no author - connections_chapter2_2

Chapter 2 Bundles Contents 2.1Vector bundles ...................17 2.2Smoothness .....................26 2.3Operations onvector bundles ..........28 2.4Vector bundles withstructure ..........33 2.4.1 Complex structures ................33 2.4.2 Bundle metrics ..................35 2.4.3 Volume forms andvolume elemen ts.......37 2.4.4 Indenite metrics .................43 2.4.5 Symplectic structures ..............45 2.4.6 Arbitrary G-structures ..............48 2.5Innite dimensional bundles ...........49 2.6General berbundles ...............56 2.7Structure groups ..................59 2.8Principal bundles .................63 2.1Vector bundles: denitions andexam- ples Roughly speaking, asmoothvectorbundle isafamily ofvectorspaces that are\attac hed"toamanifold insome smoothly varying manner. Wewill presen tthisideamorerigorously inDenition 2.8below.Firstthough, it's worthlooking atsome examples tounderstand whysuchanobjectmight beofinterest. 1718 CHAPTER 2.BUNDLES Example 2.1(Thetangen tbundle ).IfMisasmoothn-dimensional manifold, itstangen tbundle TM=[ x2MTxM associates toeveryx2Mthen-dimensional vector spaceTxM.Allof these vector spaces areisomorphic (obviously ,since theyhavethesame dimension), though it'simportanttonotethatthereisgenerally nonatural choice ofisomorphism betweenTxMandTyMforx6=y.There ishowever asense inwhichtheassociation ofx2Mwiththevector spaceTxMcan bethough tofasa\smooth"function ofx.We'llbemore precise about thislater, butintuitivelyonecanimagineMasthesphereS2,embedded smoothlyinR3.Then eachtangen tspace isa2-dimensional subspace ofR3, andtheplanesTxS2R3varysmoothly asxvariesoverS2.Ofcourse, the denition of\smoothness" forabundle should ideally notdependonany embedding ofMinalarger space, justasthesmoothmanifold structure ofMisindependen tofanysuchembedding. Thegeneral denition will havetore ect this. Instandard bundle terminology ,thetangen tbundle isanexample ofa smoothvectorbundleofranknoverM.WecallMthebaseofthisbundle, andthe2n-dimensional manifoldTMitselfiscalled itstotalspace.There isanatural projection map:TM!Mwhich,foreachx2M,sends everyvectorX2TxMtox.Thepreimages1(x)=TxMarecalled bers ofthebundle, forreasons thatmightnotbeobvious atthisstage, butwill become moresoasweseemore examples. Example 2.2(Trivialbundles ).Thisisthesimplest andleastinteresting example ofavectorbundle, butisnonetheless important.Foranymanifold Mofdimensionn,letE=MRmforsomem2N.Thisisamanifold ofdimensionn+m,withanatural projection map :MRm!M:(x;v)7!x: Weassociatewitheverypointx2MthesetEx:=1(x)=fxgRm, whichiscalled theberoverxandcarries thestructure ofarealm- dimensional vector space. ThemanifoldsEandM,together withthe projection map:E!M,arecollectiv elycalled thetrivial realvector bundleofrankmoverM.Anymaps:M!Eoftheform s(x)=(x;f(x)) iscalled asection ofthebundle:E!M;heref:M!Rmisan arbitrary map. Theterminology comes fromthegeometric interpretation oftheimages(M)Easa\cross section" ofthespaceMRm.Sections canalsobecharacterized asmapss:M!Esuchthats=IdM,a denition whichwillgeneralize nicely toother bundles. 