(S"™) and r*oi* =(ior)* =identity ofH*(R” —{0}) i*or*=(r0i)* =identity ofH*(S""). Soi*and r*are inverses ofeach other. Thus H*(S")) =HER" —{0}) forallk. Inparticular, wehave H"-1(R" —{0})*R.Agenerator ofH"~!(R" —{0})is the closed form r*o’. Wearenowgoing tocompute H*(R” —{0})forallk.Weneed onefurther observation. The manifold Mx{3}CMxR! isclearly adeformation retraction ofMxR!.SoH¥(M) =H*(M xR') for allJ, 280 Chapler 8 16.THEOREM. For0<k<n—1 wehave H*(R" —{0})=H*(S""?) =0. PROOF. Induction onn.The first case where there isanything toprove is n=3.Weclaim H!(R? —{0})=0. Letwbeaclosed }-form onR?.LetAandBbetheopen sets (0,0, 1) A=R?—{(0,0) x(—00,0)} B=R? —{(0,0) x[0,00)}. (0,0,—1) Since Aand Bareboth star-shaped (with respect tothepoints (0,0, 1)and (0,0, -1), respectively), there are0-forms fyandfgonAand Bwith w=dfy ond w=dfg onB. Now d(fa—fg)=0onANB, and ANB=[R?—{0}]xR, soclearly f4—fgisaconstant conANB.Thuswisexact,for w=d(f4—c) ond w= d(fs) onB andfy—c=fgonANB. If@isaclosed I-form onR*,there isasimilar argument, using A=R* —{(0,0,0) x(—00,0} B=R*—{(0,0,0) x[0,00)}. Ifwisaclosed2-form onR‘,thenweobtain -forms n4andngwith w=dng ond w=dng onB. Integration 281 Now d(n4—ng)=0 onANB and H1(AN B)=H1([R? —{0}]xR)©H'(R? —{0})=0. Sona—ne=dAforsome 0-form Aon ANB. Unlike theprevious case, we cannot simply consider n4—dA,since thisisnotdefined onA,Tocircumvent thisdifficulty, note thatthere isapartition ofunity {¢4,¢g} forthecover {A,B} ofR?—{0}: ba+os =) doa +doz =0 support d4CA support dgCB. Now, if éphonANB Aopidenotes{éonA-(ANB), andsimilarly for$44, then gph isaC™form onA 4h isaC® form onB. OnANBwehave na—U(r) =na—badd —dbp rd =a +(ba—I)dd+dob4ar =na—dh +d(bad) =ne+d(g,d). Sowecandefine aC® form onR"—{0}=AUB byletting itbena—d(@ad) onA,andng+($44) onB.Clearly, w=dng=d(n4—ad(opr)) onA =dng =d(ng+d(o4d)) onB, sowisexact. The general inductive step issimilar. 4 282 Chapter 8 Weendthischapter with onemore calculation, which wewillneed inChap- terll. 17.THEOREM. For0<k<1wehave H¥(R") =0. PROOF. Theproof thatH9(R") =0islefttothereader. Letwbeak-form onR”with compact support, 0<k<n. Weknow that =dnforsome (k—1)-form nonR". LetBbeaclosed ballcontaining support @.Then onA=R"—Bwehave dy=0.Since Aisdiffeomorphic to O0=w=dn R"—{0}and k—1<m—1 wehave from Theorem 16that n=di forsome (k—2)-form AonA. Letf:R"+[0,1] beaC®function with f=0inaneighborhood ofBand Sf=1 onR"—2B, where 2Bdenotes theball oftwice theradius ofB.Then d(fA) makes sense onallofR”and @=dn=d(n—d(fd)); theform n—d(fA) clearly hascompact support contained in2B. Integration 283 PROBLEMS 1.TheRiemann integral versus theDarboux integral. Letf:[a,b] >Rbebounded. Forapartition P={fo<+++<tm}of[a,6], letm;=m;(f) betheinfoff on[f-1,ti] and define M;=Mj(f) similarly. Achoice forPisann-tuple &=(,...,8%) with &©[1,4]. Wedefine the“lower sum”, “upper sum”, and“Riemann sum” forapartition Pandchoice &by 0 LS,P)=Ym) “Wi—G1) ist USP) =OMAP) 41-1) i=l n SUP.8) =YSENu —4-1). iat Clearly Lf, P)<S(/, P,&) <U(f, P).WecallfDarboux integrable ifthe supofallL(/,P)equals theinfofallU(/,P);thissuporinfiscalled the Darboux integral offon[a,b]. WecallfRiemann integrable if HnSPE) exists thelimit iscalled theRiemann integral offon[a,5]. (@)Wecan define S(f, P,£) even iffisnotbounded. Show however, that fm SU;P.$)cannot existiffisunbounded. (b)Iffiscontinuous on[a,b], then fisRiemann and Darboux integrable on [a,6],andthetwointegrals areequal. (Use uniform continuity offon[a,5].)()IffisRiemann integrable on[a,5],thenfisDarboux integrable on[a,6] and thetwointegrals areequal. @)Letm<f<Mon[a,b]. Let P={59<+++<5m}andQ={ip<++< tn}betwopartitions of[a,b]. Foreach i=1,...,n, let er=length of(4-1,4] —sum oflengths ofall[se—1, 5]which arecontained in[¢;—1,4]. 4-1 4atht —— [se—{;Se]’s contained in[f—1, te] shaded lengths =addupto¢ 284 Chapter 8 Show that, ifM;denotes thesupoffon[t--1, 4],then a UCL,P)<U(L,Q) +YM -Mier ist n <U(S,Q)+(M -m)ei. ist There isasimilar result for lower sums. {e)ShowthatS77;¢>0as||Pll>0,anddeduce Darboux’s Theorem: lim U. =inf{U 5 ition ofJE, (fP)=inf{U(J,Q):Qapartition of[a,b)} yimEUsP)=supLf,2):@apartition of(a,5}. ()IffisDarboux integrable on[a,6],then fisRiemann integrable on[a,6]. (@)(Osgood’s Theorem). Letfandgbeintegrable on[a,b]. Show thatfor choices &,£’forP, m b li“(u—4-1)=; im, Fede —H-9)[x Flint:If\g|<Mon[a,6],then|f(E')gE)—SG)gE'l <MISE)—SEI. (hb)Show thatf,fdx+gdy,defined asalimit ofsums, equals bf(Pe)e"O+sO)" Wlat. la 2.Compute f,dO=fig,¢*40,wherec(t)=(cos2zrt,sin2xt)on[0,1]. 3.Fornaninteger, andR>0,letcr,n: [0,1] >R?—{0}bedefined by CRn(t) =(Roos nat, Rsin2nzt). (2)Show thatthere isasingular 2-cube c:[0,1]? +R?—{0}such thatcz,..— Ran =OC. (b)Ifc:[0,1] +R?—{0}isanycurve with c(0)=c(1), show thatthere is some nsuch that¢—¢},,isaboundary inR?—{0}. (©)Showthat»isunique. Itiscalledthewinding number of¢around 0. Integration 285 4.Letf:C>Cbeapolynomial, f(z) =2"-+aj2"~! +-+-+a,, where n>1. Define cp,s: [0,1] >Cbycap =fcr, (a)Show thatifRislarge enough, then cr, ~Ryn istheboundary ofachain inC—{0}.Hint; Note that crnn(t) =[r,s (OI", andwrite a a, S@)a2 (144-43), (b)Showthatf(z)=0forsomez€C(“Fundamental Theorem ofAlgebra”). Hint: Wof(z) #0forallzwith |z|<R,then cr,¢—co,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. Let AyCR"bethe setofallx€R”such that Wli o<x'<1, Yess. ist As Ar Ao Ay — + 0 1 Asingular n-simplex inMisaC® function c:Ay>M,and ann-chain isaformal sum ofsingular n-simplexes. Asbefore, letJ”:An>RMbethe inclusion map. Define 4):An-1 >Anby Bo(x)=([)~DIS xa.) d(x) =(x4. tO) 0<i<n, andforsingular 2-simplexes c,define 4;¢=¢0;.Then wedefine uwi de=D(-1)'He. i=0 (2)Describe geometrically theimages 0;(An—1) inAn. (b)Show that9?=0. 286 Chapler 8 (©)Showthatifw=fdx!A...Adx!A+++Adx"isan(v=1)-form onR", then fdo=foo ae al” (imitate theproofforcubes.) @Define fwforanyk-chain ¢inMandk-form wonM,andprove that fda=f5 c ae forany(k—1)-form o. 6.Every x€Ag4; canbewritten asx’,for0<¢<1,andx’€do(Ax). be Morcover, x’isunique except when ¢=0.Foranysingular k-simplex ¢:Ag> R",define ¢:Ags >R"by ¢ (x) =1-¢(x'). é Wethen define @forchains cintheobvious way. (a)Show that dc=0implies that ¢=92. (b)Letc:[0,1] +R?beaclosed curve. Show that¢isnottheboundary of anysum 6ofsingular 2-cubes. Hint: If80=Dyayer, what canbesaidabout Dai? {)Show that wedohave c=46+c’ where c!isdegenerate, thatis,c’([0, 1])is apoint. (@)If(0) =c2(0) and c1(1) =¢2(1), show that ¢—cz isaboundary, using cither simplexes orcubes. Integration 287 7.Letwbea}-form onamanifold M.Suppose thatJ.w=0forevery closed curve ¢inM.Show that isexact. Hint: Ifwedohave w=df,then forany curve ¢we have fe=seo~ feo). c 8.Amanifold Miscalled simply-connected ifMisconnected and ifevery smooth mapf:S!—Missmoothly contractible toapoint. [Actually, any space M(notnecessarily amanifold) iscalled simply-connected ifitisconnected andanycontinuous f:S'—» Mis(continuously) contractible toapoint. Itis nothard toshow thatforamanifold wemay insert “smooth” atboth places.] {a)IfMissmoothly contractible toapoint, then Missimply-connected. (b)S?isnotsimply-connected. ()S"issimply-connected forn>}.Hint: Show that asmooth f:S!'— S" isnot onto. (@)IfMissimply-connected andp€M,thenanysmooth mapf:S'>M issmoothly contractible top. (©)IfM=UUV where Uand Varesimply-connected open subsets with UNV connected, then Missimply-connected. (This gives another proof that S"issimply-connected forn>1.)Hint: Given f:S'>M,partition S!into afinite number ofintervals each ofwhich istaken into either UorV. ()IfMissimply-connected, then H1(M) =0.(See Problem 7.) 9.(a)LetUCR?beabounded open setsuch that R?~Uisnotcon- nected, Show that Uisnotsmoothly contractible toapoint. (Converse of Problem 7-24.) Hint: Ifpisinabounded component ofR?—U,show that there isacurve inUwhich “surrounds” p. (b)Abounded connected open setUCR?issmoothly contractible toapointifandonlyifitissimply-connected. (©)This isfalseforopen subsets ofR?. 288 Chapter 8 10. Let @beann-form onanoriented manifold M”. Let &and Wbetwo partitions ofunity byfunctions with compact support, andsuppose that >f$+lol<00.geo M 2)Thisimplies thatDyeo Sy¢-@converges absolutely. (b)Show that Lfgo=yy f¥-o-w,een M gedyeu M andshow thesame result with @replaced by||.(Note thatforeach ¢,there areonly finitely many ywhich arenon-zero onsupport ¢.) (©)Show thatDyew Jy¥lol<00,andthat Dfgo=y faoegeo M vey M Wedefine thiscommon sumtobefryo. (d)LetAnC(1, +1)beclosed sets. Letf:R>RbeaC™function with L4,f= (-1)"/n andsupport fCUj,An.Findtwopartitions ofunity© andWsuchthatDyce JaSax andyew Ja¥«fdxconverge absolutely todifferent values. 11.Following Problem 7-12, define geometric objects corresponding toodd relative tensors oftype(/)andweight w(wanyrealnumber). 12.(a)LetMbe{(x,)€R?:|(x,y)| <1},together withaproper portion ofitsboundary, and let»=xdy.Show that fde+fos M 3M even though both sides make sense, using Problem 10. (No computations needed—note that equality would hold ifwehad theentire boundary.) ())Similarly, findacounterexample toStokes’ Theorem when M=(0,1) and isa0-form whose support isnotcompact. (c)Examine apartition ofunityfor(0,1)byfunctions withcompact support to seejustwhy theproof ofStokes’ Theorem breaks down inthiscase. Integration 289 13.Suppose Misacompact orientable n-manifold (with noboundary), and6 isan(n—})-form onM.Show thatd@is0atsome point. 14. Let Mi,Mz©R"becompactn-dimensional manifolds-with-boundary with MzCM,—4Mj. Show that foranyclosed (”—1)-form wonM,, aM, foe=feamy aM2 15.Account forthefactor 1/|p!" inLemma 7(wehave r4(¥p) =(1/|pD%(p)s butthisonly accounts forafactor of1/|p|"~, since there aren—}vectors Vty0025Un=1)- 16.Usetheformula forr*dx? (Problem 4-1)tocompute r*o’. (Note that ro! =rtita =(ior)*o; themap ior: R"—{0}—>R”—{0}isjust r,considered asamap into R"—{0}.) 17.(a)Let M” and N™ beoriented manifolds, and letwand 7beann-form andanm-form with compact support, onMand N,respectively. Wewill orient MxNbyagreeing thatvj,...,Un,W1,...,Wm ispositively oriented in (MxN)(p,9) *Mp®Ngifv1,...,Un andwy,...,Wm arepositivelyoriented inMy,andNg,respectively. If2:MxN—MorNisprojection ontheit factor, show that fepmoamin= fio:fn MxN M N (b)Ifkh:MxN>RisC®,then fhmtoamtn= fgo, MxN M where a=fhe,ymh(p,-)=qrh(p,q). ()Every (m-+n)-form onMxNishm*wAm2*yforsomewandn. 290 Chapter 8 18.(a)Letp€R”—{0}. Letw,...,wn-2 €R", and letv€R", be(Ap), for some A€R.Show that r*o"(v,W1,..-;Wn=2) =0. (b)LetMCR” —{0}beacompact (1—1)-manifold-with-boundary which is theunion ofsegments ofraysthrough 0.Show thatfyr*o"=0. (©)LetM¢R"—{0} beacompact (n—1)-manifold-with-boundary which inter- sects every raythrough 0atmost once, and letC(M) ={Ap: p€M,A >0}. en c(myns? vy M Show that [re=f ret M c(M)ns? The latter integral isthemeasure ofthesolid angle subtended byM.Forthis reason weoften denote r*a’ byd@,. 19.Foralll(x,y,2) €R?except those with x=0,y=0,z€(—00, 0],we define (x,',=) tobetheangle between thepositive z-axis andtherayfrom 0 through (x,y,2) Integration 291 (@y,3) > ' @y) ‘ (@)$(x,y,2) =arctan(vx? +y?/z) (withappropriate conventions). (b)Ifv(p) =|p|,and @isconsidered asafunction onR?,6(x,y,2z) = arctan y/x,then(v,8,)isacoordinate system onthesetofallpoints (x,y,2) inR3except those with y=0,x€[0,00) orwith x=0,y=0,z€(—00,0]. (©)Ifvisalongitudinal unittangent vector onthesphere S(r) ofradius r, then d¢(v) =1.Ifwpoints along ameridian through p=(x,y,z) €S(r), 4)<P then ; 40(w,)===.G/x?+y" (d)If@and¢aretakentomeantherestrictions of@and¢to[certain portions of]S?,then o=hd6 add, where h:S?>Ris h(x,y,2)=—Vx? +y?(theminus signcomes from theorientation). (©)Conclude that o'=d(—cos $d6). 292 Chapter 8 (1)Letr2:R?—{0}+S'betheretraction, sothatd@=r2*i*o, fortheform o onR*. Show that rtd0 =dé. Ifx:R3+R?istheprojection, thentheform d@on[part of]R?isjustx*d0, fortheform 6on{part of]R?.Usethistoshow that r*d@ =do. (g)Also prove thisdirectly byusing theresult inpart (¢),and thefact that ra(Yp)=Ur¢p)/Ipl forvtangent toS(|pl). (h)Conclude that d@3 =r*a' =d(—cos(g or)d0) =d(—cos $6). ()Similarly, express d@, onR”—{0}interms ofd@,—1 onR"~! —{0}. 20. Prove that aconnected manifold isthe union U;UU2UU3U--- ,where theU;arecoordinate neighborhoods, with U;NU; #G,and thesequence is eventually outside ofanycompact set. 21.Letf:M"—N”beaproper map between oriented n-manifolds such thatfa:Mp>Nycp)isorientation preserving whenever pisaregular point. Show that ifNisconnected, then either fisonto N,orelseallpoints are critical points off. 22.(a)Show thatapolynomial mapf:C>C,given byf(z) =2"-+ai2""1 + ++++n, isproper (7>1). (b)Letf(z) =12"! +n—ais"?++++an—1.Showthatwehavef"(z)= lim[/(¢+w)—f(@)]/w, where wvaries overcomplex numbers. ws (c)Writef(x+iy)=u(x,y)+iv(x, y)forreal-valued functionsuandv.Show that S'(x+iy) =ae)+1209) =Rey ey)=h XY dyx,V)s Hint: Choose wtobeareal h,and then tobeih. (d)Conclude that If"+iy)?=detDf(x,y), Integration 293 where f’isdefined inpart (b),while Dfisthelinear transformation defined foranydifferentiable f:R?+R?. (©)Using Problem 21,give another proof oftheFundamental Theorem ofAk gebra. ©There isastillsimpler argument, notusing Problem 21(which relies onmany theorems ofthischapter). Showdirectly thatiff:M—Nisproper, thenthe number ofpoints inf~!(a) isalocally constant function onthesetofregular values off.Show that thissetisconnected forapolynomial f:C+C,and conclude that ftakes onallvalues. 23.LetM"-! CR"beacompact oriented manifold. Forp€R"—M,choose an(n—})-sphere &around psuch thatallpoints inside ©areinR”—M.Let tp:R"—{p} >©betheobvious retraction. Definethewinding numberw(p) ofMaround ptobethedegree ofrp|M. (a)Show that thisdefinition agrees with thatinProblem 3. (b)Show thatthisdefinition does notdepend onthechoice of©. (c)Show thatwisconstant inaneighborhood ofp.Conclude thatwiscon- stant oneach component ofR”—M. (d)Suppose Mcontains aportion Aofan(2—1)-plane. Letpandgbepoints A eoOa) 294 Chapler 8 close tothisplane, butonopposite sides. Show that w(g) =w(p) +1. Show thatrg|M ishomotopic toamap which equals rp|M onM—Aandwhich does nottake anypoint ofAonto thepoint xinthefigure.) (ce)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 11. 24.Let MandNbecompact n-manifolds, andletf,g: M—Nbesmoothly homotopic, byasmooth homotopy H:Mx[0,1] >N. (a)Letg€Nbearegular value ofH.Let#f~(q) denote the(finite) number ofpoints inf~!(q). Show that #f"@) =#g"q)_ (mod 2). Hint: H~!(q) isacompact J-manifold-with-boundary, Thenumber ofpoints initsboundary isclearly even. (This isoneplace where weusethestronger form ofSard’s Theorem.) (b)Show, more generally, thatthisresult holds solong asqisaregular value of both fand g. 25.Fortwomaps f,g: M>Nwewillwrite f~gtoindicate that fis smoothly homotopic tog. (a)Iffxg,then there isasmooth homotopy H’: Mx[0,1] >Nsuch that H"(p,t)= {(p) fortinaneighborhood of0, H'(p,t) =g(p) for tinaneighborhood ofJ (b)=isanequivalence relation. 26.If/issmoothly homotopic togbyasmooth homotopy Hsuchthatp+> H(p,t) isadifleomorphism foreach 1,wesaythat fissmoothly isotopic tog. (a)Being smoothly isotopic isanequivalence relation. (b)Let¢:R®+RbeaC®function which ispositive ontheinterior ofthe unitball, and0elsewhere. Forp€S"~", letH:RxR”>R"satisfy OH(t, x)Hes)=o(H(1,x)) PB H(0,x) =x. (Each solution isdefined forallt,byTheorem 5-6.) Show thateach x+H({t,x) isadiffeomorphism, which issmoothly isotopic totheidentity, andleaves all points outside theunit ball fixed. Integration 295 (©)Show thatbychoosing suitable pand¢wecanmake H(/,0) beanypoint inthe interior ofthe umit ball. (@IfMisconnected andp,q€M,then there isadiffeomorphism f:M> Msuchthatf(p)=qandfissmoothly isotopic totheidentity. (c)Usepart (@)togiveanalternate proofofStep3ofTheorem9. (f)IfMand Narecompact n-manifolds, and f:M—N,then forregular values g1,g2 €Nwehave #I7"\(g1) =#L-"G2)_ (mod 2) (where #/7!(q) isdefined inProblem 24). This number iscalled themod 2 degree off. (®)Byreplacing “degree” with “mod 2degree” inProblem 23,show that if MCR" isacompact (n—1)-manifold, then R"~Mhasatleast 2components. 27.Let{X'} beaC™family ofC°vector fields onacompact manifold M. {Tobemore precise, suppose XisaC®vector field onMx[0,1]; then X‘(p) willdenote 144.(p,1).) Fromtheaddendum toChapter 5,andtheargument which wasused intheproof ofTheorem 5-6, itfollows thatthere isaC™family {¢1) ofdiffeomorphisms ofM[notnecessarily a]-parameter group], with ¢o= identity, which isgenerated by{X‘}, ie.,foranyC™function f:M>Rwe have (X'f)(p) =limL(Gr+h(P)) =LOR)40 A Forafamily w,ofk-forms onMwedefine thek-form Oph ~OFOe (a)Show thatforn{t)=¢,*@, wehave me=or"(Ly wy+@). (b)Letwoand @benowhere zero n-forms onacompact oriented n-mani- fold M,and define @=(1—1)w+fay. Show thatthefamily ¢,ofdiffeomorphisms generated by{X'} satisfies gi*ay =o forallt ifand only if Lyt a=0—wy. 296 Chapter 8 ()Using Problem 7-18, show thatthisholds ifandonly if d(X' 10) =wp—a1. (d)Suppose thatfyy@o=fyy@1,80thatwp—@=ddforsome A.Show that there isadiffeomorphism fi:M—Msuchthatw=fi*o. 28.Letf:M*->R"andg:N!+R”beC®maps, where MandNare compact oriented manifolds, n=k+1+1, andf(M)M g(N) =9.Define apg: MxN>S"™' CR"—{0} by 89)~SP) apg(p.g)=r - =f. ta(Ps4)=(8G)—f(P)) ie=F Wedefine thelinking number offandgtobe £(f,8)=degarg, where MxNisoriented asinProblem 18. @)&fg) =(1He(e,f). (b)LetH:Mx[0,1]>R®andK:Nx[0,1]>R”besmooth homotopies with H(p,0)=f(p) —-K(y,0) =8) Hip.) =f(p) —-K@1) =&@) such that {H(p,0): peMJN{KG,1):¢¢€N} =O foreveryt. Show that th9) =eh8- ()Forf,g: S!>R?show that —)fp!ftAqu,v) naaaLLtaper Integration 297 where (u,v) =|g) -S| Yo (Pw (PYw Atu,v)=det{ (gv) (Yo) (e°Y@) F~O-FM) PM-LPY) #O)-Lw {thefactor 1/4 comes from thefactthatfs20’=42[Problem 9-14]). (a)Show that £(f,g) =0iffand gboth lieinthesame plane (first doit for(x,y)-plane), The next problem shows how todetermine £(f,g)without calculating 29.(a)For(a,b,c) €R?define ra _(=a) dyAdz~(y~b)dxAdz+(2-c)dxady(a,b,c) = [oea)+(y—b+ —cpp 7 Foracompact oriented 2-manifold-with-boundary M¢R?and(a,b,c) ¢M, let Q(a,b,¢)=i}460.5.M Let(a,b,c) and(a’,b’,c’) bepoints close top€M,onopposite sides ofM. Suppose (a,6,¢)isonthesame sideasavector wp€R3,—Myforwhich the wp, (a,b,c) 2 Hal,b.c) triple wp,(1)ps (v2)p ispositively oriented inR*,,when (v;)p, (v2)p ispositively oriented inM,. Show that lim—2(@,b,c) -Q@',b',c') =ar. (a,b,c) p (al,b'c!+p Hint: First show that ifM=8N,then Q(a,b,c) =—4m for(a,b,c) €N-M and Q(a,b,c) =0for(a,b,c) ¢N. 298 Chapter 8 (b)Letf:S!+R?beanimbedding such that{(S!) =3Mforsome com- pact oriented 2-manifold-with-boundary M.(AnMwith thisproperty always exists. SeeFort, Topology of3-Manifolds, pg,138.) Letg:S!+>R?andsuppose The figure ontheleftshows anon-orientable surface whose boundary isthe“trefoil” knot, ae ey WwW ¢ Uy ) butthesurface ontheright—including the hemisphere behind theplane ofthepaper— tsorientable. thatwheng(t)=p€Mwehavedg/dt ¢Mp.Letn+bethenumber ofinter- sections where dg/dt points inthesame direction asthevector wpofpart (a), and n~ the number ofother intersections. Show that nent=n =zfe*(dQ). 4x Jt (c)Show that an_.(Q—b)dz- (2-0)dy qb [5(Se an (2-0)dx —(x-a)dz=G,6,¢)Sil*(eee ee) HH HI=JS eo) aa (x-a)dy—(y—b)dx=z(a,b)=f(Sean geOe). ae ys!f Io.y2)P (@)Show that n=&(f,g). Compute &(f,g)forthepairs shown below. cD ab CP? S J Integration 299 30.(a)Letp,g €R"bedistinct. Choose open setsA,B CR”—{p,q} so thatAandBarediffeomorphic toR"—{0},andANBisdiffeomorphic toR". Using anargument similar tothat intheproof ofTheorem 16,show that se A CORES B H*(R" —{p,g}) =0for0<k<n—1}, andthatH"~"(R" —{p,g}) has dimension 2. (b)Find thedeRham cohomology vector spaces ofR”—Fwhere F¢R’is afinite set. 31.Wedefine thecupproduct U:H*(M) xH'(M) —H*+!(M) by [o]v[n]=[oA n)- (a)Showthatuiswell-defined, i.e.,@A7isexactifwisexactand7isclosed.(b)Showthatvisbilinear. (c)Ifw¢H*(M) andB¢H'(M), thenaUB=(-1)"B Ue. (@)Iff:M>N,andw€H*(N), B€H'(N), then SHA p)= frau f*B. (e)Thecross-product x:H*(M) xH'(N) >H*+"(M xN)isdefined by (o]x[n]=[tuto xy"). Show that xiswell-defined, and that axp=ry*aunn'p. 300 Chapter 8 (f)IfA:M— MxMisthe“diagonal map”, given byA(p) =(p,p),show that aUB=AtaxB). 32. On the n-dimensional torus T"=S'x---xS} ox niimes letd6’denote 1;*d6, where x;:T”>S!isprojection ontheifactor. (a)Show thatalld6#a...4d6"representdifferentelementsofH*(7"),by finding submanifolds of7”over which they have different integrals. Hence dimH*(T") >(j).Equality isproved intheProblems forChapter 11. (b)Show thatevery map f:S”—7”hasdegree 0.Hint: UseProblem 25. CHAPTER 9 RIEMANNIAN METRICS ikprevious chapters wehaveexploited nearlyeveryconstruction associatedwith vector spaces, and thus with bundles, butthere hasbeen one notable exception—we have never mentioned inner products. The time hasnow come tomakeuseofthisneglected tool. Aninnerproduct onavector space VoverafieldFisabilinear function from VxVtoF,denoted by(v,w)>(v,w),which issymmetric, (v,w)=(w,»), and non-degenerate: ifv#0,then there issome w#0such that (w,v) #0. Forus,thefield Fwillalways beR. Foreach rwith 0<r<1, wecandefine aninner product (,)ronR”by r n (a,b), =Soa‘ —SPald; isl ter4l thisisnon-degenerate because ifa#0,then ae (@,...:4"),@),...,a",-a"1,...,-a")), =Va’)? >0. i=l Inparticular, forr=1weobtain the“usual inner product”, {,)onR", a (a,b) =Da‘! isl Forthisinnerproduct wehave(a,a)>0foranya#0.Ingeneral, asymmetric bilinear function (,)iscalled positive definite if (v,v)>0 —forallu 40. Apositive definite bilinear function (,)isclearly non-degenerate, andconse- quently aninner product. . 30) 302 Chapter 9 Notice thataninner product (,)onVisanelement ofT(V), soif J:W—Visalineartransformation, then/*(,)isasymmetric bilinear function onW.This symmetric bilinear function may bedegenerate even iff isone-one, eg.,if(,)isdefined onR?by (a,b) =a'b! —ab?, andf:R> R?is F(a)=(a,a). However, {*( ,)isclearly non-degenerate iffisanisomorphism ontoV.Also, if(,)ispositive definite, then f*( ,)ispositive definite ifand only iffis one-one. Foranybasis v1,...,Un ofV,with corresponding dual basis v*,..., 0", we can write » C25 YSgiv@v%). jel Inthisexpression. Sis=(Yi04), sosymmetry of(,)implies that thematrix (gz) issymmetric, 8ij =Bit The matrix (giy) hasanother important imerpretation. Since aninner product (,)islinearinthesecond argument, wecandefine alinearfunctional dy€V*, foreach v€V,by dv(w)=(v,w). Since (,)islinear inthefirstargument, themap v+>¢yisalinear transfor- mation from V10V*.Non-degeneracy of(,)implies that ¢y40ifv#0. Thus, ifVisfinite dimensional, aninner product (_,)gives usanisomorphism a:V>V*, with (v,w)=a{v){w). Clearly, thematrix (g;j)isjustthematrix ofa:V>V*withrespect tothe bases {v;} forVand {v*;} forV*.Thus, non-degeneracy of{,)isequivalen! tothecondition that (gij) isnon-singular, det(gij) #0. Positive definiteness of(,)corresponds tothemore complicated condition thatthematrix (g;j) be“positive definite”, meaning that " Vgiaia! >0 forallay,...,ay withatleastonea!#0. ist Riemannian Metrics 303 Given anypositive definite inner product (,)onVwedefine theassociated norm ||||by lull=Vv, v) (the positive square root istobetaken). InR"wedenote thenorm corresponding to¢,)simply by A lal=Vla,a) =|SO’? . i=] ‘The principal properties of|||arethefollowing 1.THEOREM. For allv,w€Vwehave ()llavll =lal Hell. (2)|{v,w)| <[lvl]-wl],withequality ifandonlyifvandwarelinearly dependent (Schwarz inequality). (3)|lv+wll<lull +llw]] (Triangle inequality). PROOF. (I)iswivial. (2)Ifvand warelinearly dependent, equality clearly holds. Ifnot, then 04 Av—wforallA€R,so 0<|JAv—wll?=(Av—w, Av—w) =?|jul]? —2A(v,w)+[wi Sotheright side isaquadratic equation inAwith noreal solution, and its discriminant must benegative. Thus 4(v,w)?—4llv|Pwl)?<0. (3) lv+wl?=(vt+u,v+w) =[oP+lw? +2(v, w) Sel? +tol? +2Ioll wll by@) =(lull +llwll)?. The function ||||hascertain unpleasant properties—for example, thefunc- tion ||onR”isnotdifferentiable at0€R”~which donotarise forthefunction ||?.Thislatterfunction isa“quadratic function” onV—intermsofabasis {u;}forVitcanbewritten asa“homogeneous polynomial ofdegree 2”inthe components, » A » [drew|=Veaia’.i= i,j) 304 Chapter 9 More succinctly, n WP=SOgases vy. ijal Aninvariant definition ofaquadratic function canbeobtained (Problem ])from thefollowing observation. 2.THEOREM (POLARIZATION IDENTITY). If|]||isthenorm associ- ated toaninner product (,)onV,then Q)(vw) =3[flv+wil?—ell?—[Jw] (2)(v,w)=4[llv+wll?—Jv—wi]. PROOF. Compute. ¢ Theorem 2shows that two inner products which induce thesame norm are themselves equal. Similarly, iff:V>Visnorm preserving, thatis,||/(v)l = |v]forallv€V,thenfisalsoinnerproduct preserving, thatis,(f(v), f(w))= (vw) forallv,w€V. Wewillnow seethat, “up toisomorphism”, there isonly onepositive definite inner product. 3.THEOREM. If(,)isapositive definite innerproduct onann-dimen- sional vector space V,then there isabasis vj,...,U, forVsuch that (v;,vj)= 8;j.(Such abasis iscalled orthonormal with respect to(,).)Consequently; there isanisomorphism f:R”—Vsuch that (a,b) =(f(@), f@)), a,b R". Inother words, f(,)=4) PROOF, Letwi,...,Wn beanybasis forV.Weobtain thedesired basis by applying the“Gram-Schmidt orthonormalization process” tothisbasis: Since w,#0,wecandefine n=ha? and clearly ||v||=1.Suppose that wehave constructed v},...,v% sothat (viv) =67 Isijsk Riemannian Metrics 305 and span v),..., Uk=Span wy,..., Wk. Then wg41 islinearly independent ofv1,...,vg. Let Why =Wher —(V1,Vega)VI —++~(VK,Ve+IDUE FO. Itiseasy toseethat (What) =O F=1,...,k. Sowecandefine , Vest Upqy=—EtL, eSTh ll and continue inductively. Apositive definite inner product (,)onVissometimes called aEuclidean metric onV.This isbecause weobtain ametric ponVbydefining pv, w)=|lv—wll. The “triangle inequality” (Theorem 1(3))shows thatthisisindeed ametric. We also call ||v||theJength ofv. Wehave only onemore algebraic trick toplay. Recal] thataninner product (,)onVprovides anisomorphism a:V>V*with @(v)(w) =(v,w). Using thenatural isomorphism i:V>V**, defined by i(v)A) =AC), weobtain anisomorphism an} i B:V*—> V—> (V*)". Wecannow useftodefine abilinear function (,)*onV*by (A,w)*=BAW) =foA)(W) =w(@™"(Q)). Now, thesymmetry of(,)canbeexpressed bytheequation a(v)(w) =a(w)(v). 306 Chapter 9 Letting a(v)=A, aw) =p, this can bewritten Ao "u)) =ule"), which shows that (,)*isalso symmetric, (HsA)*=(A,my”. Consequently (,)*isaninner product onthedual space V*(infact, theone which produces 6). . Toseewhat this al]means, choose abasis {v;} forV,Jet{v*;} bethedual basis forV*,and Jet a (= DOaur @v%). jel Then (gi;)isthematrix of=a:V-—>V* withrespect to{uj}and{v*)} so (gij)7? isthematrix ofa7": V*>V_—withrespect to{v";} and {vj} so. (gij)~' isthematrix of|6:V*>V**withrespect to{v*;}and{vy}. Thus, ifweletg'/betheentries oftheinverse matrix, (g!/) =(gij)~}, sothat D DYaiken; =8, kat then a : (=) se ety ijet a =0giv@vy, ifweconsider vj¢V™. ijal One can check directly (Problem 9),without theinvariant definition, that this equation defines (,)*independently ofthechoiceofbasis. Riemannian Metrics 307 Notice that if(,)ispositive definite, sothat a(v)\v) >0 forv #0, then, letting a(v) =A,wehave A@7A)) =BAJA >0 ford £0, so(,)*isalso positive definite. This canalso bechecked directly from the definition interms ofabasis. Inthepositive definite case, thesimplest way to describe (,)*isasfollows: The basis v*1,...,0*» ofV*isorthonormal with respect to(,)*ifand only ifv1,..., vnisorthonormal with respect to(,). Similar tricks can beused (Problem 4)toproduce aninner product onall thevector spaces T*(V), T(V) =T*(V*), andQ*(V). However, weare interested inonly onecase, which wewil]notdescribe inacompletely invariant way. The vector space 2"(V) is1-dimensional, sotoproduce aninner product onit,weneed only describe which twoelements, wand—w, willhave length 1. Let vj,...,U, and wy,...,W, betwo bases ofVwhich areorthonormal with respect to(,).If'we write n w=Yayiv;, ja then a a a 55;=(wi,Wj)=(Yano Dra)=>»OK5041;(VkU1)k=l 11 kl=1 0 =Varian). k=l Sothetranspose matrix A‘ofA=(aij) satisfies A-At=J,which implies that detA=+1.Itfollows from Theorem 7-5thatforanyw€Q"(V) wehave @(Y1,...,Un) =Eo(u1,..., Wn). Itclearly follows that VAs Av, =tw Ae Aw y. Wehave thus distinguished twoelements ofQ”(V); they areboth oftheform v*)A---Av*, for{u;} anorthonormal basis ofV.Wewill cal] these two elements 308 Chapter 9 theelements ofnorm 1in2”(V). Ifwealso have anorientation 2,then wecan further distinguish theonewhich ispositive when applied toany(v1,...5 Un) with [v1,...,Un] =“; wewillcal}itthepositive element ofnorm |inQ”(V). Toexpress theelements ofnorm|intermsofan arbitrary basis w1,..., Wn, wechoose anorthonormal basis v;,...,U, and write n w=airy. j=l Problem 7-9implies that det(ajj) wi)Ass Aw,=UTArAUy. Ifwe write n (.)= Voayvtieuv', ijel then a 2 Bij=(Wi,Wy)=(Seu, Yoav) kat 1 a =Varian, k=l soifA=(aij). then det(giz) =det(A'- A)=(detA). Inparticular, det(g;;) isalways positive. Consequently, theelements ofnorm 1 inQ"(V) are Vdet(giy) w"1Av Aw"n Bij=(Wi,Wy). Wenowapplyournewtooltovector bundles, If&=2:E>Bisavector bundle, wedefine aRiemannian metric on&tobeafunction (_,)which assigns toeachp€Bapositive definite innerproduct (,)ponx~!(p), andwhich iscontinuous inthesense thatforanytwocontinuous sections 5,52: B>E, the function (51,52) =pt (si(p), 2(P))p isalso continuous. If&isaC® vector bundle over aC® manifold wecan also speak ofC® Riemannian metrics. Riemannian Metrics 309 [Another approach tothedefinition canbegiven. LetExc(V) bethesetofall positive definite inner products onV.Ifwereplace each 2~"(p) byEuc(x—"(p)), and let Euc(§)=(_)Bul"(p)), peB then aRiemannian metric on&canbedefined tobeasection ofEuc(). The only problem isthatEvc(V) isnotavector space; thenewobject Euc() thatwe obtain isnotavector bundle atall,butaninstance ofamore general structure, afibre bundle.) 