2.1.VECTOR BUNDLES 19 Remark 2.3.Awordaboutnotation beforewecontinue:sincetheessential information ofavector bundle iscontained intheprojection map,one usually denotes abundle by\:E!M".Thisnotation isconvenient because italsomentionsthemanifoldsEandM,thetotalspaceandbase respectively.When there's noambiguit y,weoftenomititselfanddenote abundle simply byE!M,orevenjustE. Consider foronemore momen ttherstexample, thetangen tbundle :TM!M.Asection ofthebundle:TM!Misnowdened tobe anymaps:M!TMthatassociates witheachx2Mavector inthe berTxM;equivalently,s:M!TMisasection ifs=IdM.Thus \section ofthetangen tbundle" issimply another termforavectoreld onM.Wedenote thespace ofvector elds byVec(M). IfMisasurface embedded inR3,thenitisnatural tothink ofeach tangen tspaceTxMasa2-dimensional subspace ofR3.Thismakes: TM!Masmoothsubbund leofthetrivial bundleMR3!M,in thateachberisalinear subspace ofthecorresp onding berofthetrivial bundle.Example 2.4(Thecotangen tbundle ).Associated withthetangen t bundleTM!M,there isa\dual bundle"T M!M,called thecotan- gentbundle:itsbersarethevector spaces T xM=Hom R(TM;R); consisting ofreallinear mapsTxM!R,alsoknownasdualvectors.The sections ofTMarethusprecisely thedieren tial1-forms onM.This isourrstexample ofanimportantvector bundle thatcannot easily be visualized, though onecanstillimagine thatthevector spacesTMvary smoothly insome sense withrespecttox2M. Observ ethatTMhasthesame rankasTM,andindeed there are alwaysisomorphisms TxM=T xMforeachx,butthechoice ofsuchan isomorphism isnotcanonical. Thisisnotatrivial commen t:oncewedene what \isomorphism" means forvector bundles, itwillturnoutthatTM andTMareoftennotisomorphic asbundles, eventhough theindividual bersTxMandT xMalwaysare. Example 2.5(Tensor bundles ).Thetangen tandcotangen tbundles are bothexamples ofamoregeneral construction, thetensor bundlesTk `M! M,whose sections arethetensor eldsoftype(k;`)onM.Forintegers k0and`0,theber(Tk `M)xoverx2Misdened tobethevector space `O j=1T xM! kO j=1TxM! ;20 CHAPTER 2.BUNDLES whichconsists ofmultilinear maps TxM:::TxM|{z} `T xM:::T xM|{z} k!R: Thus(Tk `M)xhasdimensionnk+`,and Tk `M!M isarealvectorbundle ofranknk+`.Inparticular, T1 0M=TMhasrankn, andsodoesT0 1M=TM.Fork=`=0,weobtain thetrivial linebundle T0 0M=MR,whose sections aresimply realvalued functions onM.For k0,T0 kMcontainsanimportantsubbundle T0 kMkTM!M ofrankn! k!(nk)!,whose sections arethedieren tialk-forms onM.1 Notation. Inallthatfollows,wewillmakeoccasional useoftheEinstein summation convention ,whichalready made anappearance intheintroduc- tion,x1.2.Thisisanotational shortcut bywhichasummation isimplied overanyindex thatappearsasapairofupperandlowerindices. Sofor instance iftheindexjrunsfrom1ton, jdxj:=nX j=1jdxj;andXj@ @xj:=nX j=1Xj@ @xj; inthelatter expression, theindex in@xjcountsasa\lower"index by conventionbecause it'sinthedenominator. Thesummation convention andthesignicance of\upper"vs.\lower"indices isdiscussed more fully inAppendix A. Example 2.6(Distributions ).Supposeisasmooth1-form onthe manifoldMwhichiseverywhere nonzero. Then weclaim thatatevery pointp2M,thekernelofjTpMisan(n1)-dimensional subspacep TpM.Toseethis,choosecoordinates (x1;:::;xn)nearapointp0sothat canbewritten as =jdxj; using thesetofreal-valued \comp onentfunctions"1(p);:::;n(p)dened nearp0.Byassumption thevector (1(p);:::;n(p))2Rnisneverzero, andthekernelofatpisprecisely thesetoftangen tvectors X=Xj@ @xj2TpM 1Thenotions ofmultilinear maps andtensor products usedherearediscussed in detail inAppendix A. 2.1.VECTOR BUNDLES 21 suchthat(X)=j(p)Xj=0,i.e.itisrepresen tedincoordinates bythe orthogonalcomplement of(1(p);:::;n(p))2Rn,an(n1)-dimensional subspace. Theunion ofthesubspaces kerjTpMforallpdenes asmoothrank- (n1)subbundle !M ofthetangen tbundle, alsoknownasan(n1)-dimensional distribution onM.More generally ,onecandene smoothm-dimensional distributions foranymn,allofwhicharerank-msubbundles ofTM.Inthiscase, \smoothness" means thatinsome neighborhoodofanypointp2M, onecanndasetofsmoothvector eldsX1;:::;Xmthatarelinearly independen tandspanqateverypointqclosetop.Asection ofisby denition avector eldX2Vec(M)suchthatX(p)2pforallp2M. Example 2.7(Acomplex linebundle overS3).Alloftheexamples so farhavebeenrealvector bundles, butonecanjustaswelldene bundles withbersthatarecomplex vector spaces. Forinstance, deneS3asthe unitsphere inR4,andidentifythelatter withC2viathecorresp ondence (x1;y1;x2;y2) !(x1+iy1;x2+iy2): Then ifh;iC2denotes thestandard complex inner productonC2,one ndsthattherealinner producth;iR4onR4isgivenby hX;YiR4=RehX;YiC2; andinparticularhX;XiR4=hX;XiC2.Thusbythisdenition, S3=fz2C2jhz;ziC2=1g: Thetangen tspaceTzS3isthentherealorthogonal complemen tofthe vectorz2C2=R4withrespecttotheinner producth;iR4.Butthere is alsoacomplex orthogonal complemen t z:=fv2C2jhz;viC2=0g; whichisbothareal2-dimensional subspace ofTzS3andacomplex 1- dimensional subspace ofC2.Theunion ofthese forallz2S3denes a vector bundle!S3,whichcanbethough tofaseither arealrank-2 subbundle ofTS3!S3orasacomplex rank-1 subbundle ofthetrivial complex bundleS3C2!S3.2 2Thedistribution TS3iswellknownincontactgeometry asthestandar dcontact structur eonS3.22 CHAPTER 2.BUNDLES With these examples inmind, weareready forthegeneral denition. Therigorous discussion willbegininapurely topological context,where thenotion ofcontinuityiswelldened butsmoothness isnot|smo oth structures onvector bundles willbeintroduced inthenextsection. Notation. Ineverything thatfollows,wechooseaeld F=either RorC; andassume unless otherwise noted thatallvector spaces andlinear maps areF-linear. Inthiswaytherealandcomplex casescanbehandled simul- taneously . Denition 2.8.Avectorbundleofrankmconsists ofapairoftopological spacesEandM,withacontinuoussurjectiv emap:E!Msuchthat foreachx2M,thesubsetEx:=1(x)Eisavectorspace isomorphic toFm,andforeverypointx2Mthere exists anopenneighborhood x2UMandalocaltrivialization :1(U)!UFm: Hereisahomeomorphism whichrestricts toalinear isomorphism Ey! fygFmforeachy2U. WecallEthetotalspaceofthebundle:E!M,andMiscalled thebase.Foreachx2M,thesetEx=1(x)Eiscalled theberover x. Denition 2.9.Avector bundle ofrank1iscalled alinebundle. Abundle ofrankmissometimes alsocalled anm-dimensional vector bundle, oranm-plane bundle.Theuseoftheterm\linebundle" form=1 isquite intuitivewhen F=R;onemustkeepinmind howeverthatinthe complex case,thebersshould bevisualized asplanes rather thanlines. Givenavector bundle:E!MandanysubsetUM,denote EjU=1(U).Asinthedenition above,atrivialization overUisa homeomorphism :EjU!UFm thatrestricts linearly tothebersasisomorphisms Ex!fxgFm foreachx2U.Onecannot expectsuchatrivialization toexistforevery subsetUM;when itdoesexist, wesaythat:E!Mistrivializable overU.Thuseveryvector bundle isrequired tobelocallytrivializable ,in thesense thattrivializations existoversomeneighborhoodofeachpointin thebase. Thebundle iscalled (globally) trivializable ,orsimply trivial ,if 2.1.VECTOR BUNDLES 23 there exists atrivialization overtheentiretyofM.We'vealready seenthe