4.THEOREM. Let§=2:E>Mbea [C®] k-plane bundle over aC® manifold M.Then there isa[C°°] Riemannian metric on&. PROOF. There isanopen locally finite cover @ofMbysetsUforwhich there exists [C®] trivializations ty:n—(U) >UxR, OnUxR*,there isanobvious Riemannian mewic, ((p,4), (p:5))p =(a,b). Forv,w€7!(p), define (v,w)p =(tu(),tu(w))p- Then (,)¥isa[C%] Riemannian metric for&|U.Let{gu} beapartition of unity subordinate to©.Wedefine (,)by (v,w)p =D>du(p)v, w)y vwexp). UcO Then (_,)iscontinuous [C®] andeach (,)pisasymmetric bilinear function onx7(p). Toshow thatitispositive definite, note that (v,0)p =D>du(p)(v, v8; UcO eachdy(p)(v,v)¥ >0,andforsomeUstrictinequality holds.4 {The same argument shows that anyvector bundle over aparacompact space hasaRiemannian metric] Notice thattheargument inthefinal stepwould notwork ifwehadmerely picked non-degenerate inner products (,)¥.Infact(Problem 7),there isno (,)onTS?which gives asymmetric bilinear function oneach S,which is notpositive definite ornegative definite butisstillnon-degenerate. 310 Chapter 9 Asanapplication ofTheorem 4,wesettle some questions which have tillnow remained unanswered. 9.COROLLARY. If=a:E>Misak-plane bundle, then&~&*. PROOF. Let (,)beaRiemannian metric for&Then foreach p€M,we have anisomorphism Oy:(p) >[pI defined by ap(v)(w) =(v,w)p vwen'(p), Continuity of(,)implies thattheunion ofallayisahomeomorphism from E toBE’=Upemlr(p)I*. 6.COROLLARY. If§=2:E>Misa|-plane bundle, then&istrivialifandonlyif&isorientable. PROOF. The “only if”part istrivial. If€hasanorientation yuand (,)isa Riemannian metric onMthen there isaunique s(p) ex"(p) with (s(p), $(P))p =1, [s(p)] =Hp. Clearly sisasection; wethen define anequivalence f: E>MxRby S(As(p)) =(p,A). ALTERNATIVE PROOF. Weknow (see thediscussion after Theorem 7-9) that if&isorientable, thenthereisanowhere 0section of MEHR, sothat &*istrivial. But &~&*.¢ Allthese considerations take onspecial significance when ourbundle isthe tangent bundle TM ofaC®manifoldM.Inthiscase,aC°Riemannian metric (,)forTM, which gives apositive definite inner product (,)pon Riemannian Metrics 311 each Mp, iscalled aRiemannian metric onM.If(x,U)isacoordinate system onM,then onUwecan write our Riemannian metric (,)as ; (.)= 0gydx!@dx, ij=) where theC® functions gijsatisfy gij=gyi,since (,)issymmetric, and det(gij) >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 if/fisanimmersion (fp isone-one for allp€N). The Riemannian metric (,)*,which (,)induces onthedual bundle T*M, isacontravariant tensor oforder 2,and wecan write itas n ;9 a =J—_@a. (=Vosi sea ‘j=l Our discussion ofinner products induced onV*shows that foreach p,the matrix (g4/(p)) istheinverse ofthematrix (g;;(p)); thus ul ;Ydgna”=§. kal Similarly, foreach p€MtheRiemannian metric (,)onMdetermines twoclements ofQ"(Mp), theelements ofnorm 1.Wehave seen thatthey can bewritten +Vdet(gij(p)) dx'(p) A+++ Adx"(p). IfMhasanorientation s2,then jpallows ustopick outthepositive element of norm 1,andweobtain ann-form onM;ifx:U>R®isorientation preserving, then onUthis form can bewritten Vadet(gij) dx!A-..A dx". EvenifMisnotorientable, weobtain a“volume element” onM,asdefined inChapter 8;inacoordinate system (x,U)itcanbewritten as Vdet(gij) |dx!A-++adx"). This volume element isdenoted bydV, even though itisusually notdof anything (evenwhenMisorientable anditcanbeconsidered tobeann-form), 312 Chapter 9 and iscalled thevolume element determined by(,).Wecan then define the volume ofMas [aM This certainly makes sense ifMiscompact, andinthenon-compact case (see Problem 8-10) iteither converges toadefinite number, orbecomes arbitrarily Jarge over compact subsets ofM,inwhich case wesaythatMhas“infinite volume”. IfMisann-dimensional manifold (-with-boundary) inR”,with the“usual Riemannian metric” n (.)= lax‘@ax', i=l then gij=6i,so dV=|dx'a.--A dx", 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 YO=F, Myon andcantherefore use(,)todefine their length dy dy dy dy dy := (24 =(2.2), tobeprecise}. |dt|MFdt dt"atfy)?“7Precise Wecan then define thelength ofyfrom atob, big. 6 Loy)=fFaldt(-/l'on“).a t a Ifyismerely piecewise smooth, meaning thatthere isapartition a=f9<--- < tn=bof[a,b] such that yissmooth oneach [f:-1,1i] (with possibly different Riemannian Metrics 313 Jefi-andright-hand derivatives at11,...,t-1), wecandefine thelength ofyby Lay)=YoLi(vl[e-14). i= Whenever there isnopossibility ofmisunderstanding wewilldenote Lsimply byL.Aliteargument shows(Problem 15)thatforpiecewise smooth curves in R”,with theusual Riemannian metric uw: ;>»dx!@dx', i=l thisdefinition agrees with thedefinition oflength astheleast upper bound of thelengthsofinscribed polygonal curves. Wecanalso define afunction s:[a,6] >R,the“arclength function ofy” by onia1)=Lily) = =| dt.s)=Lay)[|a| Naturally, dy “(= |—]. (*) s|dt| Consequently dy/dt hasconstant length 1precisely when s(t)=¢+constant, thus precisely when s(t) =f—a.Then b—a=s(b) =LEY). Wecanreparameterize ytobeacurve on[0,5 ~a]bydefining VD =v(t—a). Forthenew curve 7wehave news(t)=Lo(V) =Lot(y)=olds(@+a) ~olds(a) 5 Ifysatisfies s(t)=¢wesaythatyisparameterized byarclength (and then often usesinstead of¢todenote theargument inthedomain ofy). 314 Chapter 9 Classically, thenorm ||||onMwasdenoted byds.(This makes some sort ofsense even inmodern notation; equation (*)says that foreach curve yand corresponding s:[a,b] +Rwehave Idsi =y*(l Wt) on[a,5].) Consequently, inclassical books oneusually sees theequation ; ds?=D>gidxidx’. j=) Nowadays, thisissometimes interpreted asbeing theequivalent ofthemodern equation (,)=D71gij4x! @dx/,butwhatitalways actually meant was n WP=YOgizax'ax, ijt Thesymbol dx‘dx/ appearing hereisnotaclassical substitute fordx!@dx/— thevalue (dxidx/)(p) ofdx‘dx/ atpshould notbeinterpreted asabilinear function atall,butasthequadratic function vesdx!(p)(v)-dx4(p)) veMy, andwewould usethesame symbol today. The classical wayofindicating dx'@dx!wasvery strange: onewrote w ; DYgiydx'6x/ where dxandxareindependent infinitesimals. j= (Classically, theRiemannian metric wasnotafunction ontangent vectors, but theinner product oftwo “infinitely small displacements” dxand 5x.) Consider now aRiemannian metric (,)onaconnected manifold M. If Pq €Mareanytwopoints, then there isatleast one piecewise smooth curve y:[a,b] Mfromptoq(thereisevenasmooth curvefrompto4).Define (p,q) =inf{L(y): yapiecewise smooth curve from ptog}. Itisclear that d(p,qg) =0and d(p, p)=0.Moreover, ifr€Misathird point, then forany¢>0,wecanchoose piecewise smooth curves y:[a,b] >Mfrom ptog with L(4)—d(p,q) <e Y2:[b,¢] >Mfrom gqtorwith L(y.) —d(g,r) <& Riemannian Metrics 315 Ifwedefine y:[a,c] +Mtobey,on[a,b] and ypon[b,c], then yisa piecewise smooth curve from ptorand L(y) =L(y) +L02) <(p,q) +4(q,7) +2. Since this istrue foralle>0,itfollows that Ap,r) Sa(p,q) +dq"). [Ifwedidnotallowpiecewise smooth curves, therewould bedifficulties infitting together y,and y2,butdwould stillturn outtobethesame (Problem 17).] The function d:MxM=Rhasallproperties forametric, except thatitisnotso clear that d(p,q) >0forp#q.This ismade clear inthefollowing. 7,THEOREM. Thefunction d:MxM—Risametric onM,andif p:MxM=Ristheoriginal metric onM(which makes Mamanifold), then(M,d)ishomeomorphic to(M,p). PROOF. Both parts ofthetheorem areobviously consequences ofthefollowing 7’,LEMMA. LetUbeanopenneighborhood oftheclosedballB={p€R": {pl<1}, let(,)ebethe “Euclidean” orusual Riemannian metric onU, a (de= Dodx!@dx’, ia and Jet(,)beanyother Riemannian metric. Let||=||lleand |Ilbethe corresponding norms. Then there arenumbers m,M>0suchthat m-list<sM-t] onB, andconsequently foranycurve y:[a,b] >Bwehave mLe(y) SL(y) sMLe(y). PROOF. Define G:BxS"-} +Rby . G(p,a) =llapllp. ThenGiscontinuous andpositive. SinceBxS”—'iscompact therearenum- bers m,M>0suchthat m<G<M —onBxs". Now ifp€Band0#by€R",, leta€S"! bea=b/|b|. Then mlb] <1b1G(p,a) <MIbI; since 101G(p,a)=[Btllapllp=Mlbla)plly=lolly, thisgives thedesired inequality (which clearly alsoholds forb=0).# 316 Chapter 9 Notice that thedistance d(p,q) defined byourmetric need notbeL(y) for anypiecewise smooth curve from ptoq.Forexample, themanifold Mmight beR?~{0},andgmightbe—p.Ofcourse, ifd(p,g) =L(y)forsomey, |p —pe thenyisclearly ashortest piecewise smooth curvefromptog(there mightbe more than oneshortest curve, ¢.g,, thetwosemi-circles between thepoints p and—ponS'). Inorder toinvestigate thequestion ofshortest curves more thoroughly, we have toemploy techniques from the“calculus ofvariations”. Asanintroduction tosuchtechniques, weconsider firstasimpleproblem ofthissort.Suppose we aregivena(suitably differentiable) function F:RxRxR-R, Weseek, among allfunctions f:[a,b] >Rwith f(a) =a’ andf(b) =d'one Pafo a a b which willmaximize (orminimize) thequantity b[Feso.srona. la Forexample, if Fux, y)=V1+y', Riemannian Metrics 317 then wearelooking forafunction fon[a,b] which makes thecurve 1+ (t,f(1)between (a,a’)and(6,6’)ofshortest length b{Ji+Lorat. a Asasecond example, if F(x, y)=20xV1 4y?, then wearetrying tominimize thearea ofthesurface obtained byrevolving thegraph offaround thex-axis, which isgiven (Problem 12)by anfsay+tPor ar.‘ J Toapproach this sort ofproblem werecall first themethods used forsolv- ingthemuch simpler problem ofdetermining themaximum orminimum of afunction f:R>R.Tosolve thisproblem, weexamine thecritical points off,ie.,those points xforwhich f’(x) =0.Acritical point isnotnecessarily amaximum orminimum, oreven alocal maximum orminimum, butcritical points arctheonly candidates formaxima orminima iffiseverywhere difler- entiable, Similarly, forafunction {:R?>Rweconsider points (x,y)€R? for which (*) Dif(x,y) =Dof(x,y) =0. (x, 3) This isthesame assaying that thecurves te fix+ny) te fytr) 318 Chapter 9 have derivative 0at0.Wemight trytogetmore information byconsidering the condition 0=(f oc) forevery curve c:(—é,¢) >R?with c(0)=(x,y),butitturns outthatthese conditions follow from (#),because ofthechain rule. Tofind maxima and minima for n=[Paseo sonat la wewishtoproceed inananalogous way,byconsidering curvesinthesetofall functions f:[a,b] +R,This canbedone byconsidering a“variation” off, that is,afunction a:(-¢,£) x[a,b] >R such that a(0,1) =f(). The functions ¢++a@(u,t) arethen afamily offunctions on(—e,e) which pass through fforu=0.Wewilldenote thisfunction by&(u). Thus @isa function from (—é,€)tothesetoffunctions f:[a,b]>R.Ifeach &(u)satisfies &(u)(a) =a’,&(u)(b) =8’,inother words if . Bt Ff @(u,a)=a’ a Uy (u,b) =o a b forallu€(—e,€), then wecall@avariation offkeeping endpoints fixed. Foravariation @wenow compute as Doo)| =ral[F(naenn, 50.9)at du yao Ax0 Ja ot Riemannian Metrics 319 ora aa =I[HP (exon een)at oTaaar , =[[Zooxesosro aa ar , +HOO OSOSO)] a. Since 0?a/dudt =d*a/dtdu, wecanapply integration byparts tothesecond term intheintegrand, thus obtaining aJ@u))| [Beear , (*)du[.=],9g|aSOLO) d (oF ,71(Fasos |at aa OF ral?$yOGOLOSol. Forvariations &keeping endpoints fixed, thesecond term is0,and weobtain dJ(&(u)) >Oa oF , to) nLOo[FUOLO) d (aF ;7(Fosos |dt. Inclassical treatments ofthecalculus ofvariations, thevariations wwere taken tobeofthespecial form a(u,t) =f(t) +unr), forsome 1:[a,b] +Rwith n(a) =(6) =0.Then weobtain as@(u))| _of?[z 1d(= ,) LEO) =[ol Feso.ro-F (Foso.sro)| Thefinalresultis,ofcourse,essentially thesame.Thederivative Z|0J(&(u)) iscalled the“first variation” ofJandisdenoted classically by +TardF a=ffeo dt. 320 Chapter 9 Asisusual inclassical notation, thearguments offunctions areeither putin indiscriminately orleftoutindiscriminately—in thiscase, notonly arethear- guments ¢and (¢,f(¢), f’(1)) omitted (resulting inthedisappearance ofthe function fforwhich wearesolving), butthedependence of6Jon@isnot indicated (which canmake things pretty confusing). Iffistomaximize orminimize J,then 6J() must be0forevery variation a offkeeping endpoints fixed, Asinthecase ofI-dimensional calculus, there is noreason toexpect that thecondition J(a) =0forallawillimply that /fis even aJocal maximum orminimum forJ,andweemphasize thisbyintroducing adefinition. Wecallfacritical point ofJ(oranextremal forJ)if5J(@) =0 forallvariations aoffkeeping endpoints fixed. The particular form (##)into which wehave put6Jnow allows ustodeduce animportant condition. 8.THEOREM (EULER’S EQUATION). The C?function fisacritical point ofJifandonly iff satisfies OF d (aF 1 Fnsonsoy= +(FUsSOL@) =o. PROOF. Clearly fmust make theintegral in(«*)vanish forevery da j=1)=77.0 which vanishes ataand6.Sothetheorem isaconsequence ofthefollowing simple 8’.LEMMA. Ifacontinuous function g:[a,b]>Rsatisfies 6froewar=o forevery C®function 7on[a,b] with n(a) =n(6) =0,then g=0. PROOF. Choose ntobe$gwhere¢ispositive on(a,b)and$(a)=$(b)=0. Asanexample, consider thecasewhere F(t,x,y) =V1+y?.The Euler cquation is ond(TAO)) avisor) Riemannian Metrics 321 CY Nig[yef?ptf. o2 (—) , hence ot sf" —fff +fF which implies that {”=0,sofislinear. Notice that wewould have obtained the same result ifwehad considered the caseF(t,x, y)=1+y’,forthentheEuler equation issimply d opt==(2 : 0=Fes) This isanalogous tothesituation in1-dimensional calculus, where thecritical points of\/farethesame asthose off,since f Wy- af Forthecase ofthesurface ofrevolution, where F(t,x,y)=xv1+)’, the Euler equation is haipope —£{L0£0 0=/1 "“®)P-—|—=—— }; F+IPOP—3 —1+[f'(] thisleads totheequation 1+f?— ff"=0, which wewill also write inthe classical form dyYd?y1+(—) -y5 ==0.*(2)2a79 Tosolve this, weuseone oftheXostandard tricks (leaving justification ofthe details tothereader). Welet dysyst.py dx Then @y_dp_dpdy_dp dx dx dydxay’ 322 Chapter 9 soourequation becomes ap 1+ p?—yp =0, +PPa 0, Pp 1—.adp=~dy,Tp PaSe 1 5log( +p?)=logy+constant y=constant -V1+p? @poeavey=1dx dy= gy vey? -1 andthus (seeProblem 20forthedefinition andproperties ofthe“hyperbolic cosine” function cosh and itsinverse) cosh7!cyBas © Replacing ¢by1/c,wewrite thisas (*) y=ccosh(+*).¢ The graph of ex+e7* coshx=————_2 isshown below; itissymmetric about they-axis, decreasing forx<0,and increasing forx>0. .coslt Riemannian Metrics 323 Sooursurface must look liketheonedrawn below. Itis,bytheway, not trivial todecide whether there aveconstants kand¢which willmake thegraph of(*)pass through (a,a’) and (b,5").Problem 2)investigates thespecial case where a’=b’. Itiseasy togeneralize these considerations tothecase where f:[a,b] >R” and b en)-fFSO SO)dt forF:RXR"XR">R. fa Inthiscase weconsider a:(—e,¢) x[a,b] >R"with &(0) =f,andcompute that seve)EON) _fyon |Zaroro) éduyaoJaKfBu Lax” d (aF7(aeSO),ro)dt “ae! aF i 1 +haaonSeso.re| . =a Thus, anycritical point fofJmust satisfy the equations ar d (arFOSOLO~F (FOSO.L0)) =0. Wearenow going toapply these results totheproblem offinding shortest paths inamanifold M.Ify:[a,b] >Misapiecewise smooth curve, with y(a) =pandy(b) =q,wedefine avariation ofytobeafunction a:(-8,£) x[a,b] >M 324 Chapter 9 forsome ¢>0,such that 0)¢@.) =yO, (2)there isapartition @=1<< +++<ty=bof[a,b] sothat@isC® oneach strip (—e,€)x[4-1, 47]. Wecall«avariation ofykeeping endpoints fixed if 3} ua)= 8) a= Pes ally(8,2).a(u,b)=¢g 2 SSS Asbefore, welet&(u) bethepath ++a@(u,). Wewould like tofind which paths ysatisfy ae)—<— =0 du |yao forallvariations @keeping endpoints fixed. However, wewilltake ahint from ourfirst example and first find thecritica] points forthe“energy” 1feiayy? 1£2fdya) E(y== =| dt=- —,—)d, ”sf\Fl al(2.2 which hasamuch nicer integrand; afterwards wewillconsider therelation between thetwointegrals. Wecan assume that cach y|[ts-1,4i] liesinsome coordinate system (x,U) (otherwise wejustrefine thepartition). If(w,)isthestandard coordinate system in(—e,£) x[a,5]wewrite da a5—(U,t)=Oezl ou Ou|(4,2) de a(ut)=a,al. ar*(3(ust), Riemannian Metrics 325 Thenda/8t(u,t) isthetangent vector attime¢tothecurve&(u).Ifweadopt the abbreviations du)=xewo), 0 =x'(¥O) =O), then aaa! a dy_Qdy'a| Fe?ace ee So 1%[dydy E(yItt-1,4)) =f.(2.2)at li < dyidy/= ij ————di.;I7Isuv Ifweusethecoordinate system xtoidentify Uwith R",andconsider thegij asfunctions onR",then weareconsidering G[reo.viond tir where re a F Ps eX pt ptF(x,y)=52008)» yi. Then ar dy\ _14agi dy!dy)5(ro.4)=3Og a and ar dy “ dy"at(ro,+)=Lang so d(aF dy\\_< Py Agr dy!dy"5(#(v0.4) =Yonge +LeOOGea 326 Chapter 9 Inorder toobtain asymmetrical Jooking result, wenote that alittle index juggling gives JoBetede!dy”Bandedy)Seydy!dy! Oxidtdt 4axidtdt~4axididt rial ijl ij=) so n a i gyi n ;agidy)dy’_1agadyidy! 1agindy!dy! >o8tray"ayLt>o8itdyay"ye>28itdyay"—axi dt dt 2 Oxi dt dt 2 4dx!dtdt nial ij ‘j=l From (##s) wenow obtain | du _ iag! n ay" =-[09] Lenn Ge “tet ral “1(agit agi 9gij dy!dyi+h5(fow+ aYO)—How) eeat "aat 0 dy"|=, vO]. +au(odaOO[.. Remember that yisonly piecewise C°. Let 1 ¥(i-1) dy.Sur)=righthandtangentvectorofyat4 at) ary=lefihandtangentvectorofyattj. 204dt v(t41) Notice thatthefinal sum intheabove formula issimply da dy _ oa dy(Fon, Le)(S040 dt(-1")). Toabbreviate theintegral somewhat weintroduce thesymbols 71 fog,gy OgiWN=>(+oxox) Riemannian Metrics 327 These depend onthecoordinate system, buttheintegral 1 da! LJ ay LJ dyédyi- a ay ijt ao“[a[ZaoGztWMOGa|te= ra ijl which appears inourresult, clearly cannot. Consequently, wewillusetheexact same expression foreach [4;-1, 4],even though different coordinate systems may actually beinvolved (and hence different gijandy'). Now wejusthave toaddupthese results. Let dy_dy dy,_. .Ay =i)-Sh =1,...,.N—-1 “dt av) a yi Ay dy_dy Ag—- == (tot0G=ae? dy dy.Awo> = :"dt aN) Then weobtain thefollowing formula (where there isaconvention being used intheintegral). 9.THEOREM (FIRST VARIATION FORMULA). Forany variation a, wehave dE(&(u)) du a= bYoa! i dy dy!dy! --[LZ[=mooGe+DuntoomP |ar1A ral ijal Ncl d -L(Fe.4),Ayey. j=o \OM gb (Inthecase ofavariation aleaving endpoints fixed, thesum canbewritten from 1toN—1.) ‘This result isnotvery pretty, butthere itis.Itshould benoted that [//,/] are notthecomponents ofatensor. Nevertheless, later onwewillhave aninvariant interpretation ofthefirst variation formula. Forthetime being wepresent, with apologies, thiscoordinate dependent approach. From thefirstvariation formula itis,ofcourse, simple toobtain conditions forcritical points ofE. 328 Chapter 9 10.COROLLARY. Ify:[a,b]>MisaC®™path,thenyisacritical point ofE®ifandonlyifforevery coordinate system (x,U) wehave tn 0 j ay . dy’dy/Vero +VwdomL =0 wrymeu.dt dtdt ral j= PROOF, Suppose yisacriticalpoint.Given1withy(0)€U,chooseapartition of[a,b] with ¢€(%—-1,4;) forsome i,and such that y|[%-1,4;] isin U.Ifa isavariation ofykeeping endpoints fixed, then inthefirstvariation formula wecanassume that thepart oftheintegral from 4-) to4iswritten interms of(x,U). The final term intheformula vanishes since yisC. Now apply themethod ofproofinLemma 8’,choosing allda’/du(0,1)tobe0,exceptone, which is0outside of(4-1, ti), butapositive function times theterm inbrackets on(4-151). Inorder toputtheequations ofCorollary 10inastandard form weintroduce another setofsymbols “ al (Ogu,aay—dBi rheDog"fijq= nie oy-#4). aueWi]ue3losttat ad Our equations cannow bewritten CY key id!a ri ae =0.at+hWYOG- Gr=o Weknow from thestandard theorem about systems ofsecond order differential equations (Problem 5-4), that foreach p€Mandeach v€Mp, there isa unique y:(—e,£) >M,forsome ¢>0,such that ysatisfies yO) =p dy77a)=v ayk nk dyidy!attBHO Grae=O Riemannian Metrics 329 Moreover, thisyisC°on(—e,). This Jastfactshows thatify,:[0,2) >M andy):(—e,0]>MareC®functions satisfying thisequation, andifmoreover 740) =0) an dr—0t) =2 5dt) dt©) then y,and ytogether give aC® function on(—¢,¢). Naturally, wecould replace 0byanyother 1,Wenow have themore precise result, 1],COROLLARY. Apiecewise C®pathy:[a,b]>Misacriticalpoint forE®ifandonlyifyisactually C®on[a,6]andforeverycoordinate system (x,U) satisfies d?yk nk dyédy!——— = = fe‘eu.a>POOGG=9fory(0) PROOF. Letybeacritical point. Choosing thesame a!asbefore (alla!are0 outside of(4-1,%)), weseethat y][f~1,tj]satisfiestheequation, becausethe final term inthe first variation formula still vanishes. Now choose asothat da dy— (0,4) =A,—, i=1,....N—-1.5,(Oot)=AnGei=l Wealready know that theintegral inthefirstvariation formula vanishes. So we obtain resyayn O=-AyAy), >a "dt which implies thatallA,,4are0.Byourprevious remarks, thismeans thaty isactually C®onallof[a,b]. + Asthesimplest possible case, consider theEuclidean metric onR”, ” ; ; (.)= dx! @dxi, i=l Heregiy=8,s0all0gij/2x* =0,andTf,=0.Thecritical points yforthe energy function satisfy d2ykfY' <0. dp 330 Chapter 9 Thusyliesalongastraight line,soyisacritical pointforthelength function aswell. The situation isnow quite different from thefirstvariational problem weconsidered, when weconsidered only curves oftheform ¢+(t,f(9). Anyreparameterization ofyisalsoacritical pointforlength, sincelength 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, andhence ymust beparameterized proportionally toarc- length. This situation always prevails. 12.THEOREM. Ify:[4,5] >Misacritical point forE,then yisparam- eterized proportionally toarclength. PROOF. Observe first, from thedefinitions, that agg A qr= +UA. Now wehave d|ayP_da< dy!dy/<Fal5a(%wr) non i 0 j_ agiz dy!dy'dy/ ay"dys= Oa arart DoBVO GEae Jal ial nial A 1dy!ay" O)ran +peBir(OGGa Replacing 4g;;/8x! bythevalue given above, thiscanbewritten as d|ayP_avi Py dy!dy!SIF]-LS(CeomSs+ Dunome®)ja ral jal n 0dy! dy | dy!dy!Tr irYO f =). +»on(XsVOGa+WAYS a) Sinceyisacritical pointfor£,bothtermsinparentheses are0(Corollary 10). Thus thelength ldy/dr]] isconstant. 4 Riemannian Metrics 331 The formula 88 ee tiki (*)aye=UeA+Lk occurring inthisproofwillbeusedonseveral occasions lateron.Itwillalsobe useful toknow aformula fordg!/8x*. Toderive one,wefirstdifferentiate DYgima™ =8 m=} toobtain ” ” ag”! 881mmj Yiame =-See UGmt Oras 8 Thus we have agi u,og") itjmjO8tmSr=e" 8mae= Digga ay lm ayk ==—ogg"! (Uk,m]+bnk,1)by(*) Lm =-vel -Ve Ties 7 7 or agi! oNtlplipi (+4) For=~DT +2!TI.ay’ i=1 Wecanfind theequations forcritical points ofthelength function Linex- actly thesame way aswetreated theenergy function. Forthemoment we consider only paths y:[a,b] >Mwith dy/dt #0everywhere. Forthepor- tion y|[%1,4] ofycontained inacoordinate system (x,U), wehave th5 dy!dyi Lorito-wi=f|2suronGeGe ~ ij= Considering ourcoordinate system asR”,wearenow dealing with thecase Foy=|YOgui@yiyi. Nis=1 332 Chapter 9 Weintroduce thearclength function s()=Lay). ‘Then did dSs ¥ y Sa] erly).a=[arl=* (ro) Sowe have 0 ; 98ij dy!dy! YY Soma] arady)_ligeOx’dtdt atVOGT) =2 ds dt n dy">sir) BF(yyav)2I ' atVOrar) = ds ; dt Afteralittlemorecalculation wefinallyobtaintheequations foracriticalpoint ofL: ds dyk + dy!dy)_dy*ae eae rk(ynyn HX zo.aa+hWW" de~“di“ds=° dt Itisclear from thisthat critical points ofEarealso critical points ofL(since they satisfy d?s/d1? =0).Conversely, given acritical point yforLwith dy/dt #0everywhere, thefunction 5:[a,b]>[0,220] isadifleomorphism, andwecanconsider thereparameterized curve yos: 0,L(y] >M. This reparameterized curve isautomatically also acritical point forL,soit must satisfy thesame differential equation. Since itisnow parameterized by arclength, thethird term vanishes, soy0s~?isacritical point forE. There isonlyonedetailwhich remains unsettled. Conceivably acritical point forLmight have akink, butbeC®because ithasazerotangent vector Riemannian Metrics 333 there, asinthefigure below. Inthiscase itwould notbepossible toreparame- terize ybyarclength. Problem 37shows thatthissituation cannot arise. Henceforth wewillcallacritical point ofEageodesic onM(fortheRie- mannian metric (,)).This name comes from thescience ofgeodesy, which isconcerned withthemeasurement oftheearth’s surface, including surveying andthemeasurement ofdegrees oflatitude andlongitude. Ageodesic onthe earth’s surface isasegment ofagreat circle, which istheshortest path between twopoints. Before wecansaywhether thisistrue forgeodesics ingeneral, which aresofarmerely known tobecritical points forlength, wemust initiate alocal study ofgeodesics. The most elementary properties ofgeodesics depend only onfacts about differential equations. Observe that theequations forageodesic, yk Sy dy!dy!Pe rp Ao,dp Tada dt ‘j=l have animportant homogeneity property: ifyisageodesic, then f+y(cr)is also clearly ageodesic. This feature oftheequation allows ustoimprove the result given bythebasic existence anduniqueness theorems. 13.THEOREM. Letp€M.Then thereisaneighborhood Uofpanda number €>0such thatforevery g€Uandevery tangent vector v€Mgwith llvll<ethere isaunique geodesic Yv:(—2,2) >M satisfying dye O)=9, 0) =v. %0(0)=9, (0) PROOF. The fundamental existence and uniqueness theorem says that there isaneighborhood Uofpand&,£2>0sothatforg€Uandv€Mgwith lvl]<e;there isaunique geodesic Yu!(—2€2, 2e2)>M 334 Chapter 9 with therequired initial conditions. Choose ¢<e1€2: Then if|v}<¢and [tl<2wehave Wv/ealt <e1 and=|egt|<2e2. Sowecandefine yy(t) tobeYyyer(€2t). Ifv€Mgisavector forwhich there isageodesic vy:[0,.1]> M satisfying dy YO=T GAO =r then wedefine theexponential ofvtobe exp(v)=exp,(v)=(1). (The reason forthisterminology will beexplained inthenext chapter.) The geodesic ycanthusbedescribed as (2) =expg(tv). Since M,isann-dimensional vector space, there isanatural way togive it aC™swucture. If©CMgisthesetofallvectors v€Mgforwhich expg(v) isdefined, then themap expg: O->M isC™, since thesolutions ofthedifferential equations forgeodesics have aC flow. Identifying thetangent space (Mg)y atv€Mgwith Mgitself, wehave an induced map (€xpg)ux? Mg>Mexpatv)- Inparticular, weclaim that themap (expg)ox! Mg>Mg_istheidentity. Infact, toobtain acurve ¢inthemanifold Mgwith de/dt(0) =v€Mg= (Mg)o, wecanletc(t) =tv,Then expg oc(t) =expg(tv), thegeodesic with tangent vector vattime 0,so d (expg)os() =FF] expg(e(t)) =v.Treo Riemannian Metrics 335 Before proving thenext result, werecall some facts about themanifold 7M. If(x,U)isacoordinate system onM,then forg€Uwecanexpress every vector v€Mguniquely as 2 ai v=odai=lax!q Wewilldenote a’byx#(v), sothat a av=x(v)a5 PBx"Lew wherex:TM—Mistheprojection. Then (xtom...,x% 07,44,00.) =(FY EE ae") isacoordinate system onx~'(U). Forv€Mg,q €Uwetherefore have tangent vectors ™ a a (TM)y; ax|,”ax], aa |= <a thevectors 9/8x|,, areallinthetangent space ofthesubmanifold MgCTM, while thevectors 4/8X‘|,, spanacomplimentary subspace. 14,THEOREM. Forevery p€Mthere isaneighborhood Wandanumber €>0such that (1)Any twopoints ofWarejoined byaunique geodesic inMoflength <eé (2)Letv(g,q’) denote theunique vector v€Mgoflength <esuch that expg(v) =9’.Then (9,9') +v(9,9') isaC™function from WxW> T™. (3)Foreach g€W,themap expg maps theopen e-ball inMgdiffeomor- phically onto anopen setU,>W. 336 Chapter 9 PROOF. Theorem 13saysthatthevector 0€Mphasaneighborhood Vinthe manifold 7Msuch thatexpisdefined onV.Define theC®function F:V> MxMby F(v)=(x(v),exp(v)). Let(x,U) beacoordinate system around p.Wewillusethecoordinate system described above, forx~!(U). If2):MxM—Misprojection onthei factor, then (xlom,...,x" 0m,x!072,...,X" 072)=(x4,...,x17,X2),2.82") isacoordinate system onUxU.Now, using thefactthat (expp)ox: Mp>Mp istheidentity, itisnothard toseethat at0€Mpwehave a a a Felss| )=7 +s(ee|)oxy!lan8xqlan n(sel)= su OX|o, 9x2! |(p,p) Consequently, F,isone-one at0€Mp, soFmaps some neighborhood V’ of0difleomorphically onto some neighborhood of(p,p)€MxM.Wemay assume that V‘consists ofallvectors v€Mgwith ginsome neighborhood U‘ ofpand [lvl]<¢.Choose Wtobeasmaller neighborhood ofpforwhich FV) DWxW. & GivenaWasinthetheorem,andq€W,considerthegeodesicsthrough¢ oftheform 1++expg(tv) forlull<e.These filloutUg.The close analysis of geodesics depends onthefollowing. |,|afexp,(v):[lull=¢} Riemannian Metrics 337 15.LEMMA (GAUSS’ LEMMA). InUg,thegeodesics through qareperpen- dicular tothehypersurfaces {expg(v) :vl]=constant <e}. FIRST PROOF, Letv:R>Mgbeasmooth curvewith|lv(t)|| =aconstant k<eforall1,and define a(u,2)=expg(u- v(t) -l<u<l. Weareclaiming that forevery such @wehave a(secu,geen)=oforall(u,1). Acalculation precisely likethatintheproofofTheorem12provesthefollowing equation, inwhich thearguments (u,2)and@(u, f)areomitted, forconvenience: afaaday ade (OO Pa aa! da!aooA)=Xa(Leos +Lwaeae) magn 0 ; dart aa” _,00doe! +L(Leet +LUG, Gy) i=l rel a) The first term ontheright is0since each curve u++a(u,f) isageodesic. Similarly, weobtain a[dadw “dai (Pam SN eedda! @(ede)=?ge(Lea tDUGG, ar) isl rel JJ=1 which isjusttwice thesecond term ontheright of(I).But8a/8u(w,) isjustthe tangent vector attime utothegeodesic u++expg(u- u(t), where lu)I]=k; so|]@c/8u|] =k,Thusthesecond termontherightof(2) isalso 0.So But@(0,1) =expg(0)=9,so8a/81(0, 1)=0.Itfollows that da da(3.3)=0forall(u,1). 338 Chapter 9 SECOND PROOF. Letv:R>M,beanysmooth curvewith|Jv(1)||=aconstant k<eforallt,and define B(u,t) =expg(t -v(u)) (note carefully theroles played by¢and u). Then8isavariation ofthegeodesic y(t)=expg(t- v(0)), defined on[0,i].By / fY- thefirst variation formula, wehave dE(Bw)| _ [a dy ap dynm|m7(Fo.ota) 5(Foo. Fo)ju=0 ap dyae(Fon. 2o), theintegral vanishing sinceyisageodesic, Buteachcurve A(u)hasenergy 1 A 2 1E(B)=f|r|dt=f{Rat=k, lo dt fo so - —EBM} __|86 dy $ar = (9D GO). * =O 16.COROLLARY. Letc:[a,b] >Uy—{g}beapiecewise smooth curve, e(t)=expg(u(r)-v4),— Riemannian Metrics 339 for0<u(t) <eand Ju(s)|| =1.Then Lic>lu(b) —ula), withequality ifandonlyifuismonotonic andvisconstant, sothat¢isaradial geodesic joining woconcentric spherical shells around 9. PROOF, Ifa(u,t) =expg(u -v(0)), then (1) =a(u(t),) and de da, da eau (N+—.ai~ou"@+ar Since da da da ooo =]=1Gea)=> [sl= we have 2dcp fae?