archetypalexample ofaglobally trivial bundle: theproductE=MFm. Nontrivial bundles areofcourse more interesting, andarealsoinsome sense more common: e.g.thetangen tbundle ofanyclosed surface other thanatorus isnontrivial, though we'renotyetinaposition toprovethis. Example 2.12belowshowsanontrivial linebundle thatisfairly easyto understand.Denition 2.10. Asection ofthebundle:E!Misamaps:M!E suchthats=Id M. Thespace ofcontinuoussections onavector bundle:E!Mis denoted by (E)=fs2C(M;E)js=IdMg: Thisisaninnite dimensional vectorspace, withaddition andscalar mul- tiplication dened pointwise. Inparticular, everyvectorbundle hasapre- ferred section 02(E),called thezerosection,whichmaps eachpoint x2Mtothezerovector inEx.Thisgivesanatural embeddingM,!E. Remark 2.11.Aswementioned already inExample 2.2,theterm\section" refers tothegeometric interpretation oftheimages(M)asasubset ofthe total spaceE.Onecanalternately dene acontinuous section ofthe bundle:E!Mtobeanysubset Esuchthatj:!Misa homeomorphism.Example 2.12 (Anontrivial linebundle ).IdentifyS 1withtheunit circle inC,anddene abundle`!S1tobetheunion ofthesetsfeig `eiS1R2forall2R,where thebers`eiarethe1-dimensional subspaces `ei=Rcos(=2) sin(=2) R2: Ifweconsider thesubset f(ei;v)2`j2R,jvj1g consisting onlyofvectors oflength atmost 1,weobtain aMobius strip. Observ ethatthisbundle doesnotadmit anycontinuoussection thatis nowhere zero.ItfollowsthenfromExercise 2.13belowthat`isnotglobally trivializable. Localtrivializations are,however,easytoconstruct: e.g.for anypointp2S1,dene :`jS1nfpg!(S1nfpg)R: ei;c cos(=2) sin(=2) 7!(ei;c): Butthistrivialization canneverbeextended continuously tothepointp. (Spendalittletimeconvincing yourself thatthisistrue.)24 CHAPTER 2.BUNDLES Exercise 2.13. Showthatalinebundle:E!Mistrivial ifandonly ifthere exists acontinuoussections:M!Ethatisnowhere zero. Theexample aboveshowswhyoneoften thinks ofnontrivial vector bundles asbeing\twisted" insome sense|this wordisoften usedinge- ometry andtopology aswellasinphysicstodescrib etheories thatdepend crucially onthenontriviality ofsome bundle. Giventwovector bundlesE!MandF!M,ofrankmand` respectively,alinearbundlemapA:E!Fisanycontinuousmapthat restricts toalinear mapEx!Fxforeachx2M.Abundleisomorphism isahomeomorphism whichisalsoalinear bundle map|in thiscase,the inverseisalinear bundle mapaswell.HenceE!Mistrivializable ifand onlyifitisisomorphic toatrivial bundleMFm!M. There issuchathing asanonline arbundlemapaswell,alsocalled a berpreserving map. Themapf:E!Fiscalled berpreserving iffor allx2M,f(Ex)Fx. Denition 2.14. Let:E!Mbeavectorbundle. Asubbund leofEis asubsetFEsuchthattherestrictionjF:F!Misavector bundle, andtheinclusionF,!Eisacontinuouslinear bundle map. Example 2.15. Thelinebundle ofExample 2.12isasubbundle ofthe trivial 2-plane bundleS1R2!S1. Realvectorbundles canbecharacterized asorientableornon-orien table. Recall thatanorientation forarealm-dimensional vector spaceVisa choice ofanequivalence classofordered bases (v(1);:::;v(m))V; where twosuchbases areequivalentifonecanbedeformed intotheother through acontinuousfamily ofbases. Abasis inthechosen equivalence classiscalled apositively oriente dbasis, while others arecalled negatively oriente d.There arealwaystwochoices oforientation, owingtothefact thatthegeneral linear group GL(m;R)hastwoconnected components, distinguished bythesignofthedeterminan