|G|-wor+|F| =wor. with equality ifandonly if8/8 =0,andhence v(t) =0.Thus b 6deLG)areflweotar=mes—wefo iat 0 with equality ifandonly ifwismonotonic andvisconstant. 17,COROLLARY. LetWand¢beasinTheorem J5,lety:[0,1]>Mbe thegeodesic oflength <¢joining 9,9’ €W,andletc:[0,1] >Mbeany piecewise C® path from gtog’.Then Ly) LO), with equality holding ifandonly if¢isareparameterization ofy. PROOF. Wecanassume that 9’=expg(rv) €Ug—{9}(otherwise break ¢up into smaller pieces). For6>0,thepath ¢must contain asegment which joins thespherical shell ofradius tothespherical shell ofradius r,and liesbetween them. ByCorollary 16,theJength ofthissegment haslength >r—6.Sothe length of¢is>r,andclearly ¢must beareparameterization ofyforequality tohold. 340 Chapter 9 Wethusseethatsufficiently small pieces ofgeodesics areminimal paths forarc- length. WecanuseCorollary 17todetermine thegeodesics onafewsimple surfaces, without anycomputations, ifwefirst introduce anotion which will play acrucial rolelater. If(M,(, ))and (M‘,(, )’)areC®manifolds with Riemannian metrics, then aone-one C® function f:M—M’iscalled an isometry ofMinto M’iff*( ,)'=(, ).Forexample, reflection through a plane E?CR"+! isanisometry J:S">S".Itisclear thatifc:[0,1] >M isaC™curve, then thelength of¢with respect to(,)isthelength offoc withrespect to(,)‘;andif¢isageodesic, thenf0cislikewise ageodesic. Fortheisometry J:S$"+S"mentioned above, thefixedpointsetisthegreat circle C=S"M E?.Letp,q€Cbetwopoints with aunique geodesic C’ ofminimal length between them. Then J(C’) isageodesic ofthesame length asC’between I(p) =pand 1(q) =q.SoC’=I(C’), which implies that C’CC, sothatCisageodesic. 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, evenamong nearby paths. Antipodal points on ._pathofsmallerlength 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 onhowmany times thegeodesic goes around thesphere andinwhich direction itstarts. Riemannian Metrics 341 Thegeodesics onarightcircular cylinder Zarethegenerating lines,the circles cutbyplanes perpendicular tothegenerating lines, and thehelices onZ.Jnfact, ifLisagenerating lineofZ,then wecansetupanisometry I:Z-—L— R*byrolling ZontoR®.Thegeodesics onZarejusttheimages —_ under J~!ofthestraight lines inR?.Two points onZhave infinitely many geodesics between them. Wearenowinaposition towindupourdiscussion ofRiemannian met- ricsonMbyestablishing animportant connection between theRiemannian metric (,)andthemetric d:MxM=Ritdetermines, d(p,q) =inf{L(y) :yapiecewise 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 ofR”,orR"—{0}. Ingeneral, amanifold MwithaRiemannian metric (,)iscalledgeodesically 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,g€Mwithd(p,q) = r>0,choose UpasinTheorem 14.LetSCUybethespherical shell ofradius 6<e,There isapoint Po=exppdv, Hohl onSsuch that d(po, 9)<d(s,q) foralls€S.Weclaim that (*) expp(rv) =95 thiswillshow thatthegeodesic y(¢)=expp(tv) isageodesic ofminimal length between pandq.Toprove thisresult, wewillprove that (%) dy(),q)=r-t te[8,r]. Firstofall,sinceeverycurvefromptoqmustintersect S,weclearly have (p,q)=min(d(p,s) +d(s,9)) =5+(po,9). Sod(po.g) =r—8. This proves that(«#)holds for1=6. Now letfo€[6,7] betheleast upper bound ofall#forwhich (##)holds. Then (**)holds forfgalso, bycontinuity. Suppose fo<r.LetS’beaspherical shell ge (2/a) Po Ss ofradius 6’around y(%o) andletpo’€S’beapoint closest tog.Then (yo). 9)=min(a(y(t0),s) +d(s,9)) =8+d(p0',9), Riemannian Metrics 343 so (st) d(po',9) =(r—%) —8. Hence (p, po!)=d(p,9) —A(po', 9)=to+8. Butthepath¢obtained byfollowing yfromptoy(t)andthentheminimal geodesic from y(t) topo’haslength precisely %+68’. Socisapath ofminimal length, andmust therefore beageodesic, which means thatitcoincides with y. Hence (lo+8) =po. Hence (#**) gives Ay(to+8'),9) =F—(0+8), showing that(#*)holds forfo+6’.This contradicts thechoice offo,soitmust bethatt=r. Inother words, (+#)holds for1=r,which proves (+). From thisresult, itfollows easily that Miscomplete with themetric d.In fact, ifACMhasdiameter D,and p€A,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:(4,6) >M,choose t>6,Clearly y(tn)isaCauchy sequence inM,soit converges tosome point p€M.Using Theorem 14,itisnotdifficult toshow thatycanbeextended past6.Consequently, byaleast upper bound argument, anygeodesic canbeextended toR. Asa particular consequeuce ofTheorem 18,note that there isalways amin- imal geodesic joining anytwopoints ofacompact manifold. 344 Chapter 9 ADDENDUM TUBULAR NEIGHBORHOODS LetM"cN*+Kbeasubmanifold ofN,withi:M—Ntheinclusion map, sothatforevery p€Mwehave i,(Mp) CNp.If(,)isaRiemannian metric forN,thenwecandefine My+ CNpas My"=v€Np:(v,i,w) =0forallw€Mp}. Let E=\j Mm," andw:E>MtakeMytop. peM Itisnothardtoseethatv=w:E>Misak-plane bundle overM,the normal bundle ofMinN. Forexample, thenormal bundle vofS"—! @R"isthetrivial 1-plane bundle, forvhasasection consisting ofunitoutward normal vectors. Ontheotherhand ifMistheMébius stripandS!CMisacirclearound thecenter, thenitis nothard toseethatthenormal bundle vwillbeisomorphic tothe(non-trivial) bundle M—S’. Ifweconsider S'CMCcP?,then thenormal bundle ofS!inP?isexactly thesame asthenormal bundle ofS!inM,soittoois non-trivial. Riemannian Metrics 345 Our aim istoprove that forcompact Mthenormal bundle ofMinNis always equivalent toabundle x:U—Mforwhich Uisanopen neighborhood ofMinN,andforwhich the0-section s:M—Uisjusttheinclusion ofM intoU.Inthecasewhere NisthetotalspaceofabundleoverM,thisopen neighborhood canbetaken tobethewhole total space. Butingeneral the neighborhood cannot beallofN.Forexample, asanappropriate neighborhood ofS'CR? wecanchoose R?—{0}. a ——~ Z ww LS, Abundle: U>MwithUanopenneighborhood ofMinN,forwhichthe O-sections:M—Uistheinclusion ofMinU,iscalledatubularneighborhood ofMinN.Before proving theexistence oftubular neighborhoods, weaddsome remarks and aLemma. Ifx:U>Misatubular neighborhood, then clearly nos =identity ofM, som issmoothly homotopic totheidentity ofU, so isadeformation retraction, andH*(U) ~H*(M); thusMhasthesame deRham cohomology asanopen neighborhood. Moreover, ifwechoose a Riemannian metric (,)for7:U>Manddefine D={e€U:(¢,e)<1},thenDisasubmanifold-with-boundary ofU,andthemap1D:D>Mis also adeformation retraction. SoMalso hasthesame deRham cohomology asadosed neighborhood. 19,LEMMA. LetXbeacompact metric space andXoCXaclosed subset. Letf:X>Ybealocalhomeomorphism suchthat{|Xisone-one. Then there isaneighborhood UofXosuch that f|U isone-one. 346 Chapter 9 PROOF, LetCCXxXbe {(x,y)€XxXix#yandf(x)=fy}. Then Cisclosed, forif(Xn,yn)isasequence inCwith x»>xandyy>y, then f(x) =limf(x,) =limfn) =f()), and also x#ysince fislocally one-one. Ifg:C>Ris g(x,y) =d(x, Xo)+d(y, Xo), then g>0onC.Since C iscompact, thereis¢>0suchthatg>2eonC.Thenfisone-one onthe e-neighborhood ofXo. 20.THEOREM. LetMCNbeacompact submanifold ofN.Then Mhas atubular neighborhood 2:U>MinN,which isequivalent tothenormal bundle ofMinN. PROOF, Choose aRiemannian metric (,)forN,with thecorresponding norm |]),and metric d:NxN—R.Let E={v:v€ Npandv€My",forsomep€M} E,= {ve E:ful <e} Ue={g€N:d(g,M) <6}. Itfollows easily from Theorem 13,andcompactness ofM,that expisdefined onE,forsufficiently small ¢>0.Weclaim thatforsufficiently small ¢,themap expisadiffeomorphism from E,onto U,.This willclearly prove thetheorem. LetVCEbethesetofanon-critical points forexp. Then V>M(consid- ered asasubset ofEviathe0-section), andVj=V1Ejiscompact; since exp isone-one onMC\,,itfollows from Lemma 19that forsufficiently small ¢ themap expisadiffeomorphism onEy. Itisclear alsothat exp(£.) CU,.Toprove that expisonto U,,choose any q€Ue,andapoint p€Mclosest toq.Ify:[0,1] >Nisthegeodesic of length <¢with y(0) =pandy(1) =4,itiseasy toseethat yisperpendicular toMatp(compare thesecond proof ofGauss’ Lemma). This means that q=exppdy/dt(0) where dy/dt(0) €Ee. One oftheinteresting features ofTheorem 20isthat alltheparaphernalia ofRiemannian metrics andgeodesics areused initsproof, while they donot even appear inthestatement. Theorem 20willbeneeded only inChapter }], where wewillalsoneed thefollowing modification. Riemannian Metrics 347 21,THEOREM. LetNbeamanifold-with-boundary, with compact bound- aryN. Then Nhas(arbitrarily small) open [and closed] neighborhoods for which there are deformation retractions onto 3. PROOF. Exactly thesame astheproofofTheorem20,usingonlyinwardpoint- ingnormal vectors. ows” &YN 348 Chapter 9 PROBLEMS 1.LetVbeavector space over afield Fofcharacteristic #2,andleth:Vx V>Fbesymmetric andbilinear. (a)Define g:V>Fbyg(v) =A(v,v). Show thatifg),...,¢n isabasis forV*, then n g=Yoayvj-v’y forsomeajj. het (b)Show that g(-») =9(v) A(u,v) =3[g(u +v)—9)—9()]- (c)Suppose g:V—Fsatisfies q(—v) =v,andthath(u,v)=g(u-+v)—g(u) — 9(v) isbilinear. Show that q(u+u+w) —qu)—q(v-+w) =g(u+v)— Gu)—9(v)—9(u+w) —9(u)—9(w). Conclude that g(0) =0,and g(2u) =4q(u). Then show that g(v) =A(y, v). 2.Let(,)beaEuclidean metric forV*. Suppose j,Wi€V*satisfy ¢)A Ade =WiNe AWK#0, and letWyand Wybethesubspaces ofV* spanned bythe¢and y;. (a)Show that w€Wgifand only if@A¢)A--- Agy=0.Conclude that Wy=Wy. (b)Leto1,...,0% beanorthonormal basis ofWe=Wy. If¢=Lyajay, show that thesigned k-dimensional volume oftheparallelepiped spanned by Gis.-+1@x isdet(aiz). (The sign is+if@1,...,@% hasthesame orientation as 01,..+,0g, and —otherwise.) (©)Using Problem 7-9,show thatthisvolume isthesame fory,.... Wk. (@)Conversely, ifWs=Wy,andthesigned volumes ofthe parallelepipeds are thesame, show that $1A+++ Ady =WiAe AVE Ifwe identify Vwith V*, sothat wehave awedge product ¥A:-«Av ofvectors uv€V,then wehave ageometric condition forequality with w)A-+-Awx. InLegons surlaGéométrie desEspaces deRiemann, E.Cartan uses thiscondition to defineQ*(¥*)asformal sumsofequivalence classes ofkvectors; hededuces geometrically thecorresponding conditions onthecoordinates ofvw,w;. 3.LetVbeann-dimensional vector space, and (,)aninner product onV which isnot necessarily positive definite. Abasis 1,...,U, forViscalled orthonormal if(vj,vy)=+5);. Riemannian Metrics 349 (a)IfV#{0},then there isavector v€Vwith (v,v)40. (b)ForWCV, JetWt={vy€V:(v,w) =0forallw©W). Prove that dimW+ >n—dimW. Hint: If{w;} isabasis forW,consider theJinear functionals A;:V>Rdefined byAj(v) =(v,wi). (©)If(,)isnon-degenerate onW,thenV=W@W+,and(,)isalso non-degenerate onW+. (d)Vhasanorthonormal basis. Thus, there isanisomorphism f:R"> Vwith f*(,) =(,)rforsome r(the inner product (,)risdefined on page 301). (e)The index of(,)istheJargest dimension ofasubspace WCVsuchthat (,)[W isnegative definite. Show thattheindex isn—r,thus showing thatr isunique (“Sylvester's Law ofInertia”). 4.Let(,)bea(possibly non-positive definite) inner product onV,and Jet Uj,-..,Un beanorthonormal basis (see Problem 3).Define aninner product (,*onQ*(V)byrequiring that Vi Ave AUG, l<i<--+<ien beanorthonormal basis, with rs 5 k (vtAveAvr UrArAV) =det(igs Yj): (a)Showthat(,)*isindependent ofthe basis v),..., vx.(Use Problem 7-16.) (b)Show that (b1A= Abg thAoAWat=det(i,¥j)")=det(Gi,Vid"). (©)If (,)hasindex j,then (VU) Ave Avty UtAve Avy)” =(-1). (@)Forthosewhoknow about @andA*.Using theisomorphisms @*V* ~ (@*V)* andA‘(V*) ©(AFV)*, define innerproducts on@*V andA‘Vby using theisomorphism V—>V*given bytheinner product onV.Show that these inner products agree with theones defined above. 5.Recall thedefinition ofv;x++xU»—;inProblem 7-26. (a)Show that {v;x«+»xUp_1,0;) =0. (b)Show that |vx---xvz] =Vdet(gij), where gij=(vi,vj).Hint: Apply theresult onpage 308toacertain (7—1)-dimensional subspace ofR”. 350 Chapter 9 6.Let§=m:E>Bbeavector bundle. Anindefinite metric on&is acontinuous choice ofanon-positive definite inner product (,)poneach x~'(p). Show thattheindex of(,)pisconstant oneach component ofB. 7.Thisproblem requires alittleknowledge ofsimple-connectedness andcov- ering spaces. (a)There isnowayofcontinuously choosing aI-dimensional subspace ofS,, foreach p€S?.(Consider thespace consisting ofthetwounitvectors ineach subspace.) (b)There isnoRiemannian metric ofindex 1onS?. 8.Let (,)and (,)/betwo Riemannian metrics onavector bundle &= m:E—B.Let Sbethe setofe€Ewith(e,e)=1,anddefine S’similarly. Show thatSishomeomorphic toS’.If&isasmooth bundle over amanifold M, show that Sisdiffeomorphic toS’. 9.Show byacomputation thatifthefunctions gj;andg’ijarerelated by ax!Axi ,806=L858 GB i withdet(gij) #0,andthefunctions g'/,g"/aredefined by aa ; u ; Veter =8, YiaMe'y =8. k=l k=l then 6 iOx’™Ox! (eBi =veax?Oxi” This, ofcourse, istheclassical way ofdefining thetensor [having thecompo- nents] g¥. 10.(a)Let(,)beaRiemannian metric onM,andAatensor oftype(1),so thatA(p): Mp—Mp.Define atensor Boftype(5)by B(p)(v1, 2)=(A(p)(%1), ¥2)- Iftheexpression forAinacoordinate system is cn aA= Aldx!@=Xees)axJ” Riemannian Metrics 351 show thatB=>,Bixdx!@dx*,where a Bix=4 Sik: yet (b)Similarly, define atensor Coftype(9)by C(p)@r, Aa)=(A(p)* A),42). Show thatifChascomponents C¥/,then n CHsy ghtal. ist Thetensors BandCaresaidtobeobtained from Aby“raising andlowering indices”. 11.(a)Let¥1,...,Xnbelinearlyindependent vectorfieldsonamanifoldM with aRiemannian metric (,).Show that theGram-Schmidt process canbe applied tothevector fields al]atonce, sothat weobtain everywhere orthonor- mal vector fields Yj,..., Yn. (b)Forthecase ofanon-positive definitemetric,findYj,...,¥nwith(Yi,¥j)= £6i;inaneighborhood ofanypoint. 12.(a)Iff:[a,b]>Rispositive, showthattheareaofthesurface obtained byrevolving thegraph offaround thex-axis is b ~ fiver.a (b)Compute thearea ofS?. 13.LetMcR"bean(#—1)-dimensional submanifold with orientation p. The outward unit normal v(p) atp€Misdefined tobethat vector inR", oflength 1such thatv(p), (¥1)p;...,(Yn—1)p ispositively oriented inR",when (Y1)p>-+ +1(Un—1)p ispositively oriented inM,. (a)IfM=ANforann-dimensional manifold-with-boundary NCR",then v(p) isoutward pointing inthesense ofChapter 8. (b)LetdV; bethevolume element ofMdetermined bytheRiemannian metric itacquires asasubmanifold ofR”.Show that ifweconsider v(p) asan element ofR”, then vp) vy AVn—a(P)((U1)py+++5@n—t)p) =det. | Un-1 352 Chapter 9 Conclude that ¢V,—1(p) istherestriction toMyof 1 YE vi(pydel(pyAvAERC) AoNdx"(p). ist ()Note that v1x++»xUn—1 =a@v(p) forsome a€R(byProblem 5).Show that for w€R”we have (w, V(p)) +(1X+++&Va-a,VEp))=(W,VX+++XVa-1). Conclude that v!(p) -dVq—1(p) =restriction toMpof (=1)!Ndx"(p) A.AdxE(p) AvAdx"(p). (d)LetMCR" beacompact n-dimensional manifold-with-boundary, with v theoutward unit normal on8M. Denote thevolume element ofMbydVp, andthatof8MbydV_—1. Let¥=));a‘/Ax! beavector fieldonM.Prove theDivergence Theorem: [axa foonaver M ‘aM (the function divXYisdefined inProblem 7-27). Hint:Consider theform onMdefined by n o=Vena! ax!AvesAGxtAveAax", = (©)LetMcR?beacompact 2-dimensional manifold-with-boundary, with orientation yz,and outward unit normal v.Let Tbethevector field on0M consisting ofpositively oriented unit vectors. Denote thevolume element ofM bydA,andthatof8Mbyds.LetXbeavector field onM.Prove (theoriginal) Stokes’Theorem: f(vxxdasf(X,T)ds M ‘aM (VxXisdefined inProblem 7-27). 14.(a)LetV¥,bethevolume oftheunit ballinR".Show that Va=i(=x2)? y,1dx. H 5 Riemannian Metrics 353 1 (b)IfJ,=f(1~x?)"-YP. dx,showthat a n~1In=—In-2. n (c)Using Vi=2,¥2=x,show that wre — neven ‘n/2)!Ya=anny 2H/2_(HDL—— dd. Taesen ™% nl? (IntermsoftheIfunction, thiscanbewritten -————.)T(i+7/2) (@)LetAn—1 bethe(n~1)-volume ofS"—!, Using themethod ofproof in Corollary 8-8,butreversing theorder ofintegration, show that 1 Vnil"A, dr=ee 0 n (e)Obtain thissame result byapplying theDivergence Theorem (Problem 13), with X(p) =pp. 15.(a)Letc:[0,1] >R”beadifferentiable curve, where R”hastheusual Riemannian metric (,)=0;[email protected] that 1 fa Lc)=fLlcoF ar.0 isl (b)Forthespecial casec:[0,1]>R?given byc(t)=(1,(1), show thatthis length, 1fVitor a, 0 istheleast upper bound ofthelengths ofinscribed polygonal curves. Hint: Iftheinscribed polygonal curve isdetermined bythepoints (17,c(t;)) for 354 Chapter 9 apartition 0=19<---<tj=1of[0,1], then wehave 2 le(i) —e-) =Vi—4-1)? +(Sa) ~SG) =VG tay t+SEU ~ta)? forsome&€[ti—1,ti]. (c)Prove thesame result inthegeneral case. Hint: Use theresults ofProb- lem81,anduniform continuity ofV~onacompact set. Itisnatural tosuppose that thearea ofasurface is,similarly, theleast upper bound oftheareas ofinscribed polygonal surfaces, butasH.Schwarz first observed, thisleastupperboundisinfiniteforabounded portion ofacylinder! Toillustrate Schwarz’s example Ihave plagiarized thefollowing picture from a book called Mamemamuueckuii Ananus xaMuozoobpasuax, written bysomeone called M. Cimpax, h hes ZhCNS SSBNE LHSSS 7] desaSF /\SNS FH SweSFA Nt————SWFSNDSSeS TopviewSST a Toincrease thenumber oftriangles, wemaintain thehexagonal arrangement,bmmovetheplanesofthe hexagons closer together, sothatthetriangles aremorenearlyinaplaneparalleltothebasesofthe cylinder. Inthisway, wecanincrease thenumber oftriangles indefinitely, while thearea ofeach approaches hl/2. ‘The topic ofsurface area fornon-differentiable surfaces isacomplex one, which wewillnotgointo here. Riemannian Metrics 355 16.Let¢:[0,1] >Mbea curve inamanifold Mwith aRiemannian metric (,).Ifp:[0,1] >[0,1] isadiffeomorphism, show that L(c)=L(eop). 17.Showthatthemetric donMmaybedefined usingC®,instead ofpiece- wiseC%curves.(Showhowtoroundoffcornersofapiecewise C®pathsothat thelength increases bylessthan anygiven ¢>0;remember thattheformula forlength involves only firstderivatives.) 18.(@)Let BCMbehomeomorphic totheball{p€R": |p|<1}andlet SCMbethesubset corresponding to{p€R”:|p|=1}.ShowthatM~S isdisconnected, byshowing that M~Band B~Saredisjoint open subsets of M-S. (b)Ifp€B~Sandg €M—B,showthatd(p,q) =mind(p,q'). Use ge thisfactandLemma 7’tocomplete theproofofTheorem7.(Inthetheoryof infinite dimensional manifolds, these details become quite important, forM~S doesno!havetobedisconnected, andTheorem 7isfalse.) 19.(a)Byapplying integration byparts totheequation onpages 318-319, show that aJ(&(u)) iPaar 7 GunegdyBun |ByLO,F') "OF -~4f —OfO.f'O) dt] dt;[Fososroral a thisresult makes sense even iffisonly C!. (b)DuBoisReymond’s Lemma. Ifacontinuous function gon[a,b] satisfies fni(g(t) dt=0 a forallC functions 7on[a,b] with n(a) =(6) =0,then gisaconstant. Hint: The constant ¢must be 1 fe ——— ac= fg(u)du Weclearly have bfn'([g@)~e]dt=0, la soweneedtofindasuitable nwithn'(1)=g()~¢. (©)Conclude thatiftheC'function fisacritical point ofJ,thenfstillsatisfies theEuler equations (which arenotapriori meaningful iffisnotC?). 356 Chapter 9 20.The hyperbolic sine, hyperbolic cosine, and hyperbolic tangent functions sinh, cosh, andtanh aredefined by x px x penx iieea re et2 2 coshx (a)Graph sinh, cosh, and tanh. (b)Show that cosh?~sinh?=1 tanh? +1/cosh® =1 sinh(x +y)=sinh xcosh y+coshxsinhy cosh(x +y)=coshxcoshy+sinhxsinhy sinh’ =cosh cosh’ =sinh. (c)Forthose who know about complex power series: sinhx=“ coshx=cosix. (@)Theinverse functions ofsinhandtanharedenoted bysinh! andtanh", respectively, while cosh! denotes theinverse ofcosh|[0,00). Show that sinh(cosh~? x)=Vx?—1 ey 1(sinh7)'(x)=Trea cosh(sinh™! x)=V1+x2 Ix 17 1 (cosh™')'(x) =———=.1= cosh(tanh™!x)=Fai Vent 21.Consider theproblem offinding asurface ofrevolution joining twocircles ofradius 1,situated, forconvenience, ataand ~a. Wearelooking forafunction 1 Riemannian Metrics 357 ofthe form F S(x)=ccosh= where¢issupposed tosatisfy cosh =1(>0). c (a)There isaunique yo>0with tanhyo=1/yo.Examinethesignof1/y~ tanhyfory>0. (b)Examine thesign ofcosh y~Jsinh yfory>0. (©)Let Aa(e)=¢cosh<c>0. Show that A,hasaminimum ata/yo, find thevalue ofAgthere, and sketch thegraph, (d)There exists ¢with ccosha/c =1ifandonly ifa<yo/cosh yo.Ifa= ‘yo/cosh yo,then there isaunique such c,namely ¢=a/yo =1/cosh yo. If 4<yo/cosh yo,thentherearetwosuchc,withc)<a/yo<cg.Itturnsout that thesurface forc2hassmaller area. [o3 ~yo/coshyo—a | a@yo/coshyo (c)Using Problem 20(d), show that te Vy.cosh yo [0~1.2,soVyo?~1~.67.] 358 Chapter-9 [These phenomena canbepictured more easily ifweusethenotion ofan envelope—c.f. Volume III,pp.176ff. The envelope oftheI-parameter family ofcurves x fe(x) =¢cosh ~ c isdetermined bysolving theequations i) A omFeO)=cosh%—%sin2. ac coe c Weobtain x 5—=+y0, y=ccosh~ =ccosh yo,¢ c sotheenvelope consists ofthestraight lines hyagO, yo The unique member ofthefamily through (yo/cosh yo,1)istangent tothe envelope atthat point, Pora<yo/cosh yo,thegraph off.,istangent to ‘NS 1raINw\ yf\. , aN)isd afe y @ fe coshYofly ayo/coshyo yey tN theenvelope atpoints P,Q €(—a,a), butthegraph offc,istangent tothe envelope atpoints outside [—a,a]. Thepoint@iscalled conjugate toPalong theextremal fe,,and itisshown inthecalculus ofvariations that theexistence ofthis conjugate point implies that theportion offafrom Pto(a,1)doesnot Riemannian Metrics 359 giveaJocalminimum for|f¥1+(f')*. (Compare withthediscussion of conjugate points ofageodesic inVolume IV,Chapter 8,andnote theremark onpg.V.396.)] 22.Allofourillustrations ofcalculus ofvariations problems involved anF which does notinvolve 1,sothat theEuler equations areactually ar ,d(# , axSOLO) >7ay(F@), £'))=0. (a)Show thatforanyfandF:R?>Rwehave d OF [ar daral'-13)-S lea) andconclude thattheextremals forourproblem satisfy OFF-f'ri0. (b)Apply thistoF(x,y)=xV1 +y?toobtain directly theequation dy/dx = Vey? ~|which weeventually obtained inoursolution totheproblem. 23.(a)Letxandx‘betwocoordinate systems, withcorresponding gijandg'i) fortheexpression ofaRiemannian metric. Showthat 8g'ap 5OgiAx*ax!axiOx'¥ ~ L+ Aaxk Ax'¥ Ax! Ax/B i,j,k=l +>-(ax!Px!ax!x! Feaxetant©axBeeax ) (b)Forthecorresponding [i,k]and[w,y]’,showthat n iaxdaxk= axlgtxd ax!ax!ax ax!—a?x: tej,k) oeOe oxox! lop.¥12(ti,KoaB08Only+han?oxox sothat [ij,k] arenotthecomponents ofatensor. (c)Also show that a i aax!axdax!¥ ax! ax'¥ yok em»TyRyeBaldaxkPBaxaxaxl” ijkl = 24,ShowthatanyC°structure onRisdiffeomorphic totheusualC™struc- ture. (Consider thearclength function onageodesic forsome Riemannian metric onR.) 360 Chapter 9 25.Let (,)=35;dx! @dx!betheusual Riemannian metric onR”,and letSj,;gi)4u!@du/beanother metric, whereu',...,u" againdenotes the standard coordinate system onR”.Suppose wearetoldthatthere isadifleo- morphism f:R"+R"suchthatDy;gijdu!@du!=f*(,).Howcanwegoaboutfinding f? (a)LetAf/du! =es:R">R".Ifwe consider e;asavector fieldonR",show that f.(8/8u') =e;. (b)Show that gi=(€,)). (©)Tosolve forfitis,intheory atleast, sufficient tosolve forthe¢;,and to solve forthese wewant tofinddiflerential equations de; “,iy=Anerr= satisfied bythee;’s. Show that wemust have a n2fl ud ae ye er af Bik=DesidlyedPBdau)due (@)Show that agiy>as!afl|es!afaut~LeGulaukGud*GuldukBul a =)gyrAy+BirAyy ral =Bix,j +Bry,i- (©)Bycyclically permuting i,j,k,deduce that Bijk =[ijk], sothatAT,=f). InLegonssurlaGéométrie desEspaces deRiemann, E.Cartan uses thisapproach tomotivate theintroduction ofthePf. (f)Deduce theresult Aj,=If,directly from ourequations forageodesic. (Note thatthecurves obtained bysetting allbutonef*constant aregeodesics, since theycorrespond tolines parallel tothex!-axis.) Riemannian Metrics 361 26.If(V’,(, ))and (V",(_, )”)aretwovector spaces with inner products, wedefine (,}onV=V/@V" by W'@v',w' ow") =(w+ uw”, (a)Show that (,)isaninner product. (b)Given Riemannian metrics onMand J,itfollows that there isanatural way toput aRiemannian metric onMxN.Describe thegeodesics onMxN forthis metric. 27.(a)Lety:[a,b] >Mbeageodesic, andletp:[w,B] >[a,b] bea diffeomorphism. Showthatc=y0psatishes ae KO, deided_dekpr)at VOTaaTO (b)Conversely, if¢satisfies thisequation, thenyisageodesic.(0)If¢satisfies Pk deidei_dek7+hCOG Gr=Gre —forusR>R, then¢isareparameterization ofageodesic. (Theequation p”()=p'(Nu() canbesolved explicitly: p(r)=f"e@ ds,where M’(s) =y(s).) 28.Let¢beacurve inMwith de/dt #0everywhere, and consider thehy- persurfaces {expeyy¥ :[lvl]=constant, where v€Mei) with (v,de/dt) =0}. Show thatforv€Mey) with (v,de/dt) =0,thegeodesic u+expegyu +v isperpendicular tothese hypersurfaces. (Gauss’ Lemma isthe“special case” wherecisconstant.) LiFananal,aXCar: 362 Chapter 9 29.Lety:[a,b] >Mbeageodesic with y(a) =p,andsuppose that expy is adifleomorphism onaneighborhood ©CMpof{ry'(0) :0<1<1}.Show that yisacurve ofminimal length between pand g=y(b), among allcurves inexp(@). (Gauss’ Lemma stillworks onexp(@).) 30. If(,)isa Riemannian metric onMand d:MxM>Ris thecorre- sponding metric, then acurve y:[a,b] >Mwith d(y(a),y(b)) =L(y) isa geodesic. 31.Schwarg’s inequality forcontinuous functions states that 6\?bb fss)<fr)f*), with equality ifandonly iffandgarelinearly dependent (over R). (a)Prove Schwarz’s inequality byimitating theproofofTheorem1(2). (b)Foranycurve yshow that LE s©-a)ER), with equality ifandonly ifyisparameterized proportionally toarclength. (©)Lety:[a,b] +Mbeageodesic withL(y) =d(y(a), y(b)). Ifc(a)= y(a) and c(b) =y(6),show that Loy_L@?ae 1ss00). EW) b-a*b-a* ©) Conclude thatE(y)<E(c)unless¢isalsoageodesic with LB(c) =d(c(a), c(b)). Inparticular, sufficiently small pieces ofageodesic minimize energy. 32.Letpbeapoint ofamanifold MwithaRiemannian metric(,).Choose abasis u,...,Un ofMp,sothatwehave a“rectangular” coordinate system x onMpgiven by>;a'uj+(a!,...,4"); letxbethecoordinate systemxoexp™!, defined inaneighborhood Uofp. (a)Show thatinthiscoordinate system wehavePh(p) =0.Hint: Recall theequations forageodesic, andnote thatageodesic ythrough pisjustexp composed withastraight linethrough 0inMp,sothateach y*islinear. Riemannian Metrics 363 (b)Let7:U>Rbe rq)=d(p,q), sothatroy =2,(y*)*. Show that @P((rey)) ayky dy!dy!meet | ~~) yee ebdt? [>(dt)uyYdtdtk ik (©)Note that dyidy)_a(“7ss —}.ara SG ij k Using part (a),conclude that if||y‘(0)|| issufficiently small, then @r oy)? RE? 0, sothatd(r0y)/dr isstrictly increasing inaneighborhood of0. (@LetBy={v€My:llull<¢}andS.={v€M,:[lull=}.Showthat thefollowing istrue forallsufficiently small ¢>0:ifyisageodesic such that y(0) €exp(Se) andsuch that y‘(0) istangent toexp(S¢), then there is6>0 (depending ony)such thaty(t)¢exp(Be) for0#1€(68,4). Hint: Ify'(0) is S,ey tangent toexp(S¢), then d(r0y)/dt =0. (ce)Letgand q’betwopoints with r(g),r(q’) <¢€and letybetheunique geodesic ofJength <2ejoining them. Show that forsufficiently small ¢the maximum ofroy occurs ateither gorq’. (f)AsetUCMisgeodesically convexifeverypairq,q’€Uhasaunique geodesic ofminimum length between them, and thisgeodesic liescompletely inU.Show thatexp({v €My:||ul|<¢})isgeodesically convex forsufficiently small ¢>0. (@Letf:U>R®beadiffeomorphism ofaneighborhood Uof0€RY into R". Show that forsufficiently small ¢,theimage oftheopen e-ball is convex. 364 Chapter 9 33,(a)There isaneverywhere differentiable curve c(t)=(1,f(1) inR?such that length of¢|(0,A] lim—281,a Hint: Make clook something likethefollowing picture. 7 ilength from a (4,0) to(1,0)is1 N oy length from aera (4,0)to(4,0)is$ (b)Consider thesituation inCorollary 15,except that¢(1)=qifandonlyif 1=a,andsuppose u’(r) >0forrnear 0.If¢isC',then u(r)approaches a limit as1—0(eventhoughu(0)isundefined), Showthatif¢isC',thenthere issome K>0such that forall¢near 0we have [%6.n| cxu|%e.0) oswss Hint: InMgweclearly have dtu-v()) |_luldv(t)dt al a |" Since expg islocally adiffeomorphism there are 0<Ky<K2such that Killull<llexpgevll <Kallull foralltangent vectors vatpoints near q. (6)Conclude that h da 2fray+|2% himLeela,A)_himfw(t)?+|awo.0|di- ho d(p,c(h))— h-+0 u(h) —— Riemannian Metrics 365 (@)If¢isC',show thatL(¢) istheleast upper bound ofinscribed piecewise geodesic curves. 34.(a)Using themethods ofProblem 33,show that if¢isthestraight line joining v,w €Mp, then _ L(expoc) _im 1@ 7) (b)Similarly, ifyy, istheunique geodesic joining exp(v) and exp(w), and Yo,w =€Xp©Cy,w, then timLow) _ vwr0 L(Co,w) (c)Conclude that tim@exPYexPw) _ vwso flu— wl 35.Letf:M—>Nbeanisometry, Show that fisanisometry ofthemet- Ticspacestructures determined onMandNbytheirrespective Riemannian metrics. 36.Let Mbeamanifold with Riemannian metric (,)and corresponding metric d.Letf:M+Mbeamap ofMonto itself which preserves the metric d. (a)Ifyisageodesic, then foyisageodesic. (b)Define f':Mp>My¢p) asfollows: Foryageodesic with y(0) =p,let 1 asivtrooy= LEO t t=O Showthat||/’(X)f =Vl],andthat’(¢X) =¢f"(X). (©Given X,Y €Mp,useProblem 34toshow that 20%,Y) XP+HYIP |ex-7¥IP [odd ee dea WX te¥Tt =WX +P lim[d(exprX,exprY)]? Wx nyt ro XH AY 366 Chapter 9 Conclude that (¥,¥)p =(/(X), f(Y)) pcp, andthen thatf((X +Y)= LX) +£'H)- (@)Part(c)shows thatf’:Mp+Mypyisadifleomorphism. Usethistoshow that fisitself adiffeomorphism, and hence anisometry. 37.(a)Forv,w €R”with w¥0,show that JimWow tel @w) im —————— = 130 t toll The same result then holds inanyvector space with aEuclidean metric (,). Hint: Ifv:R">Ris thenorm, then thelimit isDv(v)(w). Alternately, one can usetheequation (uv,v)=full-flul| -cos@ where 6istheangle between u and v. (b)Conclude that ifwislinearly independent ofv,then w Tim(eeell=Hull=few#0. v 170 t (©)Lety:[0,1]+Mbeapiecewise C!criticalpointforlength, andsuppose that y’(to*) #y’(to) forsome fo€(0,1). Choose 1<foand consider the variation @forwhich &(2) isobtained byfollowing yupton,then theunique geodesic from y(t) toy(to +), and finally therest ofy.Show that if4is yQ) y(to +2) y(to) ya) yO) closeenough toto,thendL(&(w))/du|,,_ #0,acontradiction. Thus, critical paths forlength cannot have kinks. Riemannian Metrics 367 38,Consider acylinder ZCR?ofradius r.Find themetric dinduced bythe Riemannian metric itacquires asasubset ofR?. 39.Consider acone C(without thevertex), and letLbeagenerating line. Unfolding C—Lonto R?produces amap f:C—L—R?which isalocal !; isometry, butwhich isusually notone-one. Investigate thegeodesics onacone (thenumber ofgeodesics between twopointsdepends ontheangleofthe cone, 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$”(theonethat makes theinclusion ofS”intoR"*! anisometry). (b)Show that every geodesic y:R>P”isclosed (that is,there isanumber a such thaty(+a) =y(r) forall1),andthatevery twogeodesics intersect exactly once, (Q)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 inanytriangle is>7. 41.ThePoincaré upper half-plane #?isthemanifold {(x,y)€R?:y>0} with the Riemannian metric dx@dx+dy@dy (,) =...y (a)Compute that 2 oy 1 lo! Kk TheTh=Th=->, Thep allotherPf=0. 368 Chapter 9 (b)LetCbeasemi-circle inJ¢?withcenter at(0,¢)andradius R.Considering itasacurvet+>(t,y(¢)), showthat Py) _-vO _ye? de tc y@Q” (©)Using Problem 27,show thatallthegeodesics inJt?arethe(suitably pa- rameterized) semi-circles with center onthex-axis, together with thestraight lines parallel tothey-axis. (d)Show that these geodesics have infinite length ineither direction, sothat the upper half-plane iscomplete. (©)Show that ifyisageodesic and p¢y,then there areinfinitely many geodesics through pwhich donotintersect y. (£)Forthosewhoknowalittleaboutconformal mapping (compare withProb- jem IV.7-6). Consider theupper half-plane asasubset ofthecomplex num- bers C.Show that themaps f=E48 a,b,c,d€R,ad~be >0 areisometries, and that wecan take any tangent vector atone point toany tangent vector atany other point bysome fs.Conclude that iflength AB= length A‘B’ and length AC=length A‘C’andtheanglebetween thetangent vectors ofBandyatAequals theangle between thetangent vectors off’ andy’atA’,then length BC=length BYC’ andtheangles atBandBYand atCandC’areequal (“side-angle-side”). These results show thatthePoincaré B B Bi A’c oyYNa upper half-plane isamodel forLobachevskian non-Euclidean geometry. The sum oftheangles inanytriangle is<x. Riemannian Metrics 369 42.LetMbeaRiemannian manifold such thatevery twopoints ofMcanbe joined byaunique geodesic ofminimal length. Does itnecessarily follow that theRiemannian manifold Miscomplete? 43.LetMbeamanifold withaRiemannian metric (,),andchoose afixed point p€M.Suppose that every geodesic y:[a,b] >Mwith initial value y(a) =pcanbeextended toallofR.Show that theRiemannian manifold isgeodesically complete. 44. Letpbeapoint inacomplete non-compact Riemannian manifold M.Prove thatthere isageodesic y:[0,00) +Mwith theinitial value y(0) =p,having theproperty that yisaminimal geodesic between anytwo ofitspoints. 45.Let MandNbegeodesically complete Riemannian manifolds, andgive MxNtheRiemannian metric described inProblem 26.Show thattheRie- mannian manifold MxNisalsocomplete. 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(V,( ,))iscomplete. 47.(a)IfM"cN"** jsasubmanifold ofN,show thatthenormal bundle v isindeed ak-plane bundle. (b)Usingthenotion ofWhitney sum@introduced inProblem 3-52,showthat vV@TM ~(TN)|M. 48.(a)Show thatthenormal bundles v1,v2ofM"cN*** defined fortwo different Riemannian metrics areequivalent. (b)If€=x:E>Misasmooth k-plane bundle overM",showthatthe normal bundle ofMCEisequivalent to& 49.(a)Given anexact sequence ofbundle maps pf &o>hm 0 asinProblem 3-28, where thebundles areover asmooth manifold M[or,more generally, over aparacompact space], show that E2~Ei®E3. (b)If€=x:E>Mjsasmooth bundle, conclude thatTE~m*(é)© x*(TM). 370 Chapter 9 50.(a)LetMbeanon-orientable manifold. According toProblem 3-22 there isS'CMsothat (7M)|S! isnotorientable (theProblem deals with thecase where (7M)|S! isalways trivial, butthesame conclusions willhold ifeach (TM)|S' isorientable; infact, itisnothard toshow thatabundle over S!is trivial ifandonly ifitisorientable). Using Problem 47,show that thenormal bundle vofS!¢Misnotorientable. (b)UseProblem 3-29toconclude thatthereisaneighborhood ofsomeS!cM which isnotorientable. (Thus, anynon-orientable manifold contains a“fairly small” non-orientable open submanifold.) CHAPTER 10 LIE GROUPS Ihischapteruses,andilluminates, manyoftheresultsandconcepts ofthepreceding chapters. Itwillalso play animportant role inlater Volumes, where weareconcerned with geometric problems, because inthestudy ofthese problems thegroups ofautomorphisms ofvarious structures play acentral role, andthese groups canbestudied bythemethods now atourdisposal. Atopological group isaspaceGwhich alsohasagroup structure (theproduct ofa,b€Gbeing denoted byad)such that themaps (a,b) +ab from GxGtoG area! fromGtoG arecontinuous. Itclearly suffices toassume instead thatthesingle map (a,b)> ab" iscontinuous. Wewillmainly beinterested inavery special kind of topological group. ALiegroup isagroup Gwhich isalso amanifold with a C® structure such that (x,y)>xy xe ae areC®functions. Itclearly suffices toassume that themap (x,y)hexy7! isC. Asamatter offact(Problem J),iteven suffices toassume that themap (,p)xyisC®. The simplest example ofaLiegroup isR",with theoperation +. The circle S!isalsoaLiegroup. One waytoputagroup structure onS!isto consider itasthequotient group R/Z, where ZCRdenotes thesubgroup ofintegers. The functions x+»cos27x and xr>sin2zx areC® functions onR/Z, and ateach point atleast oneofthem isacoordinate system. Thus themap -1Gy) rex-y rmxy m mM om RxR > R— S'SR/Z. which canbeexpressed incoordinates asone ofthetwomaps (x,p)>cos2x(x—y)=cos27xcos2zy+sinzxsinwy (x,y)>sin2x(x—y)=sin2xcos2xy—cos2xxsinzy, isC®:consequently themap(x,y) +>xy! from S$!x$1toS!jsalsoC™. 371 372 Chapter 10 IfGandHareLiegroups, then GxH,with theproduct C®structure, and thedirect product group structure, iseasily seen tobeaLiegroup. In particular. thetorus S!xS!isaLiegroup. Thetorus mayalsobedescribed asthequotient group PPPeTeteee thepairs (a,b) and(a',b’) represent thesame element ofS!xS!ifandonly ifa'-ae Zandb'-beZ. Many important Liegroups arematrix groups. The general linear group GL(n, R)isthegroup ofallnon-singular real nxnmatrices, considered asa subsetofR”*.Sincethefunction det:R”’—Riscontinuous (itisapolynomial map), thesetGL(n,R) =det7'(R —{0})isopen, andhence canbegiven the C®@structure which makes itanopen submanifold ofR”. Multiplication of matrices isC®, since theentries ofAB arepolynomials intheentries ofA andB.Smoothness oftheinverse map follows similarly from Cramer’s Rule: (A™)je =detAY/detA, where A’isthematrix obtained from Abydeleting rowiandcolumnj. One ofthemost important examples ofaLiegroup istheorthogonal group O(n), consisting ofallA€GL(r, R)with A-A‘=/,where A!jsthetranspose ofA.This condition isequivalent tothecondition that therows [and columns] ofAareorthonormal, which isequivalent tothecondition that, with respect totheusual basis ofR”,thematrix Arepresents alinear transformation which isau“isometry”, ie., isnorm preserving, and thus inner product preserving. Problem 2-33 presents aproof that O(7) isaclosed submanifold ofGL(n, R), ofdimension n(n—1)/2. Toshow that O(n) isaLiegroup wemust show that themap (x,»)Hxy"! which isC?onGL(n, R),isalsoC®asamap from O(@) xO(7) toO(”). ByProposition 2-1, itsuffices toshow thatitiscontinuous; butthisistruebecause theinclusion ofO(7) +GL(m, R)isahomeomorphism (since O(n)isasubmanifold ofGL(n,R)).Later inthechapter wewillhave another wayofproving that Q(7) isaLiegroup, andinparticular, amanifold. The argument intheprevious paragraph shows, generally, thatifHCGisa subgroup ofGandalsoasubmanifold ofG,then HisaLiegroup. (This gives another proofthatS!isaLiegroup, forS'CR?canbeconsidered asthegroup LieGroups 373 ofcomplex numbers ofnorm 1.Similarly, 5?istheLiegroup ofquaternions ofnorm 1.Itisknow thatthese aretheonly spheres which admit aLiegroup structure.) Itispossible forasubgroup HofGtobeLiegroup with respect to aC®™structure thatmakes itmerely animmersed submanifold. Forexample, ifLCRxRisasubgroup consisting ofall(x,cx)forcirrational, thenthe image ofLinS'xS'=RxR/(Z xZ)isadense subgroup. Wedefine aLie subgroup HofGtobeasubset HofGwhich isasubgroup ofG,and also a Liegroup forsome C®structure which makes theinclusion map i: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 will not need this fact. The group O() isdisconnected; thetwocomponents consist ofallA€O(7) with detA=+1anddetA=—1,respectively. ClearlySO(n)={A€O(n): detA=1},thecomponent containing theidentity /,isasubgroup. This isnot accidental. 1,PROPOSITION. IfGisatopological group, then thecomponent Kcon- taining theidentity e€Gisaclosed normal subgroup ofG.IfGisaLie group, then Kisanopen Liesubgroup. PROOF. Ifa€K,then a~'K isconnected, since b+»a~'b isahomeomor phism ofKtoitself. Since ¢=a~'a €a~'K,wehave a~'K CK.Since this istrue forall@€K,wehave K~'K CK,which proves that Kisasubgroup. Forany6€G,itfollows similarly thatbKb~" isconnected. Sincee€bKb™', wehave bKb~! ©K,soKisnormal. Moreover, Kisclosed since components arealways closed. IfGisaLiegroup,thenKisalsoopen,sinceGislocallyconnected, soK isasubmanifold andasubgroup ofG.Hence KisaLiesubgroup. ¢ Thegroup $O(2) isjustS!,which wehave already seen isaLiegroup. As afinal example ofaLiegroup, wemention E(n), thegroup ofallEuclidean 374 Chapter 10 motions, ice.,isometries ofR".Alittle argument shows (Problem 5)that every element ofE(n) canbewritten uniquely asA-twhere ACO(7), and Ttisa translation, t(x) =ta(x) =x+a. Wecangive E(n) theC®structure which makes itdiffeomorphic toO(”) xR". Now E(z) isnotthedirect product O(7) xR"asagroup, since translations and orthogonal transformations donotgenerally commute. Infact, AtgA7'(x) =A(A7!x 4.4) =x+A(a) =Taay(X), so AtgA7! =Tala); Ata=TaayA. Consequently, Ata(Bt5)~! =AtatB™ =Atg-5B! =AB" t8a~»). which shows that E() isaLicgroup. Clearly thecomponent ofe€E(1) is thesubgroup ofallAzwith A€SO(n). For any Liegroup G,ifa€Gwedefine theleftand right translations. La: G>Gand Rg: G>G,bv La(b) =ab Ra(b) =ba. Notice that LaandRgareboth difleomorphisms, with inverses Ly~1 and Ry~1. respectively, Consequently, themaps Lax: Gy>Gab Ras: Gp>Goa areisomorphisms. Avector fieldYonGiscalled leftinvariant if LaX =X forallaeG. Recall this means that LarXp =Xap foralla,b eG. Itiseasytoseethatthisistrueifwemerely have LanXe =Xa foralla €G. Consequently, given X-€Ge.there isaunique leftinvariant vector field Y onGwhich has thevalue X;at¢. LieGroups 375 2.PROPOSITION. Every leftinvariant vector field XonaLiegroup G isc™. PROOF. Itsuffices toprove that XisC® inaneighborhood ofe,since the difleomorphism LathentakesXtotheC®vector fieldLayXaround a(Prob- lem 5-1). Let(x,U) beacoordinate system around e.Choose aneighbor- hoodVofesothata,b€Vimplies ab!€U.Thenfora€Vwehave Xx!() =LaxXe(x') =Xe(x! 0La). Since themap (a,b) +abisC®onVxVwecanwrite x!(ab)=x!La(b)=f(x"(a)... x(a),x"(b),- (0) forsome C®function f/onx(V) xx(V). Then Xx!(a) =Xe(x! 0La) n'a j(x! oLa) a)=Yrci22had = a=De af whereXe=Dee!alja e j=l n ; =Yet DassT'(x(a), x(e)). j=l which shows that Xx! isC°°. This implies that XisC°. ¢ 3.COROLLARY. ALiegroupGalwayshasatrivialtangent bundle (andis consequently orientable). PROOF, Choose abasis X1e,..-,Xne forGe. Let X1,...,Xn betheleft invari- antvector fields with these values ate.Then 4,..., X,areclearly everywhere linearly independent, sowecandefine anequivalence f:TG>G xR" by n P(x) =ae...jan 376 Chapter 10 Aleftinvariant vector fieldXisjustonethatisLg-related toitselfforalla. Consequently, Proposition 6-3showsthat[,,¥]isleftinvariant 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 [,]onGe,by X,Y] = Fe). The vector space Ge,together with this[,]operation, iscalled theLiealgebra ofG,andwillbedenoted by£(G). (Sometimes theLiealgebra ofGisdefined instead tobethesetofleftinvariant vector fields.) Wewillalso usethemore customary notation g(aGerman Fraktur g)for£(G). This notation requires some conventions forparticular groups; wewrite ql(m,R) fortheLiealgebra ofGL(, R) o(n) fortheLiealgebra ofO(n). Ingeneral, aLiealgebraisafinitedimensional vectorspaceV,withabilinear operation [,}satisfying [¥,x] =0 (LX,¥],Z]+ (0%Z],X]+([Z,X],¥]=0 —“Jacobi identity” forallX,¥,Z eV. Since the[,]operation isassumed alternating, itisalsoskew-symmetric, [X,Y] =~[Y, X]. Consequently, wecallaLiealgebra abelian orcommutative if[X,Y] =0forallX,Y. The Liealgebra ofR”isisomorphic asavector space toR”.Clearly £(R”) isabelian, since thevector fields 8/4x! areleftinvariant and [4/dx', /4x/] =0. TheLiealgebra £(S!) ofS!is1-dimensional, andconsequently must be abelian. If¥jareLiealgebras with bracket operations [,];fori=1,2,then wecandefine anoperation [,]onthedirect sum V=Vj@V2(=Vix2as aset)by ((%1,¥2),(M1,Yay]=(1%,Yili,[X2,Yala). Itiseasy tocheck that thismakes Vinto aLiealgebra, and that £(G xH) isisomorphic to£(G) x£(H) with thisbracket operation. Consequently, the Liealgebra £(S! x---xS!)isalsoabelian, The structure ofgl(n,R) ismore complicated. Since GL(n,R) isanopen submanifold ofR””, thetangent space ofGL(7,R) attheidentity /canbe LieGroups 377 identified withR”.If'weusethestandard coordinates x/onR"’,thenannx (possibly singular) matrix M=(Mj;) canbeidentified with a M;=LXMyal,:iy LetMbetheleftinvariant vector fieldonGL(n,R) corresponding toM.We compute thefunction @x*! onGL(n,R) asfollows. Forevery A€GL(n,R), MAx!(A)=Max") =Laws(x)=My(x"0La). Now thefunction x“!oL4: GL(1,R) >GL(n,R)isthelinearfunction 1 (x0La)(B)=x""(AB) =)AteBal, a=) with (constant) partial derivatives Ag j=l oHoLa)={od axis 0 J#i. So ~ aMx(A)=My(x*!0Ly)=zMuza oLa) n n, =0MaAni=YoMatAke: iat ant Thus, ; ee ax! 0 ki. SoifNisanother nxnmatrix, wehave ~ aonkl) ep AL Ny(Mx""!)=DN50(M") " =OM My=(NM)a. j=l From this we see that . ~ a(4,My=SUMAN-NM)xa5 a ax|, 378 Chapter 10 thus,ifweidentify ql(n,R) withR’,thebracket operation isjust [M,N] =MN-NM. Notice thatinanyring,ifwedefine [a,b]=ab—ba,then[,]satisfies the Jacobi identity. Since O(7) isasubmanifold ofGL(n,R) wecanconsider O(n); asasubspace ofGL(#,R)z, andthusidentify o(m)withacertain subspace ofR””. This subspace may bedetermined asfollows. IfA:(—e,2) >O(n) isacurve with A(0) =I,andwedenote (A(Q));; byAij(2), then " YeAOARO=by: k=l differentiating gives Aix’(0)5jx +ixAjx'(0)=0, which shows that Aj;'(0) =—Aji'(0). Thus O(n); CR™cancontain onlymatrices Mwhich areskew-symmetric, 0My M3... Min Mi 0 M=) -—M3 0 . -Min 0 This subspace hasdimension n(n—1)/2, which isexactly thedimension ofO(n), soO(m); must consist exactly ofskew-symmetric matrices. Ifwedidnotknow thedimension ofO(), wecould usethefollowing line ofreasoning, Foreach i,jwith i<j, wecandefine acurve A:R>O(n) by i i 1 costsint i A) = . (rotation inthe(i,/)-plane) —sint cost j “1 LieGroups 379 with sin¢ and—sin¢ at(i,/)and(j,/), 1’sonthediagonal except at(i,i) and(j,i),and0'selsewhere. ThenthesetofallA’(0) span theskew-symmetric ma- trices. Hence O(7); must consist exactly ofskew-symmetric matrices, andO(n) must have dimension n(n —1)/2. Wedonotneed anynew calculations todetermine thebracket operation ino(7).Infact,consider aLiesubgroup HofanyLiegroup G,andleti:H> Gbetheinclusion. Sincei,:He>Geisanisomorphism into,wecanidentify Hewithasubspace ofG,.AnyX€Hecanbeextended toaleftinvariant vectorfield¥onHandaleftinvariant vectorfield¥onG.Foreacha€HC G,wehave lefttranslations La: H> H, LaiG>G and Lact =ioLg. So _ ~ eX (a)=t4LawX =Las(ieX) =X(a). Inother words, ¥and¥arei-related, Consequently, ifY€He,then[¥,7] and [X,Y] arei-related, which means that XP) =16%, Pe). Thus,HeCGe=gisasubalgebra of9,thatis,Heisasubspace ofgwhichis closed under the[,]operation; moreover, Hewiththisinduced[,]operation isjust §=£(H). This correspondence between Liesubgroups ofGandsubalgebras ofgturns out towork inthe other direction also. 4,THEOREM. LetGbeaLiegroup, andhasubalgebra ofg.Then there isaunique connected Liesubgroup HofGwhose Liealgebra is). PROOF. Fora€G,letAgbethesubspace ofGaconsisting ofall¥(a) for X€.The factthat §isasubalgebra ofgimplies that Aisanintegrable distribution. LetHbethemaximal integral manifold ofAcontaining e.If 6G, then clearly L54(Aa) =Aga, 80Lby leaves thedistribution Ainvariant. Itfollows immediately thatLypermutes thevarious maximal integral manifolds ofAamong themselves. Inparticular, ifb€H,then Lg~1 takes Htothe maximal integral manifold containing Ly~1(b) =e,soLy~1(H) =H.This implies that Hisasubgroup ofG.Toprove that itisaLiesubgroup wejust need toshow that(a,6)+»ab™! isC°°.Now thismapisclearly C°°asamap intoG.Using Theorem 6-7,itfollows thatitisC°°asamap intoH. The proofofuniqueness islefttothereader. 380 Chapter 10 There isavery difficult theorem ofAdo which states that every Liealgebra isisomorphic toasubalgebra ofGL(N,R) forsome N.Itthen follows from Theorem 4thateveryLiealgebra isisomorphic totheLiealgebra ofsomeLiegroup, Later onwewill beable toobtain a“local” version ofthis result. We will soon see to what extent theLiealgebra ofGdetermines G. Wecontinue 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 ¢4¢: Ge>He.Forany a€Gweclearly have bola =Loa od: soifX€Ge,and¥=$,eX istheleftinvariant vector fieldonHwith value $4eX ate,then aX(0)=bealarX =LocayabecX =X(¢@). Thus ¥and¥are¢-related. Consequently, themapgve: g> isaLie algebra homomorphism, that is, declaX +bY) =abueX +bbueY GrelX, Y]=[bneXsbre¥]. Usually, wewilldenote ¢sesimply by$4:g>. Forexample, suppose that G=H=R.There areanenormous number ofhomomorphisms ¢:R—R,because Risavector spaceofuncountable dimension over Q,andevery linear transformation isagroup homomorphism. But if¢isC®, then thecondition os +1) =$8) +40 implies that dg(t+s) _dos). ds ds | evaluating ats=0gives gO =4'O, which means that$(¢) =ctforsome c(=¢'(0)). Itisnothard toseethateven acontinuous ¢mustbeofthisform(onefirstshows that¢isofthisformonthe LieGroups 381 rational numbers). Wecanidentify £(R) with R.Clearly themap ¢: R>R isjustmultiplication by¢. Now suppose that G=R,butH=S!=R/Z. Aneighborhood ofthe identity e€S!canbeidentified withaneighborhood of0€R,giving risetoan identification of£(S') with R.The continuous homomorphisms ¢:R>S! areclearly oftheform xe R—>R—R/Z; once again, $,:R+Rismultiplication byc. Notice thattheonlycontinuous homomorphism ¢:S!>Risthe0map (since {0}istheonly compact subgroup ofR).Consequently, aLiealgebra ho- momorphism g—)may notcome from anyC° homomorphism ¢:G>H. However, wedohavealocalresult. 5.THEOREM. LetGandHbeLiegroups, and©:q>}aLiealgebra homomorphism. Then there isaneighborhood Uofe€Gand aC® map :U—Hsuch that ¢(ab) =$(a)o(b) when a,b,ab €U, and such that forevery X€gwehave GreX =¥(X). Moreover, ifthere aretwoC° homomorphisms ¢,: G>Hwith xe= Wae=&,and Gisconnected, then ¢=py. PROOF. Let§(German Fraktur k)bethesubset fCgx§ofall(X,&(X)), for Xeg. Since ©isahomomorphism, tisasubalgebra ofgx§=£(G xH).By Theorem 4,there isaunique connected Liesubgroup KofGxHwhose Lie algebra is#.If 71:GxH>Gisprojection onthefirstfactor, and o=7)|K, then @:K>GisaC®homomorphism. ForX€qwehave W(X, B(X)) =X, 80Wx:Kee) >Geisanisomorphism. Consequently, there isanopen neigh- borhood Vof(e,¢)€KsuchthatwtakesVdiffeomorphically ontoanopen neighborhood Uofe€G.Ifm2:GxH>Hisprojection onthesecond factor, wecandefine g=mow! on U. 382 Chapter 10 The first condition on¢isobvious. Asforthesecond, ifX€g,then a(X, O(X)) =X, so GX=Ta(X, O(X))=O(X). Given $,¥: G>H,define theone-one map 6:G>GxHby (a) =(a,w(@)). Theimage G’of@isaLiesubgroup ofGxHandforX€qweclearly have 0,X =(X,(X)), so£(G') =1.Thus G'=K,which implies that y(a) =$(@) forallae G. 6.COROLLARY. IftwoLiegroups Gand Hhave isomorphic Liealgebras. then they arelocally isomorphic. PROOF. Given anisomorphism ©:q>,tet¢bethemap given byTheo- rem 5,Since se=®isanisomorphism, ¢isadifleomorphism inaneighbor- hood ofe€G. Remark: Forthose who know about simply-connected spaces itisfairly easy (Problem 8)toconclude thattwosimply-connected Liegroups with isomorphic Liealgebras areactually isomorphic, andthat allconnected Liegroups with a given Liealgebra arecovered bythesame simply-connected Liegroup. 7.COROLLARY. Aconnected Liegroup Gwithanabelian Liealgebra is itself abelian. PROOF. ByCorollary 6,Gislocally isomorphic toR”,soab=bafora,b inaneighborhood ofe.Itfollows that Gisabelian, since (Problem 4)any neighborhood ofegenerates G.+ 8.COROLLARY. Forevery X€Ge,there isaunique C® homomorphism ¢:R=Gsuchthat diAlare di|,20 LieGroups 383 FIRSTPROOF. Define@:R>£(G)by (@) =a. Clearly ©isaLiealgebra homomorphism. ByTheorem 5,onsomeneighbor- hood (~¢,¢) of0€Rthere isamap ¢:(-€,e) >Gwith G(s +1) =$(s)o(1) Isllel.Istel <e and ie ,|o7* (F..)=* Toextend ¢toRwewrite every ¢with |t|>€uniquely as t=k(e/2)+r —kaninteger, |r|<¢/2 and define se $(6/2)+++G(E/2)-P(r) [$(€/2) appears ktimes] k>0 ~|@(-€/2)+++@(—€/2)- (7) [(—e/2) appears —ktimes] k<0. Uniqueness also follows from Theorem 5. SECOND (DIRECT) PROOF. Iff:G>Ris C®, and ¢:R>GisaC® homomorphism, then dg. SOU +h) -SOO)re rr =limLOM) —(GO) ho h d=Hl,SoLgayog dg .=Leong,l_.Y) =LownX(1) =FOO). Thus$mustbeanintegral curve of¥,which proves uniqueness. Conversely,if$:R=Gisanintegral curveof¥,then 1 $(s)- 9 isanintegral curve of¥which passes through $(s)attime¢=0.Thesame is clearly true for ire (s +0), so¢isahomomorphism. Weknow thatintegral curves ofXexistlocally; they canbeextended toallofRusingthemethod ofthefirstproof. 4 384 Chapter 10 Ahomomorphism ¢:R—Giscalleda1-parameter subgroup ofG.We thusseethatthereisaunique 1-parameter subgroup ¢ofGwithgiventangent vector dg/dt(0) €Ge.Wehave already examined theI-parameter subgroups ufRR.More interesting things happen when wetake GtobeR—{0}, with multiplication asthegroup operation. Then allC®homomorphisms ¢:R> R—{0},with 9(s+8) =6(s)6(0), must satisfy $1) =9OH) (0) =1. Thesolutions ofthisequation are P(t)=eOr, Notice that R—{0}isjust GL(1,R). AllC° homomorphisms ¢:R>GL(n,R) must satisfy theanalogous differential equation YO=9'O 90. (*)_ gO =1, where -now denotes matrix multiplication. The solutions ofthese equations canbewritten formally inthesame way Gr) $(1) =exp(to'(0)), where exponentiation ofmatrices isdefined by A A? ABepA=1+54+start This follows from thefacts inProblem 5-6, some ofwhich willbebriefly reca- pitulated here. IfA=(aj) and JA]=max layy|, then clearly 1A+ 81<1Al+181 JAB] <nlAl- 1B]: hence JAI<nk"[Ayk <nA, Consequently; ANarnANTK<(nlAD”a+(AD+*5RS Ni (W+K)f >NE (N+K)! as °° LieGroups 385 sotheseriesforexp(A)converges (the(i,/)"®entryofthe partial sums converge), and convergence isabsolute and uniform inanybounded set. Moreover (see Problem 5-6), ifAB=BA, then exp(A +B)=(exp A)(exp 8). Hence, if$(1) isdefined by(#»), then 1 (0)) —¢'()=limexp(tg’(0) +g'(0))—exp(to’(0))h—0 hk —jnlexp(eg'(0)) —1) 1=fim ; exp(eo'(0)) hg’), b?9"(0)?=limeee*oe“exp(t¢’(0)) b= h a =9'(0)6), so@does satisfy (+). ForanyLiegroup G,wenow define the“exponential map” exp:q>G asfollows. Given X€q,let¢:R>Gbetheunique C® homomorphism with d¢/dt(0) =X.Then exp(X) =$(1). Weclearly have exp( +2)X =(expan X)(expX) exp(-1X) =(exptX)7!. 9.PROPOSITION. The map exp: Ge>GisC®(note that Ge*R”hasa natural C®structure), and0isaregular point, sothatexptakes aneighborhood of0€Gediffeomorphically onto aneighborhood ofe€G.If¥:G>His anyC® homomorphism, then G,Yt He expoWa=Woexp. en| ler oon 386 Chapter 10 PROOF. The tangent space (GexG)cx,0) oftheC®manifold GexGatthe point (X,a) canbeidentified with Ge®Gg. Wedefine avector field Yon GexGby “aoeroe.0000 (Sex St Then Yhasaflow @:Rx(Ge xG)>GexG,which weknow isC®. Since expX=projection onGof#(1,0@ X), itfollows that expisC®. Ifweidentify avector v€(Ge)o with Ge,then thecurve c(t)=¢vinGehas tangent vector vat0.So dexp(c(O) da| gS] expt expag() >|,wil.” =u Soexpyg istheidentity, andhence one-one. Therefore expisadiffeomorphism inaneighborhood of0. Given ¥:G>H,andX€Ge,let¢:R>Gbeahomomorphism with dowL.7* Then y0g: R= Hisahomomorphism with dy og)—_—_— =WX. dt|.a Consequently, exp(eX) =yo$C) =wlexp X).& 10.COROLLARY. Everyone-one C®homomorphism $:G>Hisan immersion (so$(G) isaLiesubgroup of#). PROOF. If$xp(X(p)) =0forsome non-zero X€g,thenalso}ye(X) =0. But then e=expdae(IX) =o(exp(/X)), contradicting thefactthat¢isone-one. ¢ LieGroups 387 11,COROLLARY. Every continuous homomorphism $:R>GisC®. PROOF. LetUbeastar-shaped open neighborhood of0€Geonwhich exp isone-one. Foranya€exp(4U), ifa=exp(X/2) forX€U,then @=exp(X/2) =[exp(X/4)}, expX/4€exp(4U). Soahasasquare rootinexp($U). Moreover, ifa=b?forb€exp(3U), then b=exp(Y/2) forY€U,so exp(X/2) =a=b?=[exp(¥/2)]* =expY. SinceX/2,¥ €Uitfollows thatX/2=Y,soX/4=Y/2.Thisshowsthat every a€exp(}U) hasaunique square rootinthesetexp($U). Now choose €>0sothat$(t)€exp(3U) for|t|<€.Let$(e)=expX, X€exp(}U). Since [(e/2)? =$e)=[expX/27, itfollows from theabove that $(¢/2) =exp(X/2). Byinduction wehave $(e/2") =exp(X/2"). Hence $m/2"-€)=p(e/2")" =[exp(X/2")}" =exp(n/2" -X). Bycontinuity, o(se) =expsX foralls €[-1, 1].& 12.COROLLARY. Every continuous homomorphism $:G>HisC®. PROOF. Choose abasis X1,...,Xq forGe. The map t+$(exptX)) isa continuous homomorphism ofRtoH,sothere is¥;€Hesuch that (exp 1X;)=exprY;. Thus, (*) ((exp Xi)---(exptnXn))=(expaYi)+++(expYn). Now themap y:R">Ggiven by W(h,..-5t) =(exp X))++(exp Xn) isC®andclearly 3(ga),)=% sowisadiffeomorphism ofaneighborhood Uof0€R"ontoaneighbor hood Vofe€G.Then onV, g=(Gowow, and (+)shows that ¢0yisC°. So@isC™ ate,and thus everywhere. 388 Chapter 10 13.COROLLARY. IfGandG’areLiegroupswhichareisomorphic astopo- logical groups, then they areisomorphic asLiegroups, that is,there isadiffeo- morphism between them which isalsoagroup isomorphism. PROOF. Apply Corollary 11tothecontinuous isomorphism anditsinverse. ¢ The properties oftheparticular exponential map exp:R”(=gl(n,R)) >GL@,R) may now beused toshow that O(n) isaLiegroup. Itiseasy toseethat exp(M') =(expM)!. Moreover, since exp(M +N) =(exp M)(exp.N) when MN =NM, wehave (exp M)(exp —M) =1. SoifMisskew-symmetric, M=—M¢, then (expM)(exp M)'=1, ie.expM€O(n).Conversely, anyA€O(n)sufficiently closetoJcanbe writen A=expM forsome M. LetAt=expN. Then /=A-Ab= (cxp M)(exp N),soexpN=(expM)!=exp(—M). Forsufficiently smallM andNthisimpliestha.N=—M.SoexpM*=At=exp(—M); hence Mt =~M. Itfollows that aneighborhood ofJinO(n) isann(w~1)/2 dimensional submanifold ofGL(#,R). Since O(n) isasubgroup, O(n) isitself asubmanifold ofGL(n, R). Just asinGL(n, R),theequation exp(X +Y)=expXexpYholds whenever [X,Y] =0(Problem 13). Ingeneral, [X,Y] measures, uptofirst order, the extent towhich thisequation fails tohold. Inthefollowing Theorem, andin itsproof, toindicate thatafunction c:R>G,hastheproperty thatc(r)/1? is bounded forsmall ,wewilldenote itbyO(7). Thus O(13) willdenote different functions atdifferent times. 14.THEOREM. IfGisaLiegroup and X,Y €Ge,then 2 (1)exp1Xexpt¥=exp{(X +Y)+Si,Y]+ow) (2)exp(-1X) exp(-r¥) exp1Xexp1Y =exp{t?[X, Y]+O*)} (3)exptX exp1Yexp(-1X) =exp{t¥ +7[X,Y] +OW}. LieGroups 389 PROOF. We have og : dG)Xf(a)=Xa(f)=LasX(f)=X(f0La)=in|S(a-expuX).ju=0 Similarly, ii Ff(a)=a(Y) (ii) f=5]Sla-expu , For fixed s,let b(t) =flexp sXexpr). Then a d d(ii) ')==f(expsXexprY)=in|SJ(expsXexp1YexpuY) a du\yoo =(¥f)(expsX exp1Y) byii). Applying (iii)to¥finstead offgives (iv) $"(1)=[¥Ff)(expsXexprY). Now Taylor’s Theorem says that 0 90)=90)+6"+LO24O10), Suppose that f(e) =0.Then wehave w) SlexpsX exp1Y) =flexp sX)+(¥f)(expsX) ze+SPFMexpsx)+00°). Similarly, foranyF. 4rexpsx) =(FF)(expsX) Co vegaaFlexsX)=[RPPexpsX) 2 F(expsX) =F(e)+s(¥F)(e) +SPEC) +O(s°). 390 Chapter 10 Substituting in(v)forF=f,F=¥f,andF=¥(Ff) gives (vi)flexpsX expr¥) =s(¥ye)+(FN(e) 2ee Pee mcd+TENE +ZYONnie)+stX(Ye) +O(s) +OW) +O(s%t) +O(st?). Inparticular, (vil) S(exptX exptY) =[(¥+PF)fe) ey8. +2(>+9747)|(+00°), Now for small ¢we can write exptXexptY =expZ(t) forsome C® function Zwith values inGe.Applying Taylor’s formula toZ gives Z() =tZ, +PZ.+ OW), forsome Z,,Zz€Ge.Iff(e)=0,thenclearlyf(A(t)+O(83))=f(A)+ (03), soby(vi)wehave (viii) SlexpZ()) =f(expZ1+0?Z2))+OW) =(Z,fle) +(Ze) Pon iw +FlZl2i Ne)+O°). Since wecantake thef’stobecoordinate functions, comparison of(vii)and (iii) gives X+%=Z, ZZ 5 FF oy VF t+ Ba+FP+s. which gives ZaX4¥, Z=51K), 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 exptY exp(-tX)) ae ee RX FY RX ceeeayd+? Ho+e[(F+ yet #7-22-72) (e) +00°). Ifwewrite exptXexptYexp(—tX) =exp(tS; +752+O(03)), then wealso have ®) S(exptXexpr¥exp(-X)) =f(expt; +752)+O() =F NE) +PENE 2ee +15151 Me)+ow). Comparing (ix)and(x)gives thedesired result. Notice that formula (2)isaspecial case ofTheorem 5-16 (compare also with Problems 5-16 and 5-18). The work involved inproving Theorem 14isjustified byitsrote inthefol- lowing beautiful theorem. 15.THEOREM. IfGisaLiegroup andHCGisaclosed subset which is alsoasubgroup (algebraically), then HisaLiesubgroup ofG.More precisely, there isaC®structure onH,withtherelative topology, thatmakes itaLiesubgroup ofG. PROOF, Weattempt toreconstruct theLiealgebra ofHasfollows. Leth CGe bethesetofallX€Gesuch that exp1X €Hforalts Assertion I.LetX;€Gewith X;>Xand let4+0with each 440.Suppose exp4X; €Hforalti. Then X€. Proof, Wecanassume f;>0,since exp(—;X;) =(exptjXi)~! €H.For¢>0, let k,(t)=largestinteger<:.i Then fi p~—b<ki() <-, ui fy 392 Chapler 10 so uki(t) >t. Now ky) exp(ki(QuX;) =[exp(aX) |€H, kj(ttXj>Xx. Thus exptX €H,since Hisclosed and expiscontinuous. Weclearly also have exptX €Hfort<0,so X€.QED. Wenow claim that §CGeisavector subspace. Clearly X€f)implies 5X€hforalls ER.IfX,Y €b,wecanwrite by(1)ofTheorem 14 exptX exptY =exp{t(¥ +Y)+1Z(1)} where Z(t) >0.as 1—0.Choose positive 4;>0and let¥)=X¥+Y +Z(t). Then Assertion ]implies that X+Y€h.Alternatively, wecanwrite, forfixed 1. Lyepty) saplixensetinnis oun}; OP PT =exp 2nv? , taking limits asn>oogives exp1(X +Y)€H. [Similarly, using (2)ofTheorem 14weseethat [X,¥] €b,sothat isa subalgebra, butwewillnoteven usethisfact.] Now letUbeanopen neighborhood of0€G,onwhich expisadiffeomor- phism. Then exp(h U)isasubmanifold ofG.Itclearly suffices toshow thal ifUissmal} enough, then HNexp(U) =exp(h VU). Choose asubspace h’CGecomplementary to§,sothatGe=9@f. Assertion 2.The map $:Ge>Gdefined by Q(X+X")=expXexpX" Xeh, Xeh! isadiffeomorphism insome neighborhood of0. Proof. Choose abasis X1,...,X¢s-+-,Xn ofGewith Xj,...,Xk abasis for6.Then¢isgivenby n k n o(Saai) =ep(oa)on(>aii).isl i=) isk+1 LieGroups 393 Since themap 7%, aX; +(a1,...,4n) isadiffeomorphism ofGeonto R”. itsuffices toshow that k n Wi,0-65On)=o(Soarxi ex>ax)ist iak4i isadiffeomorphism inaneighborhood of0€R”.This isclear, since ug=X; E.D. Welauye QED. Assertion 3.There isaneighborhood V'of0in such that expX’¢Hif oFXeV’. Proof.Choose aninnerproduct on'andletKCfy’bethecompact setofall X'€fywith |<|X’| <2.Iftheassertion were false, there would beX;'€fy with X;/>0and expXi’€H.Choose integers n;with njXj' €K. Choosing asubsequence ifnecessary, wecanassume X;'>X'€K.Since I/nj >0, exp(1/ni)(1iX;') €H, itfollows from Assertion ]that X'€§,acontradiction. QE.D. Wecannow complete theproof ofthetheorem, Choose aneighborhood U=WxW'ofG,onwhich expisadiffeomorphism, with Waneighborhood of0€§ W’aneighborhood of0€bh such that W'iscontained inV'ofAssertion 3,and @ofAssertion 2isadiffeomor- phism onWxW’. Clearly exp(h AU)CHNexp(U) Toprove thereverse inclusion, leta€HMNexp(U). Then a=expX expX’ Xew,X'ew'. Since a,expX€Hweobtain expX'€H,so0=X',anda€exp(hOU).& 394 Chapter 10 Uptonow, wehave concentrated ontheleftinvariant vector fields, butmany properties ofLiegroups arebetter expressed interms offorms, Aform wis called Jeft invariant ifLa*w =wforalla€G.This means that w(b) =La*[w(ad)). Clearly, aleftinvariant k-form wisdetermined byitsvalue w(e) €2G) Hence, ifw!,...,w” areleftinvariant I-forms such that!(e),...,@(e) span G.*. then every leftinvariant k-form is >Gi,aigOA NOK VAre! h<oeiy r forcertain constants ay.Ifw!(e),...,w"(e) isthedual basis toX1,..., Xn€Ge. thenanyC®vector fieldX¥canbewritten n X= s/X forC® functions //. jl Then o'(X)= f', sow!isC. Itfollows thatanyleftinvariant form isC™. Ifwisleftinvariant, thenfora€Gwehave La*dw =d(Lq*w) =dw, sodwisalso leftinvariant. The formula onpage 215implies that foraleft invariant I-form wand leftinvariant vector fields ¥and ¥wehave da XP)=KwP) —¥(w(¥) —oF, ¥)) =-w([¥,¥)). Hence (*) do(e)(X, Y)=—w(e)([X, Y)), thebracket being theoperation in. The interplay between teftinvariant and right invariant vector fields isthe subject ofProblem 1}.Here weconsider thecase offorms. LieGroups 395 16.PROPOSITION. Let¥:G>Gbe(a) =a7!. (})Aform@isleftinvariant ifandonlyify*wisrightinvariant. (2)Ifwe€Q*(G,), then Wwe =(—1)kwe. (3)If@isteftandright invariant, then dw=0. (4)IfGisabelian, then gisabelian (converse ofCorollary 7). PROOF. (1)Clearly woR,=Lycy, 80 RoW aly. Tfwisleftinvariant, then Ryo) =VWLy-"0 =yo. soy*w isright invariant. The converse issimilar. (2)Itclearly suffices toprove thisfork=1.Soitisenough toshow that YWae(X) =—XforX€G,.NowXisthetangent vector at¢=0ofthecurve 1+>exptX. SoWaeX isthetangent vector at1=0off>(exptX)7! = exp(—1X); thistangent vector isjust —X. (3)Ifwisaleftand right invariant k-form, then W"(We)=(—1)hae. Since *w and areboth leftinvariant, wehave yo=(-Iko. The form dwisalso eftand right invariant, so w'(dw) =(-1)' dw. But ¥"(dew) =d(y*w) =d((=1)kw) =(-1)* dw. Sodw =0. (4)IfGisabelian, then allteftinvariant |-forms warealsoright invariant. So dw=0forallleftinvariant 1-forms. Itfollows from (*)that [X,Y] =0foralt X,Yeq. 396 Chapter 10 Alternate proofof(4).ByTheorem 14,ifGisabelian, thenforX,Y€Gewe have fa e yieyl+ oO’)=Zhx1+or). Hence 3X,¥]+0003)/0? =YY,X)+OP). Letting ¢>0,weobtain[X,Y]=[Y,X].# Since dwiseftinvariant foranyleftinvariant @,itfollows thatforabasis w', ...,@" ofinvariant I-forms wecanexpress each dointerms ofthew!Aw/. First choose Xj,...,Xn€Gedualtoo'(e),...,"(e). ThereareconstantsC/i such that 7 (XX) =DOChXe: k=l clearly wealso have a (¥i,R= och. kal Thenumbers Cj,arecalled theconstants ofstructure ofG(withrespect tothe basis X1,..., Xnofq).From skew-symmetry of[,]andtheJacobi identity we obtain (1)Cf=Ch 1 2)Drickch, +chef, +chrcl)) =0. h=1 From (#)onpage 394weobtain | } . do!=—S'Cho!nw!=—5Cholaw’.i<j i Itturns outthat(2)isexactly what weobtain fromtherelation d?w* =0.Con- dition (2)isthus anintegrability condition. Infact, wecanprove (Problem 30) thatifCfareconstants satisfying (1)and(2),thenwecanfindeverywhere linearly independent I-forms @!,...,«" inaneighborhood of0€R”suchthat 1 i ; Ao Kk dok=3Gwrw, LieGroups 397 Moreover, theexistence ofsuch w!implies (Problem 29)thatwecandefine a multiplication (a,b) +»abinaneighborhood of0which isagroup asfaras itcan beand which has the w/asleft invariant I-forms. From this latter fact and(asuitable local version of)Theorem 5wecould immediately deduce the following Theorem, forwhich wesupply anindependent proof. 17.THEOREM. LetGbeaLiegroup with abasis ofleftinvariant 1-forms w!,...,@" andconstants ofstructure Cj.LetM”beadifferentiable manifold andlet6!,...,0” beeverywhere linearly independent I-forms onMsatisfying do*=—Cha! nos. i<j Then forevery p€Mthere isaneighborhood Uand adiffeomorphism Jf:U>Gsuchthat ; ; 6!=ftw’. PROOF, Letm:MxG>Mand m2: MxG—Gbetheprojections. Let amo, OK=mtok. Then a6*—of)=—Ch (18!06)-fa!no) i<j =—Lich a@-a/)+6-6)nw), ied ByProposition 7-14, MxGisfoliated byn-dimensional manifolds whose tangent spaces ateachpointareannihilated byall6*—*.Choose a€G andletI’bethefolium through (p,a). Now 6!,...,8",@!,...,0” arelinearly independent everywhere; soon(p,q), which isthesetofvectors in(MxG)(p,a) where 6*—ok=0,thesets6!,...,6” ando!,...,0" areeachlinearly inde- pendent. Hence x: +Mand m:T >Gareeach diffeomorphisms insome neighborhood of(p,a). This means that Icontains thegraph of adiffeomorphism ffrom aneighborhood Uofptoaneighborhood ofa. : Ki M 398 Chapter 10 Letf:U>MxGbethemap AM =GSM) CF. Since 6*—a=0onIT,wehave o=Te —ok)=Pino" =frmto® =(m0f)"* ~(rz0fy*o* =O —ftok. & Itisalsopossible tosaybyhow much anytwosuch maps differ: 18.THEOREM. LetMbeaconnected manifold, letGbeaLiegroup, and letfi,fa:M>GbetwoC®mapssuchthat fi*(w) =fr*(w) forallleftinvariant I-forms w.Then f;and /2differ byalefttranslation, that is,there isa(unique) a€Gsuch that fr=Lao fi. PEDESTRIAN PROOF. Case 1.M=Randthetwomaps y;,¥2: R>Gsatisfi ¥;(0) =72(0). Wemust show that y;=y2.Forevery leftinvariant 1-form w we have dy\_., (d|\_1,(a wrson(dtJenOOler}=n'o(a; dy,=(49)(*) dy, =[romca-*o(200))] () dy =w(y2(t))[40mm], “n): Itfollows thar dn[L ]a di rekon |G Ifweregard y,asgiven, andwrite thisequation outinacoordinate system. then itbecomes anordinary diflerential equation fory,(ofthetype considered LieGroups 399 intheAddendum toChapter 5),soithas aunique solution with theinitial condition y2(0) =¥;(0). Butthissolution isclearly yy=y;. Case2.M=R,butthemaps y,,¥2avearbitrary. Choose a€Gsothat ¥2(0) =a y,(0). If@isaleftinvariant |-form, then (La0y)*(w) =1y"(La*w) =y"(w) =y2"(w). Since Lq0¥;(0)=y2(0), itfollows from Case Jthat Laoy, =¥2- Case3.General case.Letpo€M.Choose a€Gsothat S2(Po) =4filpo). Forany p€Mthere isaC® curve c: R>Mwith c(0) =poand e(1) =p. Let 7;=froc. Then V2"(w)=e*fy*(w) =c*fi"(w) =y;"(@). ByCase 2,wehave volt) =a-y,(t) forall1. inparticular for¢=1,sofo(p) =@- fi(p)- ELEGANT PROOF. Leta;:GxG—Gbeprojection onthei**factor. Choose abasis w!,...," fortheleftinvariant I-forms. For(a,b) €GxG,let a . Aco)=()ker(n*o! —m2"). i=l ThenAisanintegrable distribution onGxG.Infact,ifA(G)CGxGisthe diagonal subgroup {(a,a) :a€G},then themaximat integral manifolds ofA aretheleficosets ofA(G). Now define h:M—GxGby (p) =(h(p), Alp). Byassumption, W(t"! —m*o') =fitw! —foto! =0. Since Misconnected, itfollows that 2(47) iscontained insome left coset ofA(G). Inother words, there area,b€Gwith afi(p) =bfa(p) forall peM.& 400 Chapter 10 19.COROLLARY. IfGisaconnected Liegroup andf:G>GisaC™ map preserving leftinvariant forms, then f=Lgforaunique a€G. While leftinvariant |-forms play afundamental role inthestudy ofG,the jeftinvariant n-forms arealsovery important. Clearly, allleftinvariant n-forms areaconstant multiple ofanynon-zero one. Ifo”isaleftinvariant n-form. then o”determines anorientation onG,and iff: G>RisaC™ function with compact support, wecandefine [fo".CG Since o”isusually kept fixed inanydiscussion, thisisoften abbreviated to fforff(a)da. G G The latter notation hasadvantages incertain cases. Forexample, leftinvariance ofo”implies that [fla)da=ff(ba)da. G G inother words, ffo"=fgo",whereg(a)=f(ba);CG G |note that Lyisanorientation preserving diffeomorphism, so ffor=[Ls'(for) =f(foLs)Lsto" =f(feLs)o".G G (ol G which proves theformula]. Wecan, ofcourse, also consider right invariant n-forms. These gencrally turn outtobequite different from theleftinvariant n-forms (seetheexample inProblem 25). Butinonecase they coincide. 20.PROPOSITION. IfGiscompact andconnected andwisaleftinvariant n-form, then@isalsorightinvariant. PROOF. Suppose w#0.Foreach a€G,theform Rq* isleftinvariant. so there isaunique real number f(a) with Reto =flaw. Since Ra* oR5* =(Ras)*, wehave S(ab) =f(ba) =f(a): f(d). So/(G) CRisacompact connected subgroup ofR—{0}. Hence f(G) ={1}. LieGroups 401 ‘We can also consider Riemannian metrics onG. Inthe case ofacompact group Gthere isalways aRiemannian metric onGwhich isboth leftand right invariant. Infact, if(,)isanyRiemannian metric wecanchoose a bi-invariant n-form o”anddefineabi-invariant ((,))onGby (Wm fan Reel),LaeRoW)dad. GxG Wearefinally ready toaccount forsome terminology from Chapter 9. 21.PROPOSITION. LetGbeaLiegroup withabi-invariant metric. (})Foranya€G,themapIg:G>GgivenbyJ4(b)=ab~'aisanisometry which reverses geodesics through a,i.e.,ifyisageodesic andy(0) =a,then Ta(y(t)) =¥(—#). (2)The geodesics ywith y(0) =eareprecisely theI-parameter subgroups ofG,i.e.themaps f+ exp(tX) forsome X€4. PROOF. (t)Since Te(b) =b=", themap Jeu: Ge+Geisjust multiplication by1(see theproof ofProposi- tion 16(2)), soitisanisometry onGe.Since Te=Ry-tTeLg- foranya€G,themap Ie4: Ga>Gg-1 isalsoanisometry. Clearly Iereverses geodesics through e. Since Iq=Rale Ra, itisclear that Jaisanisometry reversing geodesic through a. (2)Lety:R>Gbeageodesic withy(0)=e.Forfixed¢,let Pu) =ye+4). Then 7isageodesic and7(0) =y(t). So Tyley(@))=Tyg(—¥))=Ly(4—9) =Wt+u) =y(t 2). But also Ty@le(b) =yoy), 402 Chapler 10 so y(t)y(u)y() =y(u +28). Itfollows byinduction that y(nt) =y(t)” foranyinteger n. Ift!=n'tand t"=n"tforintegers n’and n",then yee) =yO" =YEDYC, soyisahomomorphism onQ.Bycontinuity yisaI-parameter subgroup. These aretheonly geodesics, since there are1-parameter subgroups with anytangent vector at¢=0,andgeodesics through earedetermined bytheir tangent vectors at¢=0, Weconclude thischapter byintroducing some neat formatism which allows ustowrite theexpression fordw* inaninvariant waythatdoes notusethe constants ofstructure ofG.IfVisad-dimensional vector space, wedefine a V-valued k-form onMtobea function wsuch thateach w(p) isanalternating map o(p): M,x++-x Mp >V. ne Atimes: Ifuj,...,0gisabasisforV,thenthereareordinaryk-formsw!,...,w4 such that forX),...,X% €Mpwehave a . (P(X, ++Xe)=Do!(p(X. Xvi: ist wewillwrite simply a wo=Yo! -y. i=l ForanyV-valued k-form wwedefine aV-valued (k+1)-form dwby d dw= dw! -vj; i=l asimple calculation shows that thisdefinition does notdepend onthechoice of basis u,...ug forV. LieGroups 403 Similarly, suppose p:UxV>Wisabilinear map,whereUandVhavebases#),...,ue andti,...,0a,respectively. IfwisaU-valued k-form ‘ w=Pol -u; ist and9isaV-valued /-form a n=Yon -y, j=l then cd | | Vol an’: puny) i=l j=l isaW-valued (k+/)-form; acalculation shows thatthisdoes notdepend onthe choice ofbases w1,...,W%¢ OFU;,-..,Ug. Wewilldenote this W-valued (k+/)- form byp(w A7). These concepts have anatural place inthestudy ofaLiegroup G.Although there isnonatural way tochoose abasis ofleftinvariant 1-forms onG,there anatural -valued I-form onG,namely theform wdefined by (+) w(a)(¥(a)) =X€g. Using thebilinear map [,]:¢xq>9;wehave, forany g-valued k-form n and any q-valued /-form 4onG,anew q-valued (k+/)-form [yAA]onG. Nowsuppose thatX),...,Xn €Ge=qisabasis,andthatw',...,w" isadualbasisofleftinvariant I-forms. Theformwdefined by(x)canclearlybe written a o=Yo* Xx. k=l Then n (i) dw=do* .X, kel a ;=(Lhe! Ao!)“Xp.kai Si<j 404 Chapter 10 On the other hand, n 1%,Xj]=Ch, kel so nnn a(2) [onal= (LV choaa!Xe).kay Niet j=t Comparing (1)and(2),weobtain theequations ofstructure ofG: ello=— zlAo). The equations ofstructure ofaLiegroup willplay animportant role in Volume III.Forthepresent wemerely wish topoint outthattheterms dwand [wA@]appearing inthisequation canalsobedefined inaninvariant way. For theterm dwwejustmodify theformula inTheorem 7-13: IfUisavector field onGandfisag-valued function onG,then(Problem 20)wecandefine a q-valued function U(/) onG.Ontheother hand, w(U) isaq-valued function onG.For vector fields Uand Vwecan then define dw(U,V) =U(a(V)) —V(w(U)) —@([U, ¥)). Recallthatthevalueata€Goftherightsidedepends onlyonthevaluesUaandVaofUandVata. Ifwechoose U=¥,V=¥forsome X,Y €Ge, then des(a)(Xq, Ya)=0~0~w(a)({¥, Pa) =o )(X, Ye) since [¥,7]isleftinvariant =—w(e)([X, Y) bydefinition of[,]inGe =-[%,Y¥] naPy s\bydefinitionofw. =—[w(a)(Xa), o(@)(Ya)] Itfollows that foranyvector fields Uand Vwehave dw(U, V)=—[w(U), o(V)). Problem 20givesaninvariant definition ofp(wAn) andshowsthatthisequation isequivalent totheequations ofstructure. LieGroups 405 WARNING: Insome books theequation which wehave justdeduced appears asdw(U,V) =—}[w(U),@(V)]. Theappearance ofthefactor }herehas nothing todowiththe4intheother formofthestructure equations. Itcomes about because some books donotusethefactor (k+/)!/K!/! inthedefinition ofA.Thismakestheir4A7equalto4ofoursforI-forms Aandy.Thenthe definition ofa(Sw;dx")as da;Adx!makes their dwequal to4ofours for 1-forms w. 406 Chapter 10 PROBLEMS 1.LetGbeagroup which isalso aC®manifold, andsuppose that(x,y)>xy isC& (a)Find f-!when f:GxG>GxGisf(x,y) =(x,xy). (b)Show that(¢,e) isaregular point of/. (c)Conclude that GisaLiegroup. 2.LetGbeatopological group, andHCGasubgroup. Show thatthe closure HofHisalsoasubgroup. 3.LetGbeatopological group and HCGasubgroup. (a)IfHisopen, then soisevery coset gH. (b)IfHisopen, then Hisclosed. 4.LetGbeaconnected topological group, and Uaneighborhood ofe€G. LetU”denote altproducts a-++ayfora;€U. (a)Show thatU"*! isaneighborhood ofU". (b)Conclude that U,U"=G.(Use Problem 3.) (c)IfGislocally compact and connected, then Giso-compact. 5.Letf:R">R"bedistance preserving, with f(0) =0. (a)Show that /takes straight lines tostraight lines. (b)Show that /takes planes toplanes. (c)Show that fisalinear transformation, and hence anelement ofO(n). (d)Show that any element of£() canbewritten A-t forA€O(n) andta translation. 6.Show thatthetangent bundle 7GofaLiegroup Gcanalways bemade into aLiegroup. 7.Wehave computed that forM€gl(n,R) wehave Pa Pa a a aM=YO)MxM=. whereMx!"(A) =YMatAna.kl ont (a)Show that this means that M(A) =A-M €R™ =GL(n,R)s. (tisactually clearapriori thatMdefined inthiswayisleftinvariant, forL4.= Lasince Lyistinear) (b)Find theright invariant vector field with value MatJ. LieGroups 407 8.LetGand Hbetopological groups and ¢:U>Hamap onaconnected openneighborhood Uofe€Gsuchthat¢(ab) =¢(a)6(b) whena,b,ab€U. (a)Foreach¢€G,consider pairs(V,),whereVCGisanopenneighbor-hoodof¢withV-V7!¢U,andwhere ¥:V>Asatisfies w(a)-w(b)7) =g(ab™) fora,b€V.Define (Vi,~1)y(V2,¥2)ifvr=Woonsomesmaller neighborhood of¢.Show thatthesetofallyequivalence classes, forallc€G, canbemade into acovering space ofG. (b)Conclude thatifGissimply-connected, then ¢canbeextended uniquely toahomomorphism ofGinto H. 9.InTheorem 5,show that¢andyareequal even ifthey aredefined only onaneighborhood Uof¢€G,provided thatUisconnected. 10.Show that Corollary 7isfalse ifGisnotassumed connected. 11.IfGisagroup, wedefine theopposite group G°tobethesame setwith themultiplication «defined bya+b =b-a. IfgisaLiealgebra, with operation [,],wedefine theopposite Liealgebra q°tobethesame setwith theoperation [X,Y]? =-[¥,Y]. (a)G°isagroup, andif¥:G>Gisarsa7,thenyisanisomorphism from GtoG°. (b)9°isaLiealgebra, andX++—Xjsanisomorphism ofgonto ¢°. ()£(G°) isisomorphic to[.£(G)]° =a°. (A)Let[,]betheoperation onG,obtained byusing right invariant vector fieldsinstead ofleftinvariant ones.Then(g,[,])isisomorphic to£(G°), and hence to9°. (e)Usethistogiveanother proofthatqisabelian whenGisabelian. 12. (a)Show that wp{° 7)a( c8@ sinaPla o}~\-~sina cosa)’ (b)Use thematrices Aand Bbelow toshow that exp(A +B)isnotgenerally equal to(exp A)(exp B). ol 0 0 13. Let X,Y €Gewith [X,Y] =0. (a)UseLemma 5-13toshowthat(expsX)(exptY) =(expt (expsX). (b)More generally, useTheorem 5toshow that exp isahomomorphism onthesubspace ofGespanned byXand Y.Inparticular, exp(¥ +¥)= (expX)(expY). 408 Chapter 10 14.Problem 13implies that exp¢(X +Y) =(exptX)(expeY) if[X,Y] =0. A moregeneral resultholds. Let¥andYbevector fieldsonaC®manifold M with corresponding tocal I-parameter families oflocal diffeomorphisms {¢r}, {Ws}. Suppose that[X,Y] =0,andletm=$1©Wr=Wrgr. (a)Show that dPUP) Xenp))+$ualYHUD)- (b)Using Corollary 5-12, show that diPUP)=Xaoy)+Yeu). Inother words, {nr} isgenerated byX+Y. 15.(a)IfMisadiagonal matrix with complex entries, show that detexp M=ettaceM | (b)Show thatthesame equation holds foralldiagonalizable Mwith complex entries. (c)Conclude that itholds forallMwith complex entries. (The diagonalizable matrices aredense; compare Problem 7-15.) (a)Using Proposition 9,show that forthehomomorphism det: GL(n,R) > R—{0},themapdet,:ql(z,R) +£(R—{0})=RisjustM+ traceM. (e)Use this fact togive afancy proof that trace MN =trace NM. (Look at trace(MN —NM) =trace[M, N),) (f)Prove theresult inpart (d)directly, without using (c).(Since det, and trace arehomomorphisms, itsuffices tolook atmatrices with only onenon-zero entry.) (g)Now usethisresult andProposition 9togive afancy proofof(c). 16.(a)LetUbeaneighborhood oftheidentity (1,0) ofS?(considered asa subsct ofR?). Show thatnomatter how small Uis,there areelements a€U which have square roots outside Uinaddition totheir square root inU. (b)Show that foreach n>|,there isaneighborhood Uofe€Gsuch that every clement inUhasaunique 7"rootinU. (Q)ForG=S?,show thatthere isnoneighborhood Uwhich hasthisproperty foralln. 17.(a)Let(x,V)beacoordinate system around e€Gwith x/(e) =0. Let x!(ab)=f(x" (a),..., (a), x"(b), «4x(8)) LieGroups 409 forC®functions /’.Show that Dzf'(0) =Dns;f'(0)=8. (b)Ifa,B:(~e,) +Garedifferentiable, show that (a-BY'(0) =a'(0) +B'(0). ()AlsodeducethisresultfromTheorem 14(1).(Noteventhefullstrength of(1) isneeded; itsuffices toknow that exptX exptY =exp{t(X +Y)+O(t)}. The argument ofpart (a)isessentially equivalent totheinitial part ofthededuction of(1).) 18.LetGbeaLiegroup,andletHCGbeasubgroup ofG(algebraically), such that every a€Hcanbejoined toe¢byaC™ path lying inH.Let§CGe bethesetoftangent vectors toallC®paths lying inH. (a)Show that isasubalgebra ofGe.(Use Theorem 14.) (b)LetKCG betheconnected Liesubgroup ofGwith Liealgebra §.Show that HCK.Hint: Join any a€Htoe byaC™ curve ¢,and show that the tangent vectors of¢lieinthedistribution constructed intheproofofTheorem 4. (c)Letc1,...,¢% becurves inHwith {¢;/(0)} abasis forh.Byconsidering themap f(t',...,0%) =a(t!)-++cx(t), show thatKCH.Thus, HisaLie subgroup ofG.ItiseventruethatHCGisaLiesubgroup ifHispath connected (bynotnecessarily C® paths); seeYamabe, Onanarcwise connected subgroup ofaLiegroup,OsakaMath.J.2(1950),13-14. (d)IfHCGisasubgroup andanimmersed submanifold, thenHisaLie subgroup. 19.ForaeG,consider themap b+ aba~! =LaRa7(b). The map (LaRa at§>9 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), whereAu!(q), theautomorphism groupof4q,isthesetofallnon-singular linear transformations ofthevector space gqonto itself (thus, isomorphic toGL(n, R) ifqhasdimension n).The map Adiscalled theadjoint representation. (b)Show that exp(Ad(a)X) =a(exp X)a7'. Hint:Thisfottowsimmediately fromoneofourpropositions. 410 Chapter 10 (c)ForA€GL(n,R) andM€gl(n,R) showthat Ad(A)M =AMA™}. (Itsuffices toshow thisforMinaneighborhood of0.) (d)Show that Ad(exptX)Y =¥+t[X,Y]+ O(:). (e)Since Ad: G+g,wehave themap oat tangent space ofAut(q) attheAde: (=Ge)>identity mapIqof@toitself. This tangent space isisomorphic toEnd(q), where End(q) isthevector space of alllinear transformations ofqintoitself:If¢isacurveinAut(q) with¢(0)=Ig, then toregard ¢’(0) asanelement ofAut(q), weletitoperate onY€gqby d (0)(Y) == Y). (OY)i0) (Compare with thecase q=R",Aul(g) =GL(n,R), End(q) = xnmatrices.) Use (d)toshow that Adse(X)(Y) =[X,Y]. (Aproof mayalsobegiven using thefactthat[¥,7] =Lg¥.) Themap Y+[X,Y] isdenoted byadX€End(). (f)Conclude that 2 Ad(exp X)=exp(adX)=Igsadn4SEAN. . (g)LetGbeaconnected Liegroup andHCGaLiesubgroup. ShowthatH isanormal subgroup ofGifand only if =£(H) isanideal ofg=£(G), that is,ifand only if[X,Y] €forall¥eq, Y€§ 20.(a)Letf:M—V,whereVisafinitedimensional vectorspace,withbasis V},.-+,04. ForXp€Mp, define X,(f) €Vby dg. XN)=VOX) vi. tsi where f=4,fi-v; forf!:M—R.Show thatthisdefinition isindepen- dentofthechoiceofbasis vj,...,va forV. LieGroups 411 (b)IfwisaV-valued k-form, show that dwmay bedefined invariantly bythe formula inTheorem 7-13 (using thedefinition inpart (a)). (©)Forp:UxV—W,show that p(wA7)maybedefined invariantly by (WA NX, ..-, Xe,Xepas ses Xegd) 177 YEsgne-p(w(Xoays ++,Xo)»M(Xorkegs)s +++»Xock41))- oeSk41 Conclude, inparticular, that [wAw\(X,Y)=2[w(X), o(¥)]. (d)Deduce thestructure equations from (b)and(c). 21.(a)IfwisaU-valued k-form and7isaV-valued /-form, and p: UxV> W,then d(p( An)) =p(dw an)+(-1)k p(wadn). (b)Forag-valued k-form wand /-form wehave form =(“I aa. (c)Moreover, if4isag-valued m-form, then ("oa tad +(-D" faDaal] +(-I"— a[nAa]=0. 22.Let GCGL(n,R) beaLie subgroup. The inclusion map G>GL(n,R) > R™willbedenoted byP(for“point”). Then dPisanR”-valued 1-form (it corresponds totheidentity mapofthetangentspaceofGintoitself).Wecan alsoconsider dPasamatrix of1-forms; itisjustthematrix (dx!/), where each dx"jsrestricted tothetangent bundle ofG.WealsohavetheR"’-valued I-form (ormatrix of1-forms) P~!.dP,where -denotes matrix multiplication, andP-denotes themap A+AWonG. (a)P7).dP =p(P'adP), where p:R™xR”—R’”ismatrix multiplication. (b)LatdP =A-dP. (Use ftd =df*) (0)P~)dP isteftinvariant; and(dP)+ P~isright invariant. (a)P-. dPisthenatural q-valued I-form wonG.(Itsuffices tocheck that P.dP =wat 1) (&)Using dP=P-w, show that 0=dP-w+P-dw, where thematrix of 2-forms Pdwiscomputed byformally multiplying thematrices of1-forms dP and w.Deduce that dw+o-w =0. 412 Chapter 10 Ifwisthematrix of}-forms w=(w"/), thissaysthat deli=—ro no, k Check thattheseequations areequivalent totheequations ofstructure (use the form dw( X,Y)=—[o(X),(¥)].) 23.LetGCGL(2,R) consistofallmatrices (¢)with@#0.Forconve- nience, denote thecoordinates x!!andx!?onGL(2,R) byxandy. (a)Show that forthenatural q-valued form wonGwehave lfdx dy os=(00). sothat dx/x and dy/x areleftinvariant 1-forms onG,and aleft invariant 2-form is(dxAdy)/x?. (b)Find thestructure constants forthese forms. (c)Show that -)_ 1 (dx -ydx+xdpone y iy (dP). P=x(0 0) andfind theright invariant 2-forms. 24.(a)Show thatthenatural q{(n, R)-valued |-form wonGL(n, R)isgiven by 1Sik pk is fkaykl o>FettxeB)XYa where (98) =det(x%) -(x2)-". (b)Show thatboth theleftandright invariant ?-forms aremultiples of 1 eg (dxneNdxMY)NoA(dxANd), (det(x#4))"y 25.The special linear group SL(”,R) CGL(n,R) isthesetofallmatrices of determinant | (a)Using Problem 15,show thatitsLiealgebra sl(1, R)consists ofallmatrices with trace =0. LieGroups 413 (b)Forthecase ofSL(2,R), show that “1p =m vdx—ydu udy—ydv =IT(aes: -udy+xdv}* where weusex,y,u,vforx!!,x!?,x71,x22. Check thatthetrace is0by differentiating theequation xv—yu=|. (©)Show that aleftinvariant 3-form is udxaduAdy—ydxaduAdv. 26.ForM,N €o(n) =£(O(m)) ={M: M=—M%}, define (N,M)=—traceM -Nt (a)(,)isapositive definite inner product on0(n). (b)IfA€Of), then (Ad(A)M, Ad(A)N) =(M,N). (Ad(A) isdefined inProbtem 19.) (c)The leftinvariant metric onO(n) with value (,)atO(7); isalso right invariant. 27.(a)IfGisacompact Liegroup, then exp: ¢+Gisonto. Hint: Use Proposition 21. (b)LetA€SL(2,R). Recall thatAsatisfies itscharacteristic polynomial, soA?—(trace A)A+I=0.Conclude thattraceA?>—2. (c)Show thatthefollowing element ofSL(2,R) isnotA?foranyA.Conclude that itisnotintheimage ofexp. 20(0 -1/2 (a)SL, R)does nothave abi-invariant metric. 28. Letxbeacoordinate system around ¢inaLiegroup G,let7j: GxG>G betheprojections, andlet(y,z) bethecoordinate system around (e,¢) given byyp!=x!om,2!=x!oma. Define o!:GxG>Rby (a,b) =x!(ab), 414 Chapter 10 and letX;betheleftinvariant vector field onGwith a Xj(e)=wl, (a)Show that nogos JSaeLiax!" jar where ; ; ag!Wa)=Fae). (b)Using LaLy=Lap, show that {LavXi(b)\(x") =[Xi(a6)](2"). Deduce that Xi(b)(x! 0La)=wi(ab). and then that - SSicpy o¢! 1Dewi) Fz) =Wilad). j=l~ Letting ¥=(#})betheinverse matrix ofy=(¥)),wecanwrite ag! u .Gab= Deviled) Ho. Thisequation (oranyofnumerous things equivalent toit)isknown asLie’sfirst fundamental theorem. The associativity ofGisimplicitly contained init,since we used the fact that LaL, =Lop. (0)ProvetheconverseofLie’sfirstfundamental theorem,whichstatesthefollowing Let¢=(¢!,...,6") beadifferentiable function inaneighborhood of0€R?" [with standard coordinate system y!,...,¥",=!,...,2/] such that o(a,0) =a foraeR". Suppose therearedifferentiable functions yinaneighborhood of0€R”[with standard coordinate system x',...,x"] such that ¥}0)=8} ()28gb=vieleby)Gi) 2(@,)inaneighborhoodanf” a u of0R™, LieGroups 415 Then (a,b)+$(a,b)isalocal Liegroup structure onaneighborhood of0€R” (itisassociative and hasinverses forpoints close enough to0,which serves as theidentity); thecorresponding leftinvariant vector fields are fn aa aoeX=ovai j=l [Toprove associativity, note that a9!(a,b),2)_ ziSS [email protected])-VE) by ist andthenshowthat(a,¢(d,2))satisfies thesameequation.] 29.Lie’ssecond fundamental theorem states thattheleftinvariant vector fields X;of aLiegroupGsatisfy n (XX) = hx kal forcertain constants chin otherwords, thebracket oftwoleftinvariant vector fields isleftmvariant. The aim ofthisproblem istoprove theconverse ofLie's second fundamental theorem, which states thefollowing: ALiealgebra ofvector fields onaneighborhood of0€R",which isofdimension 7over Rand contains a basis forRo, isthesetofleftinvariant vector fields forsome local Liegroup structure onaneighborhood of0€R”. (a)Choose X;,...,X, intheLiealgebra sothatX;(0) =/x!|p andset 1j07 jX= Wa j=l If a : : ol=)yjdx!, jal then thew!arethedual forms, andconsequently dot=-cho! nw! Chconstants. i<j ()Letxj:R"xR"—R”betheprojections. Then n n mato—mo!=i)oma[ac!072)—(yj072)-nite!] ist i=l 416 Chapter 10 Consequently, theidealgenerated bytheforms d(x!o72)— 7,(W/om2)-mi*w! isthesame astheideal £generated bytheforms 22*w/ —z;*w/. Using thefact thattheCj,areconstants, show thatd(4)C4.Hence R”xR"isfoliated by n-dimensional manifolds onwhich theforms d(x!072)— Yi, (Wjom2)mito! allvanish. ()Conclude, asintheproof ofTheorem 17,that forfixed a,there isafunction q: R”—R"satisfying ,(0)=aand A dl(b)=>w}(a(b))-w'(b), ist orequivalently; ao, Ol alm= Lwio) Hj). i= Now set$(a,b) =0a(), andusetheconverse ofLie’s firstfundamental theo- rem. 30.Lie’sthirdfundamental theorem states thattheC},satisfy equations (1)and(2) onpage 396, i.e.,that theleftinvariant vector fields form aLiealgebra under [.].Theaimofthisproblem istoprovetheconverse ofLie’sthirdfundamental ‘keorem, which states that anyn-dimensional Liealgebra istheLiealgebra for some local Liegroup inaneighborhood of0€R”. LetCtbeconstants satisfying equations (1)and(2)onpage396.Wewould like tofind vector fields ¥;,..., Xnonaneighborhood of0€R®suchthat 1%).Xj]=Dhar ChXe.Equivalently, wewanttofindforms w!with dot=~ cho! rw. ty Then the result will follow from the converse ofLie’s second fundamental the- orem. (a)LetAkbefunctions onRxR”such that ant x Kind=e —Vckxin ad hk(0,x) =0. These areequations “depending ontheparameters x”(seeProblem 5-5(b)). Note thatAk(1,0)=841,sothathk(1,0) =8%.Leto*betheI-form onRxR® defined by ok=Sk dx’. 7 LieGroups 417 and write dok=i 4(dtaa’), where A*and edonotinvolve dz.Show that anknk , ke St) dst adi x»(anthe|&Adx ak=dxk—»ckx'o4, aj (b)Show that du=dtn(-Vickax aci-Yohx).aj ad (c)Let 1 :. ok=ak+5Ghatad!ay Show that d9h=arn(~Sochx' —DYchaix’a*no!)bd ij ons +terms notinvolving dr. Using ;1 ; LU GGoAo!=5VYGHG —GEGano! jones a ns 1 A ; =5LUG +AG,)oao!, andequation (2)onpage 396, show that do=din(-Vajx' +5Vici x"ofac!) ij “ij ons +terms notinvolving dr. Finally deduce that dok=den~Chx16!+terms notinvolving de. dl 418 Chapter 10 (4)Wecanwrite ok=gkdx!dx. i<j where gh(0.x)=0(Why?). Using (0),showthat agi, pe kyaeLex si Conclude that 6*=0. (c)Wenow have 1 ae=-3 Go! adi. bl A I P dot=~5Lhe! Ao!+(dtna). if Show thattheforms w*(x) =o*(1,x)satisfy 7 I dok==Vcjo'nro!. if CHAPTER 11 EXCURSION IN THE REALM OF ALGEBRAIC TOPOLOGY Tschapterexploresfurtherproperties ofthedeRhamcohomology vector spaces ofamanifold. Ourmainresults willberestatements, intermsofthe deRham cohomology, offundamental properties oftheordinary cohomology which isstudied inalgebraic topology. Because wedeal only with manifolds. manyoftheproofsbecome significantly easier.Ontheotherhand,wewillbe using some ofthemain tools ofalgebraic topology, thus retaining much ofthe flavor ofthatsubject. Along theway wewilldeduce allsorts ofinteresting con- sequences, including atheorem about thepossibility ofimbedding n-manifolds inRet LetMbeamanifold with M=UUV foropen setsU,V CM.Before examining thecohomology ofMwewillsimply lookatthevector space Ck(M) ofk-forms onM. Let iy:U>M iv:VoM ju: UNV 3U jviUNV>V betheinclusions. Then wehave twolinear maps aand f. kasiv Oi y kb=iv-iv" chm)————» c#(u) @ck(v)———> chunv) defined by a(w) =(iy*(@), iv*(@)) BlAt,A2) =ju*(As) =jv*(2). Here iy*(w) isjust therestriction ofwtoU.etc. Clearly Bow=0:Inother words, imagea CkerB.Moreover, theconverse holds: kerBCimagea. For, ifB(As,A2) =0,then Ay=A2onUNV. sowecan define wonMtobeAy onUand AgonV.and then a(w) =(As,42). The equation imagea=kerBis expressed bysaying that theabove diagram isexact atthemiddle vector space. ‘Wecanextend thisdiagram byputting thevector space containing only 0atthe 419 420 Chapter 1] ends; thearrows ateither cnd ofthefollowing sequence aretheonly possibl linear maps. 1,LEMMA. The sequence Bo>chum+ckuye@ ckiv)>chUNY) +0 isexact atallplaces. PROOF. tisclear that@isone-one. This isequivalent toexactness atC(M), sincetheimageofthefirstmapis{0}CC(M). Similarly, exactness atCK(UN V)isequivalent toBbeing onto. Toprove that isonto, let{¢u. gv}bea partition ofunity subordinate to{U,V}. Then w€CK(U NV) is o=Bigvw, ~gue). where ¢y@ denotes theform equal to¢yw onUNV, and equal to0on U-(UNV). + Byputting inthemaps d.wecanexpand ourdiagram asfollows, : : || | 0—— c#(m) —*— chu @ch(vy4chunvy-—0 la |eed la 0 chy 2cH @ch(V) 2chH(U AV)—+0 | | | i sothat therows areallexact. 1tiseasy tocheck that thisdiagram commutes. that is,anytwocompositions [rom onevector space toanother areequal: a (d@d)oa=aod laoa=4|: b. doB=Bo(d@d) la5ded|p Excursion intheRealm ofAlgebraic Topology 421 Our first main theorem depends only onthesimple algebraic structure in- herent inthis diagram. Toisolate this purely algebraic structure, wemakc thefollowing definitions. Acomplex Cisasequence ofvector spaces C’. k=0,1,2,... ,together withasequence oflinear maps dksCk cht satisfying d*+! od‘ =0,orbriefly, d?=0.Amap a:C;>CGbetween complexes isasequence oflinear maps ak:ok=ok such that thefollowing diagram commutes forallk. A ck—*— c# at] ascit attoe Themost important examples ofcomplexes areobtained bychoosing Ch= C*(M) forsome manifold M,with d*theoperator donk-forms. Another example, implicit inourdiscussion, isthedirect sum C=C;@C2oftwo complexes, defined by Cackect d=dkeds. Foranycomplex Cwecandefine thecohomology vector spaces ofCby ' kerd*AKC) =——_5-Ge)imagedé~! Naturally, ifC={C*(M)}, then H*(C) isjustH*(M). Ifa: GC,>Czisa map between complexes, then wehave amap, also denoted bya. a:HEC) >HG). Todefine awenote thatevery clement ofH*(C}) isdetermined bysome x€CKwithd(x) =0.Commutativity oftheabove diagram shows that dk(ok(xy)=akdyk(x) =0,sook(x)determines anelement ofH*(C)), which wedefine tobea(the class determined byx).This map iswell-defined. 422 Chapter 1] forifwechange xtox+di*~!(y) forsome y€C)*~!, theno*(x)ischanged to ak(x+dk") =a(x)+0dik") =a(x) +dgk-"ak“"(y)), which determines thesame element ofH*(G). When C\*=Ck(M), Gk= CK(N), andw:Ch(M) >CE(N) isf*forf:N>M,thenthismapisjust St:HE(M) >HFN). Now suppose thatwehave anexact sequence ofcomplexes a B 03 G > @— G0. which really means avast commutative diagram inwhich allrows areexact. Poof |6—— ct0oktBO ok nt far [er [at Aak kB* k ® 9———+c4 —*_+o* —£_.qt———0 lat las |ask ! !o0——> qt!att of_becet)0 | | What doesthisimply about themaps a:H*(C,) >H*(Cy) andB:Hk(G) > H*(C3)? Thenicest thing thatcould happen would beforthefollowing dia- gram tobeexact: Ak Ook Buk 0 H"(G) —H*(Q) —H*(G)> 0. This isnottrue. Forexample, ifUandVareoverlapping portions ofS?for which there isadeformation retraction ofUNV intoS!,then wehave anexact sequence 0+chs) chuyeckiy) >chunY)>0.(eee‘2 Excursion intheRealmofAlgebraic Topology 423 butnotanexact sequence 0—— 1(S?) ——H'(U) @H}(V) —— H'(U nv) ——0. z x z 0 0 R Nevertheless, something very nice istrue: 2.THEOREM. If0>C;— G—> C;>0isashort exact sequence of complexes, then there arelinear maps bh:AKC) >HVC) sothatthefollowing infinitely long sequence isexact (everywhere): 0 a0, Bo ow 0— H°(C)) — H%Ca) —H°(C3) —HVC) = a B 5vee HAC) >HNC) >HRC) >HIG) PROOF, Throughout theproof, diagram (*)should bekept athand. Letx€ CFwithd3*(x) =0.Byexactness ofthe middle rowof(#),there isy€C2 with B*(y) =x. Then 0=disk(x)=dsBK(y)=BATdsi(y). Sody*(y) €kerB**!=imagea**!; thusdy*(v) =ok+!(z) forsome (unique) =€C+!, Moreover. ahaKAN(2)=dohMaktN(2)=drtthdyk(y) =0. Since ak*+! jsone-one, thisimplies thatdj*+!(z) =0,sozdetermines anele- ment ofH*+1(C;): thiselement isdefined tobe5*oftheelement ofH*(C3) determined byx. Inorder toprove that5*iswell-defined, wemust check thattheresult does notdepend onthechoice ofx€C3*representing theelement ofH*(C3). So wehave toshow that weobtain 0¢H*+!(C)) ifwestart with anelement of theform d3k—!(x’) forx’€Csk-!. Inthiscase, letx’=B*-!(y"), Then xdN) =dk BIN) =Bhagk"(y), sowechoose d2*~!(3") as+.This means thatd;*(y:) =0,andhence >=0. 424 Chapter 11 Itisalso necessary tocheck that ourdefinition isindependent ofthechoice ofywithB*(y) =x;thisislefttothereader. The proof that thesequence isexact consists of6similar diagram chases. We willsupply theproof that kera Cimage6.Letx€Cisatisfydik(x)=0,and suppose thata*(x) €Cy*represents 0€H*(C)). Thismeans thata(x) = a-"(y) lorsome y€Gk“! Now ask BAN 3)=Bak Ny) =Bhat(x)=0. SoB*~!(y) represents attelement ofH*-!(C3). Moreover. thedefinition of5 immediately shows that theimage ofthiselement under 6isprecisely theclass represented byx.¢ Itisaworthwhile exercise tocheck that themain step intheproof ofTheo- rem 8-16 isprecisely theproof that kera Cimage 6,together with thefirst part oftheproofthat6iswell-defined. AltofTheorem 8-16canbederived directly from thefollowing corollary ofLemma |and Theorem 2. 3.THEOREM (THE MAYER-VJIETORIS SEQUENCE). IfM=UUY, where UandVareopen,thenwehaveanexactsequence (eventually ending inO89): 0HM) ~--. =HAM) =HKU) @HAV) >HKU NV)8HM) Asseveral oftheProblems show. thecohomology ofnearly everything can becomputed byasuitable application oftheMayer-Vietoris sequence. Asa simple cxamplc. weconsider thetorus T=S'xS',andtheopen setsU andVillustrated below. Since thereisadeformation retraction ofUandV u = Vv Ra: Bae as onto circles, andadeformation retraction ofUNV onto 2circles, theMayer- Vietoris sequence is Excursion intheRealmofAlgebraic Topology 425 —A(T) —HU)@HV)—HOUNV)—A(T)—HU)@HV)— & a Rr R —= HU NY) —+ HAT) — 0. a au R@R R Themap H'(U NV) >H?(T) isnot0(itisonto H?(T)), soitskernel is 1-dimensional. Thus theimage ofthemap H1(U)® H'(V) >HU NV) isI-dimensional. Sothekernel ofthismap is1-dimensional. andconsequently themap H'(T) >H'(U)@ H'(V) hasaI-dimensional image. Similar rea- soning shows that this map also hasaJ-dimensional kernel. Itfollows that dimH}(T) =2.Thereasoning used herecanfortunately besystematized. 4.PROPOSITION. Ifthesequence a Oe ee en Se ee) isexact, then 0=dim )—dim Vy+dim V3—---+(—1)'"! dim PROOF. Byinduction onk.Fork=|wehave thesequence 0+ h>0. Exactness means that {0}CVisthekernel ofthemap V;—-0,which implies that Vj=0. Assume thetheorem fork—J.Since themap V2>V3haskernel (V1), it induces amap V2/a(V,) +V3.Moreover, thismap isone-one. Sowehave an exact sequence ofk~]vector spaces 0>Va/a(V;) >Va--+>Vpn>Ve>0: hence 0=dim V2/a(Y1) ~dim V3++ =—dim +dim¥)—dimV3+--+. which proves thetheorem fork. 426 Chapter 1] Rather than compute thecohomology ofother manifolds, wewill usethe Mayer-Vietoris sequence torelate thedimensions ofHE(M) toanentirely different setofnumbers, arising from a“triangulation” ofM,anew structure which wewill now define. The standard n-simplex A,isdefined astheset n={xeR :0<x) sland De}x!=1. Az aA* J va —e —_ 0 1 (InProblem 8-5.A,isdefined tobeadifferent, although homeomorphic, set.) The subset ofA,obtained bysetting m—kofthecoordinates x!equal to0 ishomcomorphic toAg.and iscalled ak-face ofAy. IfACMisadif- feomorphic image ofsome Am, then theimage ofak-face ofAy,iscalled a k-face ofA.Now byatriangulation ofacompact n-manifold Mwemean a finite collection {0";} ofdiffeomorphic images ofA,which cover Mandwhich satisfy thefollowing condition: If0",No"; #@.then forsome ktheintersection 0”;No"; isak-Lace ofboth o”;and0”;. we57Thestandardtriangulation VASeen“XOofS?(afterSteinberg \/pauo s\e of3-simplexe op, aieKa\\31["wiangulations™ Wy(n= 2, Itisadifficult theorem that every C* manifold hasatriangulation; fora proof seeMunkres. Elementary Differential Topology, orWhitney, Geometric Integration Excursion intheRealn ofAlgebraic Topology 427 Theory. Assuming that ourmanifold Mhasatriangulation {o";} wewill call each o”;ann-simplex ofthetriangulation; anyk-face ofany0”;willbecalled ak-simplex ofthetriangulation, Weleta,bethenumber ofthesek-simplexes. Now letUbethedisjoint union ofopen balls, onewithin each -simplex 0”, andletV,—1bethecomplement ofthesetconsisting ofthecenters ofthese balls, sothatVp—1isaneighborhood oftheunionofall(n—1)-simplexes ofM.Then D ; Vaart & / J . Le" :L<\¢ M=U UV,.4 where UM Vy—1 hasthesame cohomology asadisjoint union ofdycopies ofS?-). Consider firstthecase where n>2.The Maver-Vietoris sequence breaks into pieces: (1) 0+ H°(M) —HU) ©H°(Vy1) —HU 9Vn) —HM) —WU) ®H'(Vp-1) HU V1) u 1 0 0 QQ) Forl<k<n-—-1. HEV 0Vp) >HEM) >HEU) ©HE(Vy) >HEU 0Mn) i} I L 0 cu 0 (3) H-2(U 0Vy)—HMM) —HP") ®HO"Vy) a Ul] 0 0 >HU 1Vy) —HM) —HU) ©A"(V1) Hi] 0 428 Chapter 1] Applying Proposition 4tothese pieces yields dimH*(Vp1) =dimH*(M) O<sk<n-2 dimH™! (Vp—1) =dimH"-)(M) —dimH"(M) +dn. Forthecase »=2weeasily obtain thesame result without splitting upthe sequence. Wenow introduce theEuler characteristic x(M) ofM,defined by X(M) =dimH°(M) —dimH'(M) +dimH?(M) —---+(—1)"dimH"(M). This makes sense foranymanifold inwhich allH*(M) arefinite dimensional: weanticipate herealaterresultthatH*(M) isfinitedimensional whenever Mf iscompact. The above equations then imply that nl x(Mnt) =YI dimHVy1) k=0 n-2 =0-1) dimA*(M1) k=0 +(=1)"") [dimH""!(M1) —dimH"(M) +ay] =xX(M) ~(-1)"en. or X(M) =xVn-1) +(-1)"on. 5,THEOREM. Foranytriangulation ofacompact manifold Mwehave X(M) =a—1+02 —++++(—1)"Gp. PROOF. Inthemanifold Vy—1 wedefme anew open setUwhich consists of adisjoint union ofsetsdiffeomorphic toR”,oneforeach (n—1)-face, joining the balis ofthe old U. [“——_ components of wy newU(1=2) < Re2 newU(n=3) = Excursion intheReakn ofAlgebraic Topology 429 Wewilllet2bethecomplement ofarcs,inthenewU,joiningthecenters ofthe bails inthe old U. V,-2 isthe complement of | A ¥n—2 isthe complement of (v=2) =3) Anargument precisely likethat which proves theequation XCM) =x(Vn—-1) +(-1)"0n also shows that - X(Vn—t) =X(Vn=2) +(=1)" "n= Similarly, weintroduce V,—3,..., Vosthelastofthese isadisjoint union ofa setscach ofwhich issmoothly contractible toapoint. Hence x(Vo) =a,while inallother cases wehave x(Ve) =x(Va-t) +(-haw. Combining these equations, wehave x(M) =X(Vn—1) +(-1)"in =X(Vn—2) +[(=1)""ont+(=1)"0] =x(Vo) +[(-lay++--+ (1a) =a —a+--+ +(—1)"On. & 6.COROLLARY (DESCARTES-EULER). Ifaconvex polyhedron has V vertices, Eedges, and Ffaces, then V-E+F=2. 430 Chapter 11 IfweturnfromH*toHfweencounter averydifferent situation. IfUCM jsopen, aform with compact support CMmay notrestrict toaform with compact support CU:theinclusion mapofUintoMisnotproper. Onthe ysupport@ other hand, ifwisaform with compact support CU,then wcanbeextended toMbyletting itbe0outside U;wewilldenote thisextended form by support iu"). IfCK(M) denotes thevector space ofk-forms withcompact support onM,we candefine anew sequence. 7.LEMMA. The sequence ju’®-jv’ iu!+iv' o>chu ny)== chu @ck(v) ———*s ck(m) +0 isexact. PROOF. tisclear that jy’®—jy’isone-one; infact, each map jy’and jy’ isone-one. Toprove thatiy’-+iy’ isom, letwbeak-form withcompact support onM. and let{¢u,@y} beapartition ofunity forthecover {U,V}. Then = duv +ovo isclearly theimage of(¢uw, vw) €CK(U) @CA(V). Itisclear that image (jy’ ®—jv’) Cker(iu’ +iy’). Toprove theconverse, suppose that (A1,A2) €CK(U) @CK(V) satisfies iy'(A1) +iy’(A2) =0. This means that 4)=~A2. Since supportA, CUand support ACU.this shows that support ArCUNV and supportaz CUNV. So(Ay,A2) isthe image ofAy€CK(UU NV). & Excursion intheRealmofAlgebraicTopology 431 8.THEOREM (MAYER-VIETORIS FOR COMPACT SUPPORTS). Ifthe manifold M=UUV forU,V open inM,then there isalong exact sequence 3 so HKU NV)@HEU) @HEV) >HE(M) —HEMUAV) PROOF. ApplyTheorem 2totheshortexactsequence ofcomplexes givenby the Lemma. ¢ This sequence ismuch harder towork with than theMayer-Vietoris sequence. Forexample, suppose wewanttofindH¥forR"—{0}, which isdiffeomorphic to S""'xR, IfwewriteS"=UUVintheusualway,sothatUNVisdiffeomorphic toS’! xR,then S"xR=(UxR)U(V xR),where (UxR)N(V xR) isdiffeomorphic toS"-' xR?.Theonlywaytouseinduction istofindHé forallS"xR”,starting with S'xR”.Thedetails willbelefttothereader; wewillmerely record one further result, forlater use. and then proceed toyet another application ofTheorem 2. 9.COROLLARY, IfM=UUV forU,V open inM,then there isadual long exact sequence so HEU) =HEM) =[AU @HE(Vt >BE(UAVY oo PROOF. Wejusthavetoshowthatifthesequence oflinear maps aWi—Wr4,Ws isexact atW2.then soisthesequence ofdual maps and spaces 7a Ww;&,Wot—>Wit. Forany 4€W3* wehave a*B*(A) =a*(Lo B)=Ao(Poa) =ho0=0. Soa*op*=0. Now suppose 4€W3" satisfies a*(A) =0.Then Ao@=0.Weclaim that wm, 2w, ifve R there is1:Ws>RwithA=B*(A), ie.A=Bod. Given aw€W3which is 432 Chapter 1] ofthe form (w’), wedefine Aw) =(6'). This makes sense, forifB(w’) =B(w"), then w—w”=a(z) forsome z,so A(w) —A(w”) =Aa(z) =0.This defines 4onB(W2) CW3. Now choos WCW with Ws=B(2) @W,anddefine 4tobe0onW.& Wenow consider arather different situation. LetNCMbeacompact sub- manifold ofM.Then M—N isalsoa manifold. Wetherefore have thesequence e ” CEM —Ny—CEM) —CK), where ¢is“extension”, This sequence isnotexact atCk(M): thekernel ofi* contains all»€CK(M) which are0onN,while theimage ofecontains all w€Ck(M) which are0inaneighborhood ofN. Tocircumvent this difficulty, wewill have touseatechnical device. We appeal first toaresuk from theAddendum toChapter 9.There isacompact neighborhood VofNandamapz:V-NsuchthatVisamanifold-with- boundary, and if:N—Vistheinclusion, then x0/istheidentity of1. while jozissmoothly homotopic totheidentity ofV.Wenow construct sequence ofsuch neighborhoods V=V;>V23V3D«++with Vj=N. Vy a?Sg $7Cus % y, N/A fy | yaeNee “ Now consider twoforms w;€C*(V;), a;€CK(V;). Wewillcallaandwy eguivatent ifthere is¢>i,jsuchthat o|Vi =a|Y- Tisclearthatwecanmakethesetofallequivalence classes into avector space 9*(N), the“germs ofk-forms inaneighborhood ofM”.Moreover, itiseasy todefine d:g(N) >g*+!(N), sothatweobtain acomplex g.Finally, we define amap ofcomplexes i” ck(m) —9k(N) intheobvious way: w+theequivalence class ofanyw|Vj. Excursion intheRealmofAlgebraic Topology 433 10,LEMMA. The sequence Kk ©.cki*gk 0Ch(M—N)—>Ch(M) —>gh(N)>0 isexact. PROOF. Clearly eisone-one. Ifw€CK(M —N),thenw=0insome neighborhood UofN.Since Nis compact and(),V;=N,there issome isuch that VjCU,andconsequently w=0onVj.Thismeans thati*e(w) =0.Conversely, suppose 4¢CK(M) satisfies i*(A) =0.Bydefinition ofg*(N), thismeans thatA|V;=0forsome i. Hence A|M —N hascompact support CM~N,and4=e(A|M —N). Finally, anyelement ofg*(N) isrepresented byaform 7onsome Vj.Let f:M—[0,1] beaC®function which is1onVi41, having support fC interior Vj.Then f'n€Ck(M), andfnrepresents thesame clement of$*(N) as9;consequently thiselement isi*(fn). 4% 11,LEMMA. Thecohomology vector spaces H*(G) ofthecomplex {§*(N)} areisomorphic toH*(N) forallk. PROOF. This follows easily from thefactthatj*:H*(Vi) >H¥*(N) isan isomorphism foreach V;.Details arelefitothereader. ¢ 12.THEOREM (THE EXACT SEQUENCE OFAPAIR). IfNcMisa compact submanifold ofM,then there isanexact sequence é sie HEM —N)>HEM)>BEN)>HEM -N)>. PROOF. Apply Theorem 2totheexact sequence ofcomplexes given byLemma 10,and then useLemma 11.4 Intheproofofthistheorem, thedeRham cohomology ofthe manifold-with- boundary V;entered only asanintermediary (and wecould have replaced theV;bytheir interiors). Butinthenext theorem, which wewillneed later, it istheobject ofprimary interest. 13.THEOREM. LetMbeamanifold-with-boundary, withcompact bound- ary8M. Then there isanexact sequence 6 vo HEM -aM)>HE(M) >H*(aM) —>HE(M -aM)> +.. 434 Chapter 11 PROOF. Just liketheproofofTheorem 12,usingtubular neighborhoods Vj;of aM inM. & Ny, / oe ‘am Asasimple application ofTheorem }3,wecanrederive H*(R") from a knowledge ofH*(S"-'), bychoosing Mtobetheclosed ballBinR",with HE(B) ©H*(B) =0fork#0.Thereader mayuseTheorem 12tocompute HE(S" xR"), byconsidering thepair (S”xR,{p} xR). Then Theo- rem 13may beused tocompute thecohomology ofS”xS™™! =a(S" x closed ballinR™). Forournext application wewillseek bigger game. LetMcR"* beacompact n-dimensional submanifold ofR"+! (acom- pact“hypersurface” ofR"+!), Using Theorem 8-17, thesequence ofthepair (R"*), M)gives HER") —HM) 2Heth! —M)— HEYRY) HH), li x tt 0 R 0 Itfollows that (*) number ofcomponents ofR”*+? —M=dimH"(M) +1. Butwealso know (Problem 8-25) that (#*) number ofcomponents ofR"+? —M>2. 14.THEOREM. IfMcR"*! isacompact hypersurface, then Misori- entable, andR"*!— Mhasexactly 2components. Moreover, Mistheboundary ofeach component. PROOF. From (#)and (##)weobtain dim H"(M) +1>2. Excursion intheRealm ofAlgebraic Topology 435 Since dim H"(M) iseither 0or1,weconclude that dim H”(M) =1,soMis orientable; then (*)shows that R"*! —Mhasexactly twocomponents. The proof inProblem 8-25 shows thatevery point ofMisarbitrarily close topoints indifferent components ofR”*! —M,soevery point ofMisintheboundary ofeach ofthetwocomponents. 15.COROLLARY (GENERALIZED [C®] JORDAN CURVE THEOREM). IfMCR" isasubmanifold homeomorphic toS”,then R"+! —Mhastwo components, andMistheboundary ofeach. 16.COROLLARY. Ncither theprojective plane northeKlein bottle canbe imbedded inR3. Our next main result willcombine some ofthetheorems wealready have. However, there are anumber oftechnicalities involved, which wewill have to dispose offirst. Consider abounded open setU¢R"which isstar-shaped with respect to0. Then Ucan bedescribed as / U=x: xeS" and 0<1 <p(x)} foracertain function p:S"~! +R.Wewillcallptheradial function ofU. Tf IfpisC®, then wecanprove that Uisdifleomorphic totheopen ballBof radius 1inR”.The basic idea oftheproofistotaketx€Btop(x)t-x€U. This produces difficulties at0,soamodification isnecessary. 17,LEMMA. Iftheradialfunction pofastar-shaped opensetUCR”isC™, then Uisdiffeomorphic totheopen ballBofradius |inR”. 436 Chapter 11 PROOF. Wecanassume, without lossofgenerality, that p>1onS’~). Let S:0,1] >[0,1] beaC®function with | f=0inaneighborhood of0 Wy f20 Jy f=). ! Define h:B>Uby Atty) =[1+(p(x) —S(O] x, xeS™ O<1<1. Clearly /isaone-one map ofBonto U.Itistheidentity inaneighborhood of0,soitisC®,withanon-zero Jacobian, at0.Atanyotherpointthesame conclusion follows from thefact that tH 1+(p(x) —1)f(d isaC® function with strictly positive derivative. 4 Ingeneral, thefunction pneed notbeC®; itmight noteven becontinuous. However, thediscontinuities ofpcanbeofacertain formonly. 18.LEMMA. Ateach point x€S"~!, theradial function pofastar-shaped open setUCR”is“lower semi-continuous”: forevery ¢>0there isaneigh- borhood WofxinS"-! such thatp(y) >p(x) —¢forally€W. PROOF. Choose tx€Uwithp(x)—1<¢.SinceUisopen,thereisanopen ball Bwith x€BC U.There isclearly aneighborhood Wofxwith the property that for»€Wthepoint tyisinB,and hence inU.This means that for»€Wwehave p(y) >t>p(x)—8. Excursion intheRealnofAlgebraic Topology 437 Evenwhenpisdiscontinuous, itlooksasifUshould bediffeomorphic toR”. Proving thisturns outtobequite afeat, andwewillbecontent with proving thefollowing. 19.LEMMA. IfUisanopen star-shaped setinR”,thenHk(U) =H*(R") andHk(U)=HE(R") forallk. PROOF. TheproofforH*isclear,sinceUissmoothly contractible toapoint. Wealso know that H(U) ¥R=H2(R"), ByTheorem 8-17, wejusthave to show thatHK(U) =0for0<k<n. Letwbeaclosed k-form with compact support K¢U.Weclaim thatthere isaC™ function 6:S"~! —Rsuch that f<pand KCV=(tx:x €S"" and0<1<p(x)}. This willprove theLemma, forthen Visdiffeomorphic toR",andconsequently w=dnwhere nhascompact support contained inV,andhence inU. For eachx€S"~",choose tx<p(x)suchthatallpointsinKofthe form ux for0<u<p(x)actually haveu<1,.Since Kisclosed andpislowersemi- (BO QV ( continuous, there isaneighborhood W,ofxinS"~! such that t,may also beused astyforally€W.LetWyy,...,We,coverS!,letgry...s¢rbea partition ofunity subordinate tothiscover, and define P=lxQitere+tebre Any point x€S~! isinacertain subcollection oftheWs, sayWyyy-.-5Wx forconvenience. Then pj41(X),..+; r(x) are0.Each ty,,-..5tx; 18<p(x). Since$1(x)+++++(x) =1,itfollows thatA(x)<p(x).Similarly, KCV.& 438 Chapter 11 Wecanapply thislastLemma inthefollowing way. LetMbeacompact manifold, andchoose aRiemannian metric forM.According toProblem 9-32, every point hasaneighborhood Uwhich isgeodesically convex; wecanalso choose Usothatforanyp€Uthemap exp, takes anopen subset ofMp diffeomorphically onto U.Let{Ui,...,U;} beafinite cover bysuch open sets. IfanyV=U;,M:--MUj, isnon-empty, thenVisclearlygeodesically convex. If p€V,thenexp, establishes adiffeomorphism ofVwithanopen star-shaped setinMp.Itfollows from Lemma 19thatVhasthesame H*andHkasR". Ingeneral, amanifold Mwillbecalled offinite type ifthere isafinite cover {U;,...,U;} suchthateach non-empty intersection hasthesame H*andH* asR";such acover willbecalled nice. Itisfairly clear thatifweconsider N={1,2,3,...} asasubset ofR?,then M=R?—N isnotoffinite type. Toprove thisrigorously, wefirstusethe Mayer-Vietoris sequence forR?=MUV,whereVisadisjoint unionofballs around 1,2,3,.... We obtain ee H}(R?) ——> H'(M) @H'(V) ——> H'(M NV) ——> H?(R2), ul 4 i 0 0 0 whereMV hasthesameH?asadisjoint unionofinfinitely manycopies ofS?;thisshows that H(A) isinfinite dimensional (seeProblem 7formore information about thecohomology ofMM). Ontheother hand, 20.PROPOSITION. IfMhasfinite type,thenH*(M) andHS(M) arefinite dimensional forallk. PROOF. Byinduction onthenumber ofopen sets7inanice cover. Itis clear forr=1.Suppose itistrue foracertain r,and consider anice cover {Uj,...,U;,U} ofM. Then thetheorem istrue forV=UyU--» UU; and Excursion intheRealmofAlgebraic Topology 439 forU.Itisalso true forUAV, since thishasthenice cover {UNU},...,U NU}. Now consider theMayer-Vietoris sequence kA 8yk Ok k see HIV NV) >HAM) —HEU)@HAV)> Themapamaps H*(M) ontoafinite dimensional vector space, andthekernel of@isalsofinite dimensional. SoH*(M) must befinite dimensional. Theproof forH*(M) issimilar. 4 Foranymanifold Mwecandefine (seeProblem 8-31) thecupproduct map HEM) xHI(M) >HY(M) by ([w], [n]) >[wAn). We can also define HEM) xHLM) >HEH(M) bythesame formula, since wAnhascompact support if»does. Now suppose that M”isconnected andoriented, with orientation 4.There isthen aunique element ofH"(M) represented byany n€C2(M) with fnel.(Mn) Itisconvenient toalso use44todenote both thisclement ofH7(M4) and the isomorphism H?(M) —Rwhich takes thiselement to1€R.Now every a€H*(M) determines anelement ofthedualspaceH2-*(M)* by Brave Hi(M)—> RB. Wedenote thiselement ofH2-*(M)* byPD(q), the“Poincaré dual” ofa,so thatwehave amap PD:H*(M) >He-*(My", PD(a)(B) =n(@vB). One ofthefundamental theorems ofmanifold theory states that PDisalways anisomorphism, Weareallsetuptoprove thisfact, butweshall restrict thetheorem tomanifolds offinite type, inorder nottoplague ourselves with additional technical details. Aswith most bigtheorems ofalgebraic topology, themain partoftheproofiscalledaLemma, andthetheorem itselfisasimple corollary 440 Chapter 11 21.LEMMA. IfM=UUVforopensetsUandVandPDjsanisomorphism forallkonU,V,andUNV, then PDisalsoanisomorphism forallkonM. PROOF, Let/=n-—k. Consider thefollowing diagram, inwhich thetoprow istheMayer-Vietoris sequence, and thebottom row isthedual oftheMayer- Vietoris sequence forcompact supports. HEV) @HEM) —BEM OV) —HRM) —HKU)@HEV)—HKU) |poee |p | [roe | THAW) @Hyp HU Ay >Hlimy >[HU @Hy >Hay Byassumption, allvertical maps, except possibly themiddle one, areisomor- phisms. Itisnothard tocheck (Problem 8)that every square inthisdiagram commutes uptosign, sothat bychanging some ofthevertical isomorphisms totheir negatives, weobtain acommutative diagram. Wenow forget allabout ourmanifold anduseapurely algebraic result. “THE FIVE LEMMA”. Consider thefollowing commutative diagram ofvec- torspaces andlinear maps. Suppose thattherows areexact, andthat$1,¢2. 4,@sareisomorphisms. Then ¢3isalsoanisomorphism. VY,hh,4%, 4,24vy,4.Vs | |o | lo |e |e mBiW,BaWsBsWeBaWs PROOF. Suppose $3(x) =0forsome x€V3.Then B3¢3(x) =0,so$4a3(x) = 0.Hence a3(x) =0,since ¢4isanisomorphism. Byexactness atV3,there is y€V2with x=a2(y). Thus 0=$3(x) =g302(y) =Brg2(y). Hence $2(») =Bi(s) forsome z€Wi. Moreover, ==¢1(w) forsome w€Vi.Then G2(¥) =Bi(=) =Bidi(w) =grer(w), which implies that y=a1(w). Hence X=@2(y) =a2(0)(w)) =0. So¢3isone-one. Theproofthat$3isontoissimilar, andislefttothereader. Thisproves the original Lemma. Excursion intheRealmofAlgebraic Topology 441 22,THEOREM (THE POINCARE DUALITY THEOREM). IfMisa connected oriented n-manifold offinite type, then themap PD:HE(M) >HE-*(M)* isanisomorphism forallk. PROOF. Byinduction onthenumber rofopen setsinanice cover ofM.The theorem isclearly trueforr=1.Suppose itistrueforacertain r,andconsider anice cover {Uj,...,U,,U} ofM,Let V=U,U---UU,. The theorem istrue forU,V,and forUN V(asintheproofofProposition 19).BytheLemma, it istrueforM.This completes theinduction step. 23,COROLLARY. IfMisaconnected oriented n-manifold offinite type, thenH*(M) andH2-*(M) havethesame dimension. PROOF. UsetheTheorem andProposition 19,noting that V*isisomorphic toVifVisfinite dimensional. ¢ EventhoughthePoincaréDualityTheoremholdsformanifoldswhicharenot offinite type, Corollary 23does not.Infact, Problem 7shows thatH!(R? —N) andH}(R? —N) have different (infinite) dimensions. 24.COROLLARY. IfMisacompact connected orientable -manifold, then H*(M) andH"-*(M) have thesame dimension. 25.COROLLARY. IfMisacompact orientable odd-dimensional manifold, then x(M/) =0. PROOF. Intheexpression forx(M), theterms (—1)* dimH*(M) and (-1)"* dimH"-#(M) =(=1)!dimH”-#(aM) cancel inpairs. Amore involved useofPoincaré duality willeventually allow ustosaymuch more about theEuler characteristic ofanycompact connected oriented man- ifold M@”. Webegin byconsidering asmooth k-dimensional orientable vector bundle §=x:E>MoverM.Orientations 4forMandvfor&givean orientation 2@vforthe(n+k)-manifold E.since Eislocally aproduct. If {U,,...,U,} isanicecover ofMbygeodesically convex setssosmall thateach bundle &|U; istrivial, then aslight modification oftheproof forLemma 19 442 Chapter 11 shows that {2—"(U,),...,77!(U;)} isanice cover ofE,soEisamanifold of finite type. Notice alsothatforthemaps 5=O-section M————E 7 we have mos =identity ofM som issmoothly homotopic toidentity ofE, sox*:H'(M) -H!(E) isanisomorphism forall/.ThePoincaré duality theorem shows thatthere isaunique class U€H¥(£E) suchthat mou =pheve HEE), This class Uiscalled theThom class of&.Our first goal will betofind a simpler property tocharacterize U. LetFp=271(p) bethefibreof&overanypoint p€M,andletjp:Fp>E betheinclusion map. Since jpisproper, there isanelement jp*U €Hé(Fp). Ontheother hand, theorientation vfor§determines anorientation vpforFp, andhence anelement vp€Hk(Fp). 26.THEOREM. Let(M,14)beacompact connected oriented manifold, and &=m: E>Manoriented k-plane bundle over Mwith orientation v.Then theThom classUistheunique element ofH*(£) withtheproperty thatfor allp€Mwehave jp*U =vp.(This condition means that (Fp) where Uistheclassoftheclosed formw.) PROOF. Picksome closed formw€Ck(E) representing U,andlet»€C”(M) beaformrepresenting 14,sothatfiyy_,,) 1=1.Ourdefinition ofUstates that (Q) fmnaw=1.E LetACMbeanopen setwhich isdiffeomorphic toR”,sothatAissmoothly contractible toanypoint p€A.Also choose Asothat there isanequivalence fin"A)> AxR*. Excursion intheRealm ofAlgebraic Topology 443 Thisequivalence allows ustoidentify x—1(A) withAxFp.Under thisidentifica- tion,themapjp:Fp>2~'(A) corresponds tothemape>(p,e)fore€Fp, whichwewillcontinue todenote byjp.Wewillalsouse2:AxFy—>Fpto denote projection onthesecond factor. Let ||||beanorm onFp. Bychoosing asmaller Aifnecessary, wecan assume thatthere issome K>0such that, under theidentification of27!(A) withAxFp,thesupport ofw|z~'(A) iscontained in{(g,e) :g€A,llell<K}. support A Using thefactthat Aissmoothly contractible to7,itiseasy toseethatthere isasmooth homotopy H:(AxFp)x[0,1] >AxFpsuch that He,0) =e H(e,1) =(p,m2(e)) =jp(m2e)); wejustpul]thefibres along thesmooth homotopy which makes Acontractible AaCheewa cu toe. FortheHconstructed inthisway itfollows that H(e,1) ¢support @ifllell >K. Consequently, theform H*w on(AxFp)x[0,1] hassupport contained in {(g,e,1) :Nell<K}.Aglance atthedefinition of7(page 224) shows that the 444 Chapter 11 form JH*w onAxFphassupport contained in{(g,e) :llell <K}. Theo- rem 7-14 shows that (ip072)" ~w=iy*(H*w) —io”(H*w) =d(JH*w) +1(dH*w) =d(]H*w). Thus Oy m2"jptw —w=dh, support dC{(g,e) :flell<K}. So (3i)nao=[ayamtino —fm'nadn. AxFp AxFp AxFp Now, ontheone hand wehave (Problem 8-17) (4) fnnAma"jp'w=[an-fjp’.AXFp lA Fy Ontheother hand, weclaim that thelastintegral in(3)is0.Toprove this, it vlearly suffices toprove that theintegral is0over A’xFpforany closed ball A'CA. Since a*pAdh=+d(n* Add). wehave where 2* AAhas (5)fnwAd= +fd(x*ad)compactsupportonAxFy AIxFy, A!xFpby(2) =+f mA byStokes’TheoremaAxFp =0. because theform x*p A}isclearly 0on8A!xF,(since 3A!is(n—1)-dimen- sional). Combining (3),(4),(5)weseethat [tases fanffipto. [AxFp lA Fp Excursion intheRealmofAlgebraic Topology 445 Thisshows thatf;,,jp*wisindependent ofp,forp€A.Using connectedness. itiseasy toseethat itisindependent ofpforallp€M,sowewilldenote it simply byf,j*@. Thus fmanos fxnfjo. m~"(A) A F Comparing with equation (1),andutilizing partitions ofunity, weconclude that ftw=1, F which proves thefirst part ofthetheorem. Now suppose wehaveanother classU!€Hk(E). Since HE(E) ©H"(E) ©H"(M) &R, itfollows that U'=cUforsome c€R.Consequently. Jp"Ul=jp*cU=C+Up. Hence U’hasthesame property asUonly if¢=1. The Thom class Uof§=2:E+Mcan now beused todetermine an element ofH*(M). Lets:M—>Ebeanysection; there always isone(namely, the0-section) andanytwoareclearly smoothly homotopic. Wedefine theEuler class x(&)€H*(M) of&by x@) =s°U. Notice thatif€hasanon-zero section s:M—E,ando€CK(E) rep- resents U,then asuitable multiple ¢-sofstakes Mtothecomplement of support w.Hence, inthis case x) =(¢-s)*U =0. The terminology “Euler class” isconnected with thespecial case ofthebundle TM, whose sections are, ofcourse, vector fields onM. IfXisavector field onMwhich hasanisolated 0atsomepointp(thatis,X¥(p)=0,butX(q)#0 forg#pina neighborhood ofp),then, quite independently ofourprevious considerations, wecan define an“index” ofXatp.Consider first avector 446 Chapter 11 fieldXonanopen setU¢R"with anisolated zeroat0€U.Wecandefine afunction fy:U—{0}>S"! byfx(p) =X(p)/IX(p)|. IffsS*-1 >U isi(p)=ep,mapping S"-' intoU,thenthemapfyof:S"-! >S*-! hasa certain degree; itisindependent ofe,forsmall ¢,since themaps f1,#2:S’-! > Ucorresponding to;and£9willbesmoothly homotopic. This degree iscalled the index ofXat0. index 0 index 0 index 1 index | index ~1 index 2 index —2 index |inR” index (—1)" inR” Now consider adiffeomorphism h:U>VCR"with h(0)=0.Recall that A,X isthevector field onVwith (A,X)() =ha(Xp-1)- Clearly 0isalso anisolated zero ofhyX. 27.LEMMA. Ifh:U>VCR®isadiffeomorphism with (0) =0,andX hasanisolated 0at0,then theindex ofh,X at0equals theindex ofXat0. Excursion intheRealm ofAlgebraic Topology 447 PROOF. Suppose firstthat hisorientation preserving. Define H:R" x[0,1]>R" by h 0 1H(x,)={(cx) sues Dh(O)(x) 1=0. This isasmooth homotopy; toprove that itissmooth at0weuseLemma 3-2 (compare Problem 3-32), Each map H,=x++H(x,1) isclearly adiffeomor- phism, 0<1<1.Note that Hy€SO(”), since Aisorientation preserving. There isalsoasmooth homotopy {H;}, 1<t<2with each H;€SO(n) and Hy=identity, since SO(n) isconnected. So(seeProblem 8-25), themap his smoothly homotopic totheidentity, viamaps which arediffeomorphisms. This shows thatfj,,xissmoothly homotopic tofyonasufficiently smallregion of R"—{0}. Hence thedegree offy,x 07isthesame asthedegree offyof. Todeal with non-orientation preserving h,itobviously suffices tocheck the theorem forh(x) =(x!,...,x"7!, —x”).Inthiscase Snax =ho fyoh™, which shows that degree f,,x of=degree fyof. Asaconsequence ofLemma 27,wecannowdefinetheindexofavector field ona manifold. IfXisavector field onamanifold M,with anisolated zero at p€M,wechoose acoordinate system (x,U) with x(p) =0,anddefine the index ofXatptobetheindex ofxX at0. 28.THEOREM. LetMbeacompact connected manifold with anorien- tation 44,which is,bydefinition, also anorientation forthetangent bundle &=2:7M >M. Let X¥:M>TM bea vector field with only afinite number ofzeros, andletobethesumoftheindices ofXatthesezeros, Then x(§) =o-e H"(M). PROOF. Letpi,..., prbethezeros ofX.Choose disjoint coordinate systems (Ui,1),+25(UrsXr)withx1(p1)=0,andlet Bi=x({p ER": |p|<1). Ifw€C2(E) isaclosed form representing theThom class Uof&,then we aretrying toprove that fX*(w)=0.(Mu) 448 Chapter 11 Wecanclearly suppose that X(g) ¢support wforg¢Uj;Br.So r X*(w)=fX*(w); thus itsuffices toprove that (*) fX*(w)=indexofXatpj.Bi Itwillbeconvenient todrop thesubscript ifrom now on. Wecanassume that 7M istrivial over B,sothat 77}(B) canbeidentified with BxMy. Letjpandm2have thesame meaning asintheproof ofThe- orem 26. Also choose anorm ||||onMy. Wecanassume that under the identification of2~"(.B) with BxMp,thesupport ofw|zx~"(B) iscontained in{(g,v) :4€A,llvl]<1}.Recall from theproofofTheorem 26that myjp? —w=dd support AC{(g,v):[lull<3. Since wecanassume that X(g) ¢support Aforg€4B,wehave ofxr@=fpxmute -fxn'B 'B B =fX*19"(jp*w) —fX*(X) byStokes’Theorem'B ‘aB =fX*m2"(jp"), B Onthemanifold M,wehave ae pan (n—1)-form onMyJe=dp (withnon-compact support). P ppor IfDC Mpistheunitdisc(with respect tothenorm ||||)andS”~! denotes 4DCMp, then 2 fPefPefdp ‘gut aD ID D =1 byTheorem 26,andthefact "= that support jp*w CD. Excursion intheRealm ofAlgebraic Topology 449 Now, forg€B—{p}, wecandefine ¥q) =XQ/IX@I: and¥:4B+TMissmoothly homotopic toX:8B+TM.So @ fXtx7"jp")=fXtmtdp'B ‘B =fX"m*p byStokes’Theorem 2B =fX10"p OB =f(12.0X)*p. ‘OB From thedefinition oftheindex ofavector field, together with equation (2),it follows that (4) f(120¥*)p=indexofXatp. aB Equations (1),(3),(4)together imply (#). 29,COROLLARY. IfXandYaretwovector fieldswithonlyfinitely manyzerosonacompact orientable manifold, thenthesumofthe indices of¥equals the sum ofthe indices 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 Xwith just onezero ineach k-simplex ofthetriangulation. We begin bydrawing theintegral curves ofXalong the1-simplexes, with azero at cach 0-simplex andatonepoint ineach I-simplex. Wethen extend thispicture 450 Chapter 11 toinclude theintegral curves ofXonthe2-simplexes, producing azero atonc point incach ofthem. Wethen continue similarly until then-simplexes are filled. 30.THEOREM (POINCARE-HOPF). Thesumoftheindicesofthis vector field (and hence ofany vector field) onMistheEuler characteristic x(M) Thus, for§=2:TM—Mwehavex(§)=x(M)+p. PROOF. Ateach 0-simplex ofthetriangulation, thevector field looks like with index 1. Now consider thevector field inaneighborhood oftheplace where itiszero onaI-simplex. The vector field looks likeavector field onR"=R!xR"! which points direcfly inwards onR!x{0}anddirectly outwards on{0}xR"-! (ayn =2 (b)n=3 Excursion intheRealm ofAlgebraic Topology 451 Forn=2,theindex isclearly —1.Tocompute theindex ingeneral, wenote that fytakes the“north pole” N=(0,...,0,1) toitself and noother point goes toN.ByTheorem 8-12 wejusthave tocompute signy fy.Now atNwe canpickprojection onR"-! x{0}asthecoordinate system. Along theinverseimageofthex!-axis thevector field looks exactly likefigure (a)above, where we already know thedegree is—1,sofytakes thesubspace ofS"“!y consisting of tangent vectors tothiscurve into thesame subspace, inanorientation reversing way.Along theinverse image ofthex?-,... x"-!-axes thevector fieldlooks like 50fytakes thecorresponding subspaces ofS"~y intothemselves inanori- entation preserving way. Thus signy fy=—1,which istherefore theindex of the vector field. Ingeneral, near azero within ak-simplex, Xlooks likeavector field on R"=R‘xR"-* which points directly inwards onRkx{0}anddirectly outwards on{0}xR"-*, Thesame argument shows thattheindex is(-1)*. Consequently, thesum oftheindices is @o—1+a2—+++ =X(M). & Weendthischapter with onemore observation, which wewillneed inthe lastchapter ofVolume V!Let&=x:E>Mbeasmooth oriented k-plane bundle over acompact connected oriented n-manifold M,and let(,)bea Riemannian metric for .Then wecan form the“associated disc bundle” and “associated sphere bundle” peweasn _(TTh Itiseasy toseethat Disacompact oriented (n+k)-manifold, with 2D=S; moreover, theDconstructed foranyother Riemannian metric isdifleomorphic tothisone. Welet79: S>Mbe2|S. 452 Chapter 1] 31.THEOREM. Aclass@€Hk(M) satisfies 0"(a)=0ifandonlyif@isa multiple ofx(é). PROOF. Consider thefollowing picture. The toprow istheexact sequence HE(D—8)—*—» Hk(p) —"—., ns) eoNGsjea HEM) for(D,S) given byTheorem 13,The map s:M>D—S isthe0-section, while§:M—Disthesame0-section. Notethateverything commutes. no"=i* o(x|D)* since xo=(x|D) of, nis were since extending aform toD=See doesnotaffect itsvalue ons(M), and that 5*0(|D)* =identity ofH*(M), since (2|D) o§issmoothly homotopic totheidentity. Now leta€H*(M) satisfy zo*(a) =0.Then i*(x|D)*a =0,so(2t|D)*a € imagee. Since D=Sisdiffeomorphic toE,andevery element ofH*(D—S) isamultiple oftheThom class Uof&,weconclude that (x|D)’a =c-e(U) forsomece R. Hence a=5*(x|D)'a =c-5*(e(U)) =c-s*U =c-x(é). Theproof oftheconverse issimilar. Excursion intheReabn ofAlgebraic Topology 453 PROBLEMS 1.Find H*(S! x... xS!)byinduction onthenumber xoffactors. [Answer: dimH*=(?)] 2.(a)UsetheMayer-Vietoris sequence todetermine H*(M —{p})interms ofH*(M), foraconnected manifold M. (b)IfMand Naretwo connected n-manifolds, letM#N beobtained by joining MandNasshown below. Findthecohomology ofM#Nintermsof that ofMand N. M “wan\ N ‘ASG SAAS (9)Findxforthen-holed torus.[Answer: 2—21] 3.(a)Find H*(Mobius strip). (b)Find H¥(P2), ()Find H*(P"), (UseProblem 1-15(b); itisnecessary toconsider whether a neighborhood ofP"~! inP”isorientable ornot.) [Answer: dimH*(P") =1 ifkevenand<n,=0otherwise.] (d)Find H*(Klein bottle). (e)Find thecohomology ofM#(Mébius strip) and M#(Klein botde) ifM isthe n-holed torus. 4.(a)Thefigurebelow isatriangulation ofarectangle. Ifweperform the indicated identifications ofedges wedonotobtain atriangulation ofthetorus. Why not? A —___4 rs A 454 Chapter 11 (b)Thefigurebelowdoesgiveatriangulation ofthe torus when sides are iden- tified. Find ao,a1,@forthistriangulation; compare with Theorem 5and Problem }. 5.(a)Foranytriangulation ofacompact 2-manifold M,showthat 3a =2a a=3(ao —x(M)) ao(ao-1) eka 1 a>37+V49=24x(M)). (b)Show thatfortriangulations ofS?andthetorus T?=S'xS!wehave S?: a24 m26 24 T?: a>7 a221 a2 ld. Find triangulations forwhich these inequalities areallequalities. 6.(a)FindHk(S" xR") byinduction onn,using theMayer-Vietoris sequence forcompact supports. (b)Usetheexact sequence ofthepair (S”xR™,{p} xR”) tocompute the same vector spaces. (c)Compute H*(S" xS”—), using Theorem 13. 7.(a)The vector space H}(R? —N)may bedescribed asthesetofallse- quences ofrealnumbers. Using theexact sequence ofthepair (R?,N), show thatH}(R? —N)maybeconsidered asthesetofallrealsequences {aq}such that dy=0forallbutfinitely many 1. (b)Describe themap PD:H'(R? —N) >H}(R? —N)*interms ofthese descriptions ofH'(R?—N) andH}(R?—N). andshow thatitisanisomorphism. (c)Clearly H}(R? -N)hasacountable basis. Show thatH1(R? —N)does not.Hint: Ifvj;={a;/} €H'(R? —N),choose (b1,b2) €R?linearly indepen- dent of(ay!,a12); then choose (b3,b4,s) €R°linearly independent ofboth (a3,ar4,a13) and(423, a2,a2°); etc. Excursion intheRealmofAlgebraic Topology 455 8.Show that thesquares inthediagram intheproof ofLemma 21commute, except forthesquare Hu nV) ——> Hh) |» |» HAW Av)*—— Him) which commutes uptothesign(-1)*, (Itwillbenecessary torecall howvarious maps aredefined, which isagood exercise; theonly slightly difficult maps are theones involved intheabove diagram.) 9.(a)LetM=M\UM2UM3U--- bea disjoint union oforiented n-manifolds. Show thatHk(M) +@;Hk(.M;), this“direct sum” consisting ofallsequences (a1,02,03,...) withay€Hk(M;) andallbutfinitely many a;=0€HE(Mi). (b)Show thatH*(M) ~[],H*(M;), this“direct product” consisting ofall sequences (01,02,03,...) withay€H*(M;). (2)Show that ifthePoincaré duality theorem holds foreach Mj,then itholds for M. (d)The figure below shows adecomposition ofatriangulated 2-manifold into three open setsUp,Uj,and Uz.Useananalogous decomposition in»dimen- sions toprove thatPoincaré duality holds foranytriangulated manifold. Upisunionofshaded! a .g U:;nofshaded£2 U1isunionofunshaded7 2mumonors’Ga. 456 Chapter 11 10.Lecé=a:E>Mand’ =2': E’>Mbeoriented k-plane bundles. over acompact oriented manifold M,and (f,f)abundle map from &to& which isanisomorphism oneach fibre. (a)IfU€HE(E) andU'€HK(E’) aretheThom classes, thenf*(U)=U’. (b)*(x(€)) =x(é’). (Using thenotation ofProblem 3-23, wehave f*(x(&)) = x(F*8))) 11.(a)Let §=2: E>Mbeanoriented k-plane bundle over anoriented manifold M,with Thom class U.Using Poincaré duality, prove theThom Isomorphism Theorem: ThemapH!(E) >Hi+*(E) given byataUUis anisomorphism forall/. (b)Since wecanalsoconsider Uasbeing inH*(E), wecanform UUU€ H2k(E), Using anticommutativity ofa,show thatthisis0forkodd. Conclude thatUrepresents 0€H*(E), sothatx(£)=0.Itfollows, inparticular, thai x(&)=0when&=2:TM>MforMofodddimension, providing another proof that x(M7) =0inthiscase. 12.Ifavector field Xhasanisolated singularity atp€M", show that the index of—X atpis(—1)” times theindex ofXatp.This provides another proof that x(M4) =0forodd n. 13.(a)Letpi,...,pr €M.Using Problem 8-26, show that there isasubse1 DCM difleomorphic totheclosed ball, such that allp;€interior D. (b)IfMiscompact, then there isavector field ¥onMwith only onesingu- larity. (0Inisafactthat aC®map f:S"-! >S"-! ofdegree 0issmoothly ho- motopic toaconstant map. Using this, show that ifx(/) =0,then there isa nowhere 0vector field onM. (d)1fMisconnected and notcompact, then there isanowhere 0vector field onM.(Begin withatriangulation toobtain avectorfieldwithadiscrete setof zeros. Join these byaraygoing toinfinity, enclose thisrayinacone, and push everything offtoinfinity.) SS (¢)IfMisaconnected manifold-with-boundary. with 4M+,then there isa nowhere zero vector field onM. Excursion intheRealm ofAlgebraic Topology 457 14.This Problem proves deRham’s Theorem. Basic knowledge ofsingular cohomology isrequired. Wewilldenote thegroup ofsingular k-chains ofX bySi,(X).Foramanifold M,weletSp°(44) denote theC®singular k-chains, andlet7:Sf°(M) —Sj(M) betheinclusion. 11isnothard toshow thatthere isachain map t:Sx(M) >Sp°(M) sothattof=identity ofS2°(M), whilei07ischainhomotopic totheidentity ofS,(M) [basically, tisapproximation byaC® chain]. This means that weobtain thecorrect singular cohomology ofMifweconsider thecomplex Hom(SP°( M7),R). (a)Ifwisaclosed k-form onM,letRh(w) €Hom(Sf°(M), R)be Rh(w)(c)=[o.F Show thatRhisachain mapfrom {Ck(M)} to{Hom(Sp°(M),R)}. (Hint: Stokes’ Theorem.) Itfollows that there isaninduced map Rhfrom thedeRham cohomology ofMtothesingular cohomology ofM. (b)Show thatRfisanisomorphism onasmoothly contractible manifold (Lem- mas 17,18,and 19willnotbenecessary forthis.) (c)Imitate theproof ofTheorem 21,using theMayer-Vietoris sequence for singular cohomology, toshow that ifRiisanisomorphism forU,V,and UNV, then itisanisomorphism forUUV. (a)Conclude that Rhisanisomorphism ifMisoffinite type. (Using the method ofProblem 9,itfollows that Rhisanisomorphism foranytriangulated manifold.) (e)Check thatthecupproduct defined using Acorresponds tothecupproduct defined insingular cohomology. APPENDIX A CHAPTER 1 Following thesuggestions inthischapter, wewillnow define amanifold tobe atopological space Msuch that (1)M4isHausdorff, (2)Foreach x€Mthere isaneighborhood Uofxandaninteger n>0 such that Uishomeomorphic toR". Condition (1)isnecessary, forthere iseven a1-dimensional “manifold” which is notHausdorff. Itconsists ofRU{+}where «¢R,with thefollowing topology: Aset Uisopen ifandonly if (1)UNR isopen, (2)If*€U,then (UNR) U{0}isaneighborhood of0(inR). Thus theneighborhoods of*look justlikeneighborhoods of0.This space may also beobtained byidentifying allpoints except 0inonecopy ofRwith the corresponding point inanother copy ofR.Although non-Hausdorff manifolds areimportant incertain cases, wewill notconsider them. ‘Wehave justseen that theHausdorff property isnota“local property”, but local compactness is,soevery manifold islocally compact. Moreover, aHaus- dorfflocallycompact spaceisregular, soeverymanifold 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 thusamanifold itself. Before exhibiting non-metrizable manifolds, wefirstnote that almost all“nice” properties ofamanifold areequivalent. THEOREM. The following properties areequivalent foranymanifold M: (a)Each component ofMiso-compact. (b)Each component ofMissecond countable (hasacountable base forthe topology). (©)Mismetrizable. (a)Misparacompact. (Inparticular. acompact manifold ismetrizable.) 459 460 Appendix A FIRST PROOF. (a)=>(b)follows immediately from thesimple proposition that ao-compact locally second countable space issecond countable. (b)=>(c)follows from theUrysohn metrization theorem. ()=(A)because anymetric space isparacompact (Kelley. General Topology. pg.160). The second proof does notrelyonthisdifficult theorem. (d)=f@)isaconsequence ofthefollowing LEMMA. Aconnecied. locallycompact, paracompact spaceiso-compact. Proof. There isalocally finite cover ofthespace byopen setswith compact clo- sure.IfUpisoneofthese. thenUocaninterseci onlya finite number Ui,...,Un, oftheothers. Similarly Up UU,U---UG,, imersects only Un4i....5Ungi and soon. The union ToUUGgUsUDypgU+=UpUeUUp,U2UUpgUe isclearly open. Itisalso closed. forifxisintheclosure, then xmust bein theclosure ofafiniteunionoftheseU;.because xhasaneighborhood which intersects only finitely many, Thus xisintheunion. Since thespace isconnected, i1equals thiscountable union ofcompact sets. This proves theLemma and theTheorem. SECOND PROOF. (a)=(b)=(0)and(a)=(a)asbefore. (c)>(a)isTheorem 1-2. (a)=(a).LerM=C)UG,U---.whereeachC;iscompact. Clearly Cyhas anopen neighborhood U;with compact closure. Then UjUC2hasanopen neighborhood U,with compact closure. Continuing inthisway,weobtain open setsU;withU;compact andU;CUi41. whose union contains allC;,andhence isM.11iseasy toshow from thisthat Misparacompact. 4 Tt1umis outtharthere areeven I-manifolds which arenoiparacompact. The construction oftheseexamples requires theordinal numbers, whicharebriefly explained here. (Ordinal numbers will noibeneeded fora2-dimensional ex- ample tocome later. ORDINAL NUMBERS Recall thatanordering <onasetAisarelation such that (i)a<band b<cimplies a<¢foralla,b,¢€A(transitivity: Appendix A 461 (2)Foralla,b€A,oneandonly oneofthefollowing holds: (ash Gi)a<h (trichotomy). (iii)6<a(also written a>b+ Anordered setisjustapair (A,<)where <isan ordering onA.Two ordered sets(A,<)and(B.<)areorder isomorphic ifthere isaone-one onto function f:A >Bsuch that a<bimplies f(a) <f(b): themap fitselfiscalledan order isomorphism. and f—' iseasily seen tobeanorder isomorphism also. Anordering <onAisawell-ordering ifevery non-empty subset BCA hasafirs! clement. thatis.anelement 5such that }<4!forallb’€B,Some well-ordered setsareillustrated below: inthisscheme wedonot listanyofthe< relations which areconsequences oftheones already Jisted. ® {0} o<1 (A={0,1}) 0<1<? (4={0,1,2) 0<1<2<3 ete. 0<1<2<3<-:- 0<1<2<--<w (wissomeset#0,1,2,3....) (w+1is,forthepresem 0<1<2<+.-<w<wt) justasetdistinctfrom those already mentioned: 0<1<2<--<w<wtl<w+2<-- O0<1<2<--<w<wtl<wt+2<---<w-2 0<1<2<---<@<w+]1<W4+2<+.-<@-2<w-241<-: 0<1<2<---<w<wt+1 <wt+2<---<w-2<W-24+1<--<H O<1<2<- Wc CO 2c KOI K O<1<2 <0 <M <6 <2 Ke <3 << cw 462 Appendix A Any subset ofawell-ordered setis.ofcourse, alsoawell-ordered setwiththe same ordering. Inparticular. asubset Bofawell-ordered setAiscalled an (initial) segment ifb€Banda<bimply a€B.Itiseasy toseethat ifBis a segment ofA.then either B=Aorelsethere issome a€Asuch that B={aeA:a' <a): infact. aisthe first element ofA—B. Notice that each setonour listis« segment ofthesucceeding ones. Itisnothard toseethatnotwosetsonourHist areorder isomorphic. Forexample. 0<I1<-.-<w and 0<I1<---<w<w+! arenolorder isomorphic because thesecond hasboth alastand anext tolas! element. while thefirs:does not. But there isamuch more general proposition which will settle allcases atonce: 1,PROPOSITION. IfB#4isasegment ofA.thenBisnotorderisomoi- phictoA.Infact,theonlyorderisomorphism fromBtoasegmen! ofAisthe identity. PROOF. Iff:B>B’CAisanorderisomorphism andB’isasegmem ofA.then forthefirstclement 6ofB(and hence ofA)weclearly musi have JS(b)=b.Then f(b’) must be5’.where b’isthesecond element. And soon. even forthe“w""” element (thefirstoneafter thefirst,second, third, etc.)! The way weprove thisrigorously isamazingly simple: If/(b) #bforsome b€B. justconsider thefirstelement of{b€B:f(b) #6):anoutright contradiction appears almost immediately. Proposition }hasacompanion. which makes thestudy ofwell-ordered sets simply delightful. 2.PROPOSITION. If(A,<) and (B,~<) arewell-ordered sets, then one is order isomorphic toasegment oftheother. PROOF. Wematch thefirst element ofAwith thefrstofB,thesecond with thesecond, ...,the“w*"”’ with the“w""”, etc.,until werunoutofoneser. Todothisrigorously, consider order isomorphisms from segments ofAonto segments ofB.Itiseasytoshow thatanytwosuch order isomorphisms agree onthesmaller oftheir twodomains (just consider thesmallest element where Appendix A 463 they don’t). Soallsuch order isomorphisms canbeputtogether togiveanother. which isclearly thelargest ofall.1fitisdefined onallofAwearedone, Ifit isnot, then itsrange must beallofB(orwecould easily extend it)and weare still done. & Suppose wedefine arelation <between well-ordered setsbystipulating that (A.<) <(B,<) when (A.<) isorder isomorphic toaproper segment ol (B.<).Transitivity of<isobvious, and Propositions }and 2show that weal- most have trichotomy. “Almost”, because thecondition “(A, <)=(B,<)*must 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 arebeautiful:* 3.PROPOSITION. <isawell-ordering oftheordinal numbers. PROOF. Given anon-empty set#ofordinal numbers, let(4,<)beawell- ordered setrepresenting oneofitselements a.Toproduce asmallest elementofAwecanobviously ignoreelements >a.Everyelement <aisrepresented byanordered setwhich isorder isomorphic tosome proper segment ofA: eachofthese isthesegment consisting ofelements ofAlessthatsomea€A. Consider theleastofthesea's.Itdetermines asegment which represents some BeA.ThisBisthesmallest element ofA, Notice that ifaisanordinal number, represented byawell-ordered set (A.<),then thewell-ordered setofallordinals 6<ahasaparticularly simple representation: itisorder isomorphic totheset(A.<)!Roughly speaking: An ordinal number isorder isomorphic tothesetofallordinals lessthan it If@isanordinal number, wewilldenote bya+1thesmallest ordinal after @ (if@isrepresented bythewell-ordered set(A,<).then a+1isrepresented byawell-ordered setwith just one more element. larger than allmembers ofA). Notice that some ordinals arenotoftheform a+1forany a;these arecalled limit ordinals, while those oftheform @+1arecalled successor *Only onefeature mars thebeauty oftheordinal numbers aspresented here. Each ordinal numberisahorriblylargeset;itwouldbemuchnicertochooseonespecificwell- ordered setfrom each order isomorphism class, anddefine these specific setstobethe ordinal numbers. There isaparticularly elegant waytodothis,duetovonNeumann, which canbefound intheAppendix toKelley, Genera/ Topology. 464 Appendix A ordinals. Wewillalsodenote some ordinals bythesymbols appearing before: 0,1,2.3....,0,0 +1,..., ete. Our listofwell-ordered setsonly begins tosuggest thecomplexity which well ordered setscanachieve. Withalittlethought, onecanseehowthesymbols w?,w....would appear (symbols likew?+w?-3+w-4+6would beused somewhere between w?andw*):after allthese onewould need wo” wo” andafter allthese thesymbol €9pops up.After 602EO. 608s EOyo OMe one comes to EVED Eee reEWM sre Legyee Leper eet andthisisonly thebeginning! Al} the well-ordered sets mentioned sofarare countable, There are indeed an cnormous number ofcountable well-ordered sets: 4.PROPOSITION. LetQbethecollection ofallcountable ordinals (ordinals represented byacountable well-ordered set). Then &isuncountable. PROOF. ByProposition 3,(@.<)isawell-ordered set.Ifitwere countable. 1 would represent acountable ordinal a€2.Bytheremark after Proposition 3. thiswould mean that &isorder isomorphic tothecollection ofordinals <a. i.c..10aproper segment ofitself, contradicting* Proposition J.4% Wehave thus established theexistence ofanuncountable ordinal. Our spe- cihe example. represented byQ,isclearly thefirst uncountable ordinal; any member ofQiscountable, and consequently hasonly countably many pre- decessors. (Itishopeless totryto“reach” &bycontinuing 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 ofcom- plexity: they areeach simple inoneway: *Bydeleting thewords countable anduncountable inthisproof oneobiains the“Burali- Forti Parados’: these1Ordofallordinal numbers iswell-ordered, soitrepresents an ordinal @€On.andheuce isorder isomorphic toaninitial segment ofOn. For# resohnion ofthisparadox, seeKelley's Appendix. Appendix A 465 5.PROPOSITION. If@€Qisalimit ordinal, then there isasequence By< By<By<+++<a,suchthateveryB<asatisfies B<Bnforsomen(wesay that {Bn} is“cofinal” ina). PROOF. Sinceaiscountable, allitsmembers canbelisted(innot-necessarily increasing order) 7),72,73..-.- LetBt=y%;and letBn4t bethefirst yinthe listwhich comes after B,.& 6.COROLLARY. Ifa€Q.thenaisrepresented bysomewell-ordered subset ofR.However, nosubset ofRBisorder isomorphic to2. PROOF. Suppose there were one, andhence asmallest, a€&notrepresented bysome subset ofR.}1cannot happen that a=8+1,forthen Bwould be represented byasubset ofR,thus also byasubset of(—0o,0) and acould byrepresented byasubset ofR.SobyProposition 5,there isasequence By<Bo<B3<++»<@cofinal ina.Then 8;isrepresented byasubset of {—o0,/), andwecaneasily arrange thatthesubset representing f;isasegment ofthesubset representing 8;fori<j.The union ofallthese setswould then represent a,aContradiction. Ifasubset ofRwere order isomorphic to2,then there would beuncountably many disjoint intervals inR,namely those between thepoints representing o anda+1foralla€Q.This isimpossible. The first example ofanon-metrizable manifold isdefined interms ofQ. Consider Qx{0,1),with theorder <defined asfollows: (a,s)<(B.t) ifa<Borifa =Bands <t. ‘This canbepictured asfollows: 0.0) 0.0) 2.0) (@.0) +10) (@42.0) {w2,0) The setQx[0,1)with theorder topology (asubbase consists ofsetsoftheform {xx<xo}and{x:x>No}) iscalled theclosed long ray(with “origin” (0,0). andL*=&x{0,1)—{(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-long line” and“long line” may alsobe used. The Corollary toProposition 5implies easily that thelong rayandthe long lineare1-dimensional manifolds; aside from thelineandthecircle, there are noother connected |-manifolds. 466 Appendix A Quite afewnew 2-manifolds cannow beconstructed: L*xS' (half-long cylinder), =LxS'(longcylinder). L+xR (half-long strip,. LxR (long strip), LxL (bigplane}, LxL* (bighalf-plane), L*xL* (bigquadrant). Identifying allpoints ((0,0),4) intheproduct oftheclosed longrayandS? produces another 2-manifold. which might becalled the“big disc”. There isanother way ofproducing anon-metrizable 2-manifold which does notuseQatall,Webegin withtheopen upper half-plane R4.={(x,y)€R?: }*>0}andanother copy R?x{0}oftheplane: wewilldenote thissetbyR2. anddenote thepoint(x,}',0)by(x,})o.Define amapfo:(R2)4.=RRby Consider thedisjoint union ofR3andR},withp€(R3)4 andfo(p) €RZ identified. This isaHausdorff manifold; thefollowing diagram shows two opensetshomeomorphic toR?,Themanifold itselfis,infact,homeomorphic Ri “see” RG=(RG) toR?;wecould havethrown away R3.tobegin withsince itisidentified bya homeomorphism with(I3)+- Butconsider now, foreach @€R,another copy ofR?,sayR?x{a},which wewilldenote byR2.Define fa:(R3)4 >4by @ Appendix A 467 Inthedisjoint union ofR3,andallR2,a €Rwewishtoidentify eachp€(R2), withfo(p) €R2.Wemaydispense with R2completely, andinthedigoint union ofallR2identify each (x,yaand(x’,)"), forwhich »=3">0and xy-+a =x'y-+b, Theequivalence classes, ofcourse, areaspace homeomorphic toR4,sowewillconsider R3.asubset oftheresulting space. This space is still aHausdorff manifold, but itcannot besecond countable, forithas an uncountable discrete subset, namely theset{(0,0)a}. This manifold, thePriifer manifold, andrelated manifolds, have some very strange properties, developed intheproblems. PROBLEMS 1.(a)Awell-ordered setcannot contain adecreasing infinite sequence xq> XQ >Xz doe. (b)Ifwedenote (a+1)+1by@+2, (@+2) +1bya+3, etc., then anya equals 8+1 foraunique limit ordinal 6and imeger n>0.(Thus one can define even and oddordinals.) 2.Let¢bea“choice function”, i.e.,c(A) isdefined foreach setA#9,and c(A) €AforallA.Given asetX,awell-ordering <ona subset YofXwill becalled “distinguished” ifforally€Y. yee(¥-Q" EY: <y}) (a)Show that ofanytwodistinguished well-orderings, oneisanextension of the other. (b)Show that there isawell-ordering onX.(Zorn’s Lemma may bededuced from thisfactfairly easily.) (c)Given twosets,showthatoneofthem isequivalent to(canbeputinone-one correspondence with) asubset oftheother. (d)Show that onanyinfinite setthere isawell-ordering which represents a limit ordinal. (e)From (d),and Problem 1,show that ifXand Yaredisjoint equivalent infinite sets, then XUYisequivalent toY. 3.(a)Ltand Larenotmetrizable. (b)Ifxy<x2<x3S++» is.asequence inL+.then {xn} converges tosome point. Consequently, any sequence hasaconvergent subsequence (but L+is notcompact!). 468 Appendix A ()If{xn} and{yn} aresequences inL+with xn<Yn<Xnq1 foralln,then both sequences converge tothesame point. (a)L+(and alsoL)arenormal. (Use (c)). (e)More generally, anyorder topology isnorma] (completely different proof). (f)Iff:L+=Riscontinuous, andr>s,thenoneofthesetsf~?((—00, s]) and7?({r,00)) iscountable. (g)Iff£:L+>Riscontinuous, thenfiseventually constant. 4.(a)L*isnot contractible. Hint: Given H:L*x{0,1] >L*with H(x,0) = xforallx,show thatforevery twehave {H(x,1)} =Lt. (b)m(L+) =2)(L) =0.Similarly forL+xR, LxR, LxL,LxL+,L+xL*. (C)m(L* xS!)=m(Lx S')=Z. 5.(a)LtandLarenothomeomorphic. Hinl: Imitating Problem 1-19, define “paracompact ends”. (b)L+xRand LxBRarenothomeomorphic; L*xS'andLxS?arenow homeomorphic, (c)Ofthe2-manifolds constructed from L+orLwith 7=0andonepara- compact end, only L+xRhasthehomotopy type ofL*. (a)TheStone-Gech compactifications ofLxL,LtxL,L+xLt,andthe bigdisc arcalldistinct. (Using Problem 3(g), one canexplicitly construct thesc Stone-Cech compactifications. 6.(a)Show that thePrifer manifold PisHausdorfi. (b)Pdoes nothave acountable dense subset. (0)LetUbeanopen setinR2.which istheunion of“wedges” centered at (a.0)foreveryirrationa] a.ShowthatUincludes awhole rectangle ofthe form —_{|t—a (a.b)x(0.e).Hint:LetA,={a:thewedgecenteredatahaswidth>1/n}. Since R= QUU,, An.some Ayisnotnowhere dense Appendix A 469 (d)LetG1,C. cPbe Ci={(0,0)q:@irrational} Cz={(0, 0)e:@rational}. Show that Cjand Careclosed, butthat they arenotcontained indisjoint open sets. (e)Define H:Px[0,1] >Pby (:firsts, [tee)ty>o H(@, Ya) = Ue Iostsy], (xvi=5%, vi=5?) ify<0. a Show thatHiswell-defined andthatH(p, 1)€R4.U{(0,0)q} forallp€P. Conclude that Piscontractible. (f)P-{(%, yo! »<0} isamanifold-with-boundary P’,whose boundary isa disjoint union ofuncountably many copies ofR. (@The disjoint union oftwocopies ofP’,with corresponding points onthe boundary identified, isamanifold which isnotmetrizable, butwhich hasa countable dense subset. Itsfundamental group isuncountable. 7.Itisknown thatevery second countable contractible 2-manifold isS?orR? Hence theresult ofconstructing thePriifer manifold using onlycopies R2for rational @must behomeomorphic toR?.Describe ahomeomorphism ofthis manifold onto R?. 8.LetMbeaconnected Hausdorff manifold which isnotapoint. (a)IfACMhascardinality ¢(thecardinality ofR),then theclosure Ahas cardinality ¢. (b)IfC¢Misclosed andhascardinality c,then Chasanopen neighborhood with cardinality ¢. (c)Letp€M.There isafunction f:2—>(setofsubsets ofM)such that J(a)hascardinality ¢foralla€Q,and such thar I(0) ={p} (a) isanopen neighborhood oftheclosure ofUpea /(B)- (Consider functions defined oninitial segments of2with these same properties. andapply Zorn’s Lemma. Alternatively, onecanrequire f(a) tobetheresult of applying thechoicefunction tothesetofallopenneighborhoods ofthe closure 470 Appendix A ofUpee £(6) with cardinality ¢,Then there isaunique fwith therequired properties. This isanexample ofdefining afunction by“transfinite induction”.) (@)Afunction f:2—(setofsubsets of{0,1])with theproperties ofthefune- tion inpart (c)iseventually constant. (©)Mhascardinality c,(Givenp’€M,consider anarcfromptop’) 9.(a)Aconnected I-manifold whose topology istheorder topology forsome order, ishomeomorphic toeither therealJine, thelong line, orthehalf-long line. (b)Every I-manifold Mcontains amaximal open submanifold Nwhose topol- ogyistheorder topology forsome order. (©)IfMisconnected andN#M,then Mishomeomorphic toS?. Appendix A 47) CHAPTER 2 The long rayL*canbegiven aC®structure, andeven aC®structure. Toseethis weneed theresult ofProblem 9-24—any C® [orC®] structure onamanifold Mhomeomorphic toRisdiffeomorphic toRwith theusual structure. This implies that itisalso diffeomorphic to(0,1), and consequently thatthestructure onMcanbeextended ifMisaproper subset ofLt. An easy application ofZorn’s Lemma then shows that C® and C®structures exist onLt. Idonotknow whether allC® structures onL+arediffeomorphic. Itis known thatthere areuncountably many inequivalent C®structures onL*,If péL+,andLp denotes allpoints <p,then L+—Ly isclearly homeomor- phic toL+. If©isaC®structure forL*,then ityields aC®structure for Lt—L+y, andhence forL+. These arealldistinct, inother words, there is noC®map SiL*t-L*,>Lt-L*, g>p with aC®inverse. Infact, wemust have f(g) >g,and then itiseasy tosee P q _ ,£9) q that wemust also have f(/(q)) >S(q), SL(L(g))) >S(L(g)), etc. The increasing sequence 4,f(y), S(/(q))s-.- hasalimit point xo€Lt—Ly. and f(xo) =Xo. Now /cannot betheidentity onal]points >xo(forthen itwould betheidentity everywhere, since itisC®). Soforsome g,>x0we have f(gi) #41;wecanassume f(g1) >41,since wecanconsider f~? in thecontrary case. Reasoning asbefore, weobtain x)>Xowith f(x1) =x1. Continuing inthisway, weobtain x9<x1<x2 <--> with f(&n) =xn.This sequence hasalimit inL+—L+p, butthisimplies that f(x) =xforallx,a contradiction. AC®structure exists onthePriifer manifold; thisfollows immediately from thefactthatthemaps fg.used foridentifying points invarious (R2)4 with points inR4,areallC®.Idonotknow whether every 2-manifold hasaC°% structure. Using theC®structure onL*,wecangetaC®structure onL+xL+.How- ever, themethod used forobtaining aC®structure onL+willnotyield acomplex anahiic structure onL*xL*;theproblem isthatacomplex analytic structure onR?maybeconformally equivalent tothedisc, andhence extendable, butit 472 Appendix A may alsobeequivalent tothecomplex plane, andnotextendable. Infact, i isaclassical theorem ofRado that every Riemannian surface (2-manifold with acomplex analytic structure) issecond countable. Ontheother hand, amod- ification ofthePriifer manifold vields anon-metrizable manifold ofcomplex dimension 2.References tothese matters are tobefound in Calabi and Rosenlicht, Complex Analytic Manifolds without Countable Base, Proc. Amer. Math. Soc. 4(1953), pp.335-340. H.Kneser. Anahrtische Strucktur undAbzahlbarkeit, Ann. Acad. Sic.Fennicae Se- ries A,125]/5 (1958), pp.1-8 PROBLEMS 10.Prove thatforg>pthere isnonon-constant C®map f:L+-Lt,> L*- L+y. 11.Let(1',p) beametric space andletf:X+Ybeacontinuous locally one-one map. where XisHausdorff. connected, locally connected, and locally compact. (a)Every twopoints x,)'€Xarecontained inacompact connected CCcX. (b)Letd(x, +)bethegreatest lower bound ofthediameters off(C) (inthe p-metric) forallcompact connected Ccontaining xand):.Show thatdisa metic onXwhich gives thesame topology forX. 12.Ofthevarious manifolds mentioned intheprevious section, trytodeter- mine which can beimmersed inwhich. Appendix A 473 CHAPTER 6 Problem A-6(g) describes anon-paracompact 2-manifold inwhich twoopen half-planes areadense set.Wewillnow describe a3-dimensional yersion with atwist. LetA={(x,¥,z)€R?:»#0}, andforeach a€RletR}beacopy ofR°. points inR3being denoted by(x,y, z)a.Inthedisjoint union ofAandallR3. a€Rweidentify Q,y,2)¢ fory>O0 with (a+ yx,y,2+a) (%,9,2)a for y<0 with (@+ px,y,z-a). Theequivalence classes forma3-dimensional Hausdorff manifold M.Onthis manifold there isanobvious function “z”, and thesets z=constant form a foliation ofMbya2-dimensional manifold N.The remarkable factabout this 2-dimensional manifold Nisthatitisconnected. For,thesetofpoints(x,y,¢)€ Awithy>0isidentified withthesetofpoints (x,».¢—@)q€R23withy>0. Now thefolium containing {(x,y,¢—@)a} contains thepoints (x,»,¢—@)a with »<0,and these areidentified with thesetofpoints (x,y,¢—2a)€A with y<0. Since wecanchoose a=c/2, weseethat allleaves ofthefoliation arethesame astheleafcontaining {(x,y,0): y<0} CA This example isdue toM.Kneser, Beispie/ einer dimensiinserhohenden anajytis- chenAbbildung zwischen tiberibzahlbaren Mannigfaltigheiten. Archiy, Math. 11(1960), pp.280-281. 474 Appendix A CHAPTERS 7,9,10 1.Wehave seen thatanyparacompact C®manifold hasaRiemannian metric. The converse also holds, since aRiemannian metric determines anordinary metric. 2.Problem A-1] implies thatamanifold Nimmersed inaparacompact mani- fold Misparacompact, butamuch easier proof isnow available: Let(,)be aRiemannian metric onM;iff:N>Misanimmersion, thenNhasthe Riemannian metric f*(, ). Wecannow dispense with theargument intheproof ofTheorem 6-6which wasusedtoshowthateachfoliumofadistribution onametrizable manifold is alsometrizable, forthefolium isasubmanifold, andhence paracompact. 3,Since there isnoRiemannian metric onanon-paracompact manifold M. thetangent bundle 7M cannot betrivial, Thus thetangent bundle ofthelong lineisnottrivial, noristhetangent bundle ofthePriifer manifold, eventhough 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 line Lisclearly orientable, sothere can- nol beanowhere zero 1-form won L,for wand the orientation would de- termine anowhere zero vector field, contradicting thefact that thetangem bundle isnot trivial. Thus, Theorem 7-9 fails forL.Notice also that ifM isnon-paracompact, then 7M isdefinitely notequivalent to7*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, they cannot beextended tonon- paracompact manifolds, ascanbeseen byconsidering the0-dimensional sub- manifold {(0.0)a} ofthePriifer manifold. 6.ALiegroup isautomatically paracompact, since itstangent bundle istrivial More generally, alocally compact connected topological group iso-compact (Problem 10-4), Appendix A 475 7,1isnotclear that anon-paracompact manifold cannot have anindefinive metric (anon-degenerate inner product oneach tangent space). This willbe proved inVolume II(Chapter 8,Addendum 1) PROBLEM 13,Isthere anowhere zero 2-form onthevarious non-paracompact 2-mani- folds which haye been described? NOTATION INDEX CHAPTER 1 Mp 76 &(X) 23 (M,)p 68 H” 19 R, 64 me 4 T™ 75 p? i T(M,i) 68 pA 19 TR* 64 R 1 T 103 ca 6 Xp 82se 7 ¥ 83 aM 19 x 81 bx, 76 CHAPTER 2 %y, 64 a 61 uf) 80 A aa fers. nd 84 co 34 eA 72 ee aa fey 101 ps aa hx& 102 Difia) 35 e 81GLo,R) 61 oaOm 6] Fal 80Ram) 62 a (R",U) 29 = 83SL(n,B) 61 Ee SO) 62 Pa a CHAPTER 4 vosa, afi t ‘JLo,| 35 dy’ 110 ai End(V) 121 al,39 rr 107, 116,119 A tt 113 re6 36 Hom(V, W) 131 TM 109 CHAPTER 3 T@S 116 of ea TeV) 116 ex) 72 TRE) V7 83 T(V) 120 te65,75 Tv) a Sep 65 Ty) 121 S78) 101 TV) 122 478 Notation Index The) 123 ahv) 23) w 10: Filmy(Y) 23) v n hry) 23}= 10; TianY) 23) gilode 129 THV) 20) Finds 134 vio 227 plot 134 4 206 o 10& +(Vy... Ue) 202 w(X) 109 Oo#(0),.... UK) 227 i" 10; Q(M) 215ak(V) 201 QV) 201 CHAPTER 5h aaa lal 71 161 exp 171 CHAPTER 8 LyA 174 BK(M) 263 Lxf 150) BK(M) 268 LxY 150 Cte) 250 Lyw 150 Rn 284 at?) Wi deg/ 275 Wy] 153 Idx? Asn dx"| 258 ax(t)=a(t.x) 143 de 252 (0X )q 135 d®, 290 w 144 dra,b.0 297 ft 292 CHAPTER 7 ip 274 Ab 202 HRM) 263 Rb 205 HE(M) 268curly 238 us 246 div¥ 238 Nien 249 de 219, 235 ha) 296 dw 210,213,215,234 M, 283 grad 237 mi 283 Iw 224 [Pal 239 4(4) 215 r 264 iyw 227 sign,/ 215 Lyw 234 w(p) 293 Sh 202 Zk(M) 263 Notation Indes 479 ZE(M) 268 s 313 dn 285 sinh 356 6 29) tanh 356 ’ 264, 291 we 349 D 264 5"(v) 335 o! 264 a 335 {e] 263 alu) 318 a 248, 285 as 319 ae 285 on 314 au 260 r 353 rk 328 Iau 246 C.) 301 dx a Cs) 315 ffdxtgdy 23 ve 38 f5 943, 245, 246, 248. (1 301 F c.y 349 fie 257,259,288 (.)" 305 4/9) 294 aa ae(0,19° 246 «3 ;D pag i 303 wl 303, * one wt 315 A 266 fe CHAPTER 10 CHAPTER 9 14I 384 cosh322,356 yi 372 cosh™?356 Ad(a) 409 d(p.9) 314 a An ae 314 Aut(9) 409 av 31) c a EuclV) 309 dw 402,404 Eucl) 309 En) 373 Ew) 324 End(g) 410exp Bed exp 385 Lt) aoe exp) 385 (e") 306 GL(,R) 372 Ui0 326 c 407 Lh 312 a? 407 M,t 344 al(”,R) 376 480 Notation Index Ie 401 CHAPTER 1} fh 374 fp 379 ch) 419 L(G) 376 Jx 446 00) 388 g*N) 432o(”) 376 M#N 453 P 4) PD 439 Po 411 Rl 457 Ro 374 u 442 SO(n) 373 4, 426 y 376 é 423 ¥ 379 6 435 be 380 x(M) 428 y 395 x) 445 plwrn) 403,410 — 439 w(natural g-valued i-form, 403 to 376 fad 403 APPENDIX A ffo” 400 Ond 464 atl 463 ft 400 bo 464 fflayda 400 e ae d < 46) ots, 403 < 463 im JNDEX Abelian Liealgebra, 376, 382, 395 quadrant, 466 Adams, J.F,100 Bi-invariant metric, 401 Adjoint 7"ofalinear transformation Boundary, 19,248, 252 T,103 Bounded manifold, 19 Ado, 1.D., 380 Boy’s Surface, 6() Alexander's Horned Sphere, 55 Bracket, 154 Algebra. Fundamental Theorem of. inqli,R), 378 285, 293 ino(7,R), 379 Algebraic inequalities, principle of Bundle irrelevance of,233 cotangent, 10°Alternating dual,108 covariant tensor field, 207 fibre, 309 multilinear function, 201 induced, 101 Alternation, 202 map, 73 Analytic manifold, 34 r-plane, 7] Annihilator, 228 normal, 344 Annulus, & ofcontravariant tensors, ]20 Antipodal ofcovariant tensors, ]]7 map, 278 tangent, 77 point, 1) trivial, 72,210 Arclength, 312 . vector, 7] function, 313, 332 Burali-Forti Paradox, 464 Arewise connected, 20 subgroup ofaLiegroup,409 Area, generalized, 246 Associated - 7discbundle, 451 eaeen aasphere bundle, 451 aleulus ofvariations, Atlee, 28 Cartan, Elie, 39,348,360 mneximal, 26 Cartan’s Lemma, 230 ‘Auslande,, L,106 Cauchy-Riemann equations, 200 a Cayley numbers, 100 Chain, 248, 285 Chain Rule, 35,38 Change, infinitely small, 111 Banach space. 145 Chart, 28 Base space, 7) Choice, 283 Basis Choice function, 467 dual, 107 Circe, 6 forMp*, 208 Closed forQF(p), 208 form, 218, 252 Belongs toadistribution, 19] geodesic, 367 Besicovitch, A.S., 179 half-space, 19 Bie long ray, 465 disc, 466 manifold, 19 half-plane. 466 subgroup ofaLiegroup,391 plane, 466 submanifold, 49 482 Index Closed (continued, Cup product, 299, 439 uptofirst order, 16() Curl, 238 Cofinal, 465 Cylinder, 8 Cohomology, 419 C?manifold, 34 deRham, 263 C°manifold, 34 group ofMwith realcoefficients. ce 263 distribution, 179 ofacomplex, 421 form, 207 Commutative diagram, 65,420 function, 32 Commutative Liealgebra. 376 manifold, 29 Complete, geodesically. 341 manifold-with-boundary, 32 Complex, 421 Riemannian metric, 308 analytic structure, 47) structure onTM, 82 numbers ofnorm |,373 C™-related, 28 Conjugate, 358 Constants ofstructure, 390 Continuous homomorphism, 387 Contractible, 220, 225. 236 Darboux Contraction, 12],139.227 integrable, 283 Lemma, 139 integral, 283 Contravariant Darboux’s Theorem, 284 functor, 130 Debauch ofindices, 39,123 tensor field, 120 Decomposable, 228 vector field, 113 Definition, invariant, 214 Convex Deformation retraction, 279 geodesically, 363 Degenerate, 286 polyhedron, 429 Degree, 275 Coordinate lines, 159 mod 2,295 Coordinate system, 28.158 Density Coordinates, 28 even scalar, 133, 209 Cotangent bundle, 10° oddscalar, 133, 259Covariant relativescalar,231functor, 130 scalar, 133 tensor field, 117 Derivation, 39,78 vector field, 113 ofa ring, 83 Cover Derived set,25 locally finite, 50 Descartes-Euler Theorem, 429 point-finite, 60 Determinant, 232 refinement of,50 Difleomorphic, 30 Cramer's Rule, 372 Dilleomorphism, 30 Critical poim, 40 one-parameter group of,148 inthecalculus ofvariations, 320 Diflerentiable, 27,28,31,32 Critical value, 40 ata point, 31 Cross section, 227 manifold, 29 Cross-cap, 14 structure, 30 Cross-product, 299 onthelong line, 471 Cube, singular, 246 onP*, 32 Inder 483 Differentiable (continued Euler, 429 (structure continued, characteristic, 428 onR", 29 class, 445 onS", 30 Euler’s Equation, 320 Differential, 210 Even equation, 136, 164 ordinal, 467 depending onparameters, 169 relative scalar, 231 linear, 165 relative tensor, 134, 231 forms, 201 scalar density, 133, 209 ofafinction, 109 ExactDimension, 4 form, 218 Direct sum, 421 sequence, 419, 422 Disc bundle, associated, 451 ofapair, 433 Discriminant, 233 ofvector bundles, 103 Disjoint union, 4,20 Exponential map, 334, 383 Distribution, 179, 181 Exponential ofmatrices, 384 ideal of,215 Extension, 432 ontorus, 180 Extremal, 320 Divergence, 238 Theorem, 352 Domain, 3 aay aDuBoisReymond’s Lemma, 355 orb op Fibre, 64,68,71 basis, 107 Finite characteristic, 205 space,107 one 438 vectorbundle, 108 en First variation, 319, 327 Five Lemma, 440 Einstein summation convention, 39 Fixed point, 139 Elements ofnorm |,308 Foliation, 194 Elliptical non-Euclidean geometry, 367 Folium, 194 Embedding, 49 Force field, 240 End, 23 Form, 207 paracompact, 466 differential, 201 Endomorphism, 121 leftinvariant, 374 Energy, 324 right invariant, 400 Envelope, 358 f-related, 190 Equations depending onparameters. Frobenius Integrability Theorem, 192, 169 215 Equations ofstructure. 404 Fubini’s theorem, 254 Equivalence (ofvector bundles), 72 Functor, 130 weak, 96 Functorites, 89Euclidean Fundamental Theorem ofAlgebra. metric, 305, 315 285, 293 motion. 374 Fundamental Theorem ofCalculus, n-space, | 254 484 Index Gauss’s Lemma, 337 Hyperbolic General linear group, 61,372 cosine, 356 Generalized area, 246 sine, 356 Geodesic, 333 tangent, 356 closed, 367 reversing map, 401 Geodesically complete, 341 Geodesically convex. 363 Ideal ofaLiealgebra,410 Geodesy, 333 Identification, 10 Germs ofk-forms, 432 Imbedding, 49 Globaltheoryofintegral manifolds. topological, 14 193 Immersed submanifold, 47 Gradient, 237 Immersion, 46 Gram-Schinidt orthonormalization ropological, 14,46 process, 304 Implicit function theorem, 60 Grok. 84 Indefinite metric, 350 Group Independent infinitesimals, 314Lie, 371 Index ofinner product, 349 matrix, 372 Index ofvector field opposite, 407 ona manifold, 447 orthogonal, 372 onR",446 topological, 371 Indices Guillemin, V.W.,106 debauch of,39,123 raising and lowering, 351 Induced bundle, 101 orientation, 260 Hahn-Banach theorem, 145 Inequalities, principle ofirrelevance of Hair. 69 algebraic, 233 Half-lone Inertia, Sylvester’s Lawof,349cylinder. 466 Infinite volume, 312 Tine, 465 Infinitely small change, 111 strip, 466 Infinitely small displacements, 314 Halfspace, 19 Infinitesimal generator, 148 Handle, & Infinitesimals, independem, 314 Hardy, G.H., 179 Initial conditions, 136 Hasoneend,23 ofintegral curve, 136 Heinlein, Robert A.,84 Initial segment, 462 Hausdorff, 459 Inner product, 227, 301 Homogeneous, 7 preserving, 304, 372Homomorphism usual,301 continuous, 387 Inside, 21 ofLiealgebras, 380 Integrability conditions, 189 Homotopic, 104, 277 Integrable distribution, 192 Homotopy, 104, 277 Imegrable function Hopf, H.,342, 450 Darboux, 283 Hopf-Rinow-de Rham Theorem, 342 Riemann, 283 Index 485 Integral vectorfield,374 curve, 136 Left translation, 374 Darboux, 283 Length, 243, 305, 312 line, 239, 243 ofacurve,59 manifold, 179,181 Liealgebra, 376 maximal, 194 abelian, 376, 382, 395 ofadifferential equation, 136 commutative, 376 Riemann, 283 homomorphism of,380 surface, 245 ideal of,410 Integration, 136, 226, 239 opposite, 407 Invariance ofDomain, & Liederivative, 150 Invariant, 128, 232 Liegroup, 371 definition, 214 arewise connected subgroup of,409Irrelevance ofalgebraic inequalities, closed subgroup of,391 principle of,233 local, 415 Isometry, 340 normal subgroup of,410 Isomorphic Liegroups. locally, 382 topologically isomorphic, 388 Jsomorphism, natural. 10? Liesubgroup, 373 Isotopic, 294 Lie’s fundamental theorem first, 414 second, 415 third. 416 Jacobi identity, 155,376 Limit forthebracket inanyring,378 ordinal, 463 Jacobian matrix, 40 st.GU ;Jordan Curve Theorem, 21,435 Lineintegral, 239,243 .Linear differential equations, 165 systems of,17] Linear transformation adjointof,103 Kelley,J.460,463,464 aeof,121 Kink,366 positivedefinite, 104 Kleinbottle, 18,435 positive semi-definite, 104 Kneser, H.,472 Linking number, 296Kneser, M.,473 Lipschitz condition, 138 Littlewood,J.E.,179 Lives atpoints, 119 Lobachevskian non-Euclidean geome- Lang, S,145 uy,368 Laplace’s expansion, 230) Local Laplacian, 58 flow, 144LawofInertia, Sylvester's, 349 Liegroup, 415 Leaf, 194 one-parameter group oflocal diffeo- Leap offaith, 464 morphism, 148 Left invariant spanned locally, 179 form, 394 uriviality, 71 nform, 400 Local theory ofintegral manifolds, 190 486 Index Locally Mayer-Vietoris Sequence, 424 compact, 20 forcompact supports, 431 connected, 20 Measure zero, 40,41 finite cover, 50 Mesh, 239 isomorphic Liegroups, 382 Meuic Lipschitz. 139 bi-invariant, 401 one-one, 13 Euclidean, 305, 315 pathwise connected, 20 indefinite, 350Long Riemannian, 308,311 cylinder, 466 usual, 312 line, 465 spaces, disjoint union of,4,20 ray, 469 Milnor, J.W, 42 closed. 465 Mod 2degree, 295 open, 465 Mobius strip, 10 Lower sum, 283 generalized, 100 Mult-index, 208 Multilinear function. 115 Munkres, J.R., 34.106 MacKenzie. R.E., 106 Magic, 214 Manifold, 1,459 analytic, 34 n-dimensional, 4 atlas for. 28 n-forms, leftinvariant, 400 boundary of.16 n-holed torus, 9 bounded, 19 n-manifold, 4 closed. 19) n-plane bundle, 71 cr, 34 n-sphere, 7 C°, 34 n-torus, 7 c™, 24 Natural g-valued I-form, 403 differentiable, 20 Natural isomorphism, 108 dimension of.4 Neighborhood, tubular, 345 imbedding inR¥. 52 Newman, M.H.A.. 3 integral. 179, 181 Nice cover, 438 maximal, 194 Non-bounded, 19 non-nictrizable, 465, 466 Non-degenerate, 301 orientation of.86 Non-Euclidean geomeuy smooth, 29 elliptical, 367 Manifold-with-boundary. 19 Lobachevskian. 368. Ce, 32 Non-meurizable manifold, 465, 466 Map Non-oriemable between complexes. 421 bundle, 86 bundle. 73 manifold, 80 rank of.40 Norm, 303 Massey. WS. 3 preserving, 304, 372 Mawix groups. 372 Normal Maximal imicgral manifold, 194 bundle, 344 Index 487 Normal (continued) Outside, 21 space, 459 Outward pointing, 260 subgroup ofaLiegroup,410 Outwardunitnormal,351 outward unit, 351 Nowhere zero section, 209 Palais, R.S., 100, 225 Oda Paracompact, 210,459 ordinal, 467 ence) ciesrelativetensor,134,288 Parameter curves,special,167scalardensity, 133,259 Parameterized byarclength, 313One-dimensional distribution, 179 Be ee One-dimensional sphere,6 ees 245 oflocaldiffeomorphisms, local.148 ‘Piecewise smooth, 312One-parameter subgroup, 384 Pig,yellow, 434Open Poincaré, H.,450Tongray,465 Poincaré dual,439 map,60 Poincaré Duality Theorem, 441submanifold, 2 Poincaré-Hopf Theorem, 450Opposite PoincaréLemma,225roup, 407 Poincaré upper halftplane, 367Liealgebra, 407 PointOrder inward, 98 isomorphic, 461 outward, 98,260 isomorphism, 461 Point-derivation, 39 topology, 465 Point-finite cover, 60 Ordered set,461 Polar coordinates, 36 Ordering, 460 integration in,266 Ordinal numbers, 463 Polarization, 304Orientable Pollack,A.,106bundle, 86 Positive definite, 104, 301 manifold, 86 Positive element ofnorm 1,308 Orientation Positivesemi-definite, 104ofabundle,85 Productofa manifold, 86 ofvector bundies, 102 ofavectorspace,84 tensor,116preserving, 84,85,88,105,248 Projection, 7,30,32 reversing, 84,88,248, Projective Orthogonal group, 61,372 plane, 11,435, Osthonormal, 304, 348 space, 19,88 Orthonormalization process, Gram Proper map, 60,275 Schmidt, 304 Prifer manifold, 467 Osgood’s Theorem, 284 Pseudometric, 95 488 Inder Quaternions, 100 Sard’s Theorem, 42,294 ofnorm 1,373 Scalar, relative, 134, 231 Scalar density, 133 Schwarz, H., 354 Schwarz inequality, 303, 362 Second countable, 459 Section ofavectorbundle, 73 zero, 96 Radial function, 435 Segment, initial, 462 Rado, T,472 Selfadjoint linear transformation, 104 Rank Semi-definite, positive, 104 ofaform,229 Separatepointsandclosedsets,95 equence7Cae exact,419,422 ectifiable, 59 Refinement ofacover,50 olectoaaneB Mayer-Vietoris, 424 Sie 40forcompact supports,431 point,ofa pair, 433space,459 Shrinking Lemma, 51 value, 40 Shrinking Lemma, 60 Related vector fields, 190 Shuffle permutation, 227Relative Simplexscalar, 134,231 ofatriangulation, 427 tensor, 134,231,288 _Singular, 285 . Reparameterization, 244,248 Simply-connected, 287Retraction, 264 Liegroup,382deformation, 279 SingularP . cube, 246 Revolution, surface of,8,321 simplex, 285 deRham, G.,342 Skew-symmetric, 201,378 deRham cohomology vector spaces. Slice, 194 263 Slicemaps, 54 with compact supports, 268 Smooth, 28 deRham’s Theorem, 263, 457 homotopy, 27 Riemann manifold, 29 integrable, 283 piecewise, 312 integral, 283 Smoothly _ sum,283 contractible, 220 Riemannian metric, 308,31) homotopic, 277 usual, 312 isotopic, 294 Rightinvariant n-form, 400 Solidangle,290Righttranslation, 374 Spacefillingcurve, 56ight translation, Spanned locally, 179Rinow W.,342 Special linear group, 61 Roman surface, 17,26 Special orthogonal group, 62 Rosenlicht, M.,472 Sphere, 7 Rotation group. 62 Sphere bundle, associated, 451 Index 489 Standard Tensor n-simplex, 426 contravariant, 120 singular cube, 246 covariant, 113 Star-shaped, 221 even relative, 134,231 Steiner’s surface, 17,26 odd relative, 134, 288, Sternberg, S.,42,106 Tensor field Stokes’ Theorem, 253, 261, 285, 352 classical definition of,123 Stone-Gech compactification, 468 contravariant, 120 Structure constants, 396 covariant, 113 Subalgebra ofaLiealgebra,379 mixed,121,122 Subbundle, 198 Tensor product, 116 Subcover, 50 Thom class, 442Subgroup ThomIsomorphism Theorem, 456Lie,373 Topological one-parameter, 384 group, 371Submanifold, 49 imbedding, 14 arn immersion, 14,46 closed, 49 Topologically isomorphic Liegroups, immersed, 47 eee open, 2 Torus, 7,8Successor ordinal, 464 eeeSumofvectorbundles, Whitney, 101 ee 1Sut,33,147 ‘otallydisconnected, 25port, 2¥s Transitivity, 460Surface, 7 4Beate Translationpees left, 374 integral, 239 tight, 374 ofrevolution, 8,321 Triangle inequality, 303Syivester’s LawofInertia, 349 Triangulation, 426S icbilinearform,301 beers ty ymmetric bilinear form, ; simplex of,427Systemoflineardifferential equations. Trichotomy, 461 71Trivial vector bundle, 72 o-compact, 4,458 Tubular neighborhood, 345 Two-holed torus, 8 Tangent bundle, 77 ,Tangent spaceofR",64 Vick,J.W,3 Tangent vector inward pointing, 98 of'a manifold, 76 ofR",64 Wedge product, 203 outward pointing, 98,260 Whitney, H., 106 toacurve, 63,66 Whitney sum, 101 9booksweretypesetusingDonaldE,Knuth’sTEXtypesettingsystem, together with Berthold Horn’s DVIPSONE PostScript driver. The figures were produced with Adobe Illustrator, and new ormodified fonts were created using Fontographer. The textfont is1]point Monotype Baskerville—though theem-dash hasbeen modified—together with itsitalic. The elegant swashes oftheitalic »andf cause problems inwords liketopology andapology, soaspecial gyligature was added; special geandgfligatures were alsorequired. Although aBaskerville bold faceisunhistorical, bold type wasuseful inspecial circurnstances—mainly forindicating defined terms, The bold face supplied by Monotype, even the“semi-bold”, isobtrusively extended, soanon-extended version was created. The somewhat bold appearance ofchapter headings, in16point type, results from thelinear scaling, aswell asthefactthat theupper case Baskerville 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 setofnumerals wasconstructed forusewith theupper case lettering inchapter headings and statements oftheorems, and special parentheses and other punctuation symbols were also required. Numerous other modifications ofthis sort, including additional kerns andalterations ofsetwidths, were made forvarious purposes. 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CORRECTIONS FOR VOLUME I pg3,line3—:change di(x,y)<1todi(x,y)<1. pg;14:relabel thelower leftpartofthecentral figure as, Aa| [ar Bi Ve Atlfar pg19:replace thenext-to-last paragraph with thefollowing: ‘The setofpoints inamanifold-with-boundary thatdonothave aneighborhood homeomorphic toR"(but only onehomeomorphictoHI")iscalledtheboundary ofMandisdenotedby4M.Equivalently, x¢8Mifandonlyifthereisaneighbothood Vofxandahomeomorphism @:V—>Hsuchthat@(x)=0.IfMisactuallyamanifold, then9M=@,andMitselfisalwaysamanifold (without boundary} pg22:Replace thetopleftfigure with i) i ff ) pg43:Replace thelastlineanddisplayed equation with thefollowing: Since rank f=kinaneighborhood ofp,thelower rectangle inthematrix (sD)- Dav, Da“"\X Davt*!...Daw pg.60,Problem 30:Change part(f)andaddpart(g): (0)IfMisaconnected manifold, there isaproper map f:M—+R;thefunction fcanbemade C®ifMisaC®manifold. ()The same istrue ifMhasatmost countably many components, pg61,Problem 32:Forclarity, restate part(das follows: (¢)This isfalse iff:Mi>Risreplaced with f:M>Nforadisconnected manifold N. Pg70:Replacethelasttwolinesofpage70andthefirsttwolinesofpage 71with thefollowing: theoremoftopology). IftherewereawaytomapT(M,i),fibrebyfibre,homeomorphically ontoMxR?,theneachvpwouldcorrespond to(p,v(p)) forsome v(p)€R?,andwecould continuously pickw(p) €R?,corresponding toadashed vector, byusing the criterion that w(p) should make apositive angle with v(p). pg78:thethird display should read: 0=£(0) =(fh) =S(pelh) +h(p)e(f) =0+E(/). pg.103,Problem 29(d). Add thehypothesis thatMisorientable. Pg117:After thenexttolastdisplay, A(X1,..+)Xx)(P)=A(P)(Xi(P)s- ++»Xe(P)),add: IfAisC™,thenAisC®,inthesensethatA(Xi,...,X4) isaC?functionforallC%vectorfieldsX,...,Xe. Pg118: Add thefollowing tothestatement ofthetheorem: IfAisC*,then Aisalso. pg.119:Addthefollowing attheendofthe proof: Smoothness ofAfollows from thefactthatthefunction Ajy..jg isA(8/Xiq,.-.8/8%i,)- pg131, Problem 9:LetFbeacovariant functor from V,... pg.133.Though there isconsiderable variation interminology, what areherecalled “odd scalar densities” should probably simply be called “scalar densities”; what arecalled “even scalar densities” might bestbecalled “signed scalar densities”Inpart(¢)ofProblem 10,weshouldbeconsidering theAofpart(a),nottheAofpart(b)!Thusconclude thatthebundleofsignedscalar densities (notthescalar densities) isnottrivial ifMisnotorientable. pg.194.Extending thechanged terminology from pg,133,weshould probably speak ofthebundle of“signed tensor densities oftype (j)andweight w”(though sometimes thetermrelative tensor isusedinstead, restricting densities tothoseofweight 1),when the transformation ruleinvolves (detA)", omitting themodifier “signed” when itinvolves |detA|®. pg.143. Thehypothesis ofTheorem 3should bechanged sothatitreads: Letx€Uandleta,a2betwomapsonsomeopenintervalJsuchthat(J),(I)CU, a= fa) i=1,2 ane (Fo)=aaa)forsometp€J Andthefirstsentence oftheproofshouldbedeleted. pg-177. Problem 17,part(d)should begin: (d)Letf:M>N,andsupposethatfop=0.ForXp,¥p€Mpand... Pg,198, InProblem 5,wemust alsoassume thateach A,@Ayisintegrable. Pg:226. Inthecomutative diagram, thelower right entry should be“I-forms onN”. pg:233. Thereference “pg.V375” refers topg.375ofVolume V. pg237. InProblem 26,replace parts (b)and(c)with: {b)Determine theicomponent ofvjx--+xvq—1intermsofthe(m—1)x(n—1)submatrices ofthematrix (:) Tnparticular, forR?,show that vxw=(v'w? —Pw, vw! —vlw,vw? —vw), pg292. InProblem 20,thecondition UjMUj#@should beU;NUi: #8. pg408. Problem 16(b)should read: “For anyLiegroup G,show that...”. pp.408-410. Forconsistency withstandard usage, Autshould bereplaced withAut,andthenreplace EndwithEnd. Inpart(g)of Problem 19,addthehypothesis thatHisaconnected Liesubgroup. pg.411. The display inProblem 21,part (c)should read: Cylon Aad +(-IMinaAol+(-1!"[A.A[wAn]=0.