t.3IfVandWareoriented vector spaces, thenanisomorphism A:V!Wiscalled orientation preserving ifitmaps everypositivelyorientedbasis ofVtoapositively orientedbasis ofW;otherwise, itisorientation reversing .Forthespace Rmwithitsnatural orientation givenbythestandard basisofunitvectors (e(1);:::;e(m)),aninvertible matrixA:Rm!Rmisorientation preserving ifandonlyifdetA>0. Thisnotion canbeextended toarealvector bundle:E!M bychoosing orientations oneachberExsothattheorientations \vary continuously withx".More precisely: 3Bycontrast, GL(m;C)isconnected, whichiswhyorientation makesnosense for complex vectorspaces. 2.1.VECTOR BUNDLES 25 Denition 2.16. Anorientation oftherealvectorbundle:E!Misa choiceoforientation foreveryberEx,suchthatforanylocaltrivialization :EjU!URm,theinduced isomorphisms Ex!fxgRmforx2U arealleither orientation preserving ororientation reversing. Noteveryrealbundle admits anorientation; those thatdoarecalled orientable . Exercise 2.17. Showthatareallinebundle isorientable ifandonlyifit istrivializable. Inparticular, thebundle ofExample 2.12isnotorientable. AsmoothmanifoldMiscalled orientable ifitstangen tbundle isori- entable. Thisisequivalenttoamore general notion oforientation which isalsodened fortopologicalmanifolds, evenwithout awelldened tan- gentbundle. Themore general denition isframed interms ofhomology theory; seeforexample [Hat02 ]. Example 2.18. Using again thenontrivial linebundle`!S1ofExam- ple2.12,let =f(ei;v)2`jjvj1g: Thisisawellknownmanifold withboundary: theMobiusstrip.Tosee thatitisnon-orien table, imagine placing yourthumbandindex nger onthesurface toform anorientedbasis ofthetangen tspace atsome point:assume thethumbistherstbasis vector andtheindex nger is thesecond. Nowimagine movingyourhand oncearound, deforming the basiscontinuously asitgoes.When itcomes backtothesame point,you willhaveabasis thatcannot bedeformed backtotheoriginal one|y ou wouldhavetorotate yourhandtorecoveritsoriginal position, pointingin directions nottangen ttoalong theway. Other popular examples ofnon-orien tablesurfaces include theprojective plane andtheKlein bottle,cf.[Spi99]. Denition 2.19. Let:E!Mbeavector bundle ofrankm,and chooseasubsetUMofthebase. Aframe (v(1);:::;v(m)) overUisanordered setofcontinuoussectionsv(j)2(EjU),suchthatfor everyx2Utheset(v(1)(x);:::;v(m)(x))forms abasisofEx. Aglobalframeisaframe overtheentirebaseM;asthenextstatemen t shows,thiscanonlyexistifthebundle istrivializable. Proposition 2.20. Aframe(v(1);:::;v(m))overUMforthebundle :E!Mdetermines atrivialization :EjU!UFm suchthatforeachx2Uandj2f1;:::;mg,(v(j)(x))=(x;e(j)),where (e(1);:::;e(m))isthestandar dbasisofunitvectorsinFm.26 CHAPTER 2.BUNDLES Proof.Onesimply denes (v(j)(x))=(x;e(j))andextends totherestof theberExbylinearit y:thususing thesummation conventiontoexpress arbitrary vectorsX=Xjv(j)2Exinterms ofcomponentsXj2F, (X)=(x;Xje(j))=0B@x;0B@X 1 ... Xm1CA1CA: Theresulting maponEj Uisclearly continuous. Obvious asthisresult seems, it'suseful forconstructing andforvisual- izingtrivializations, sinceframes areinsome waysmore intuitiveobjects. 2.2Smoothness Sofarthisisallpurely topological. Tobring things intotherealm of dieren tialgeometry ,some notion of\dieren tiabilit y"isneeded. Todene this,wenoterstthateveryvector bundle admits asystem oflocaltrivializations ,i.e.asetoftrivializations :EjU!UFmsuch thattheopensetsfUgcoverM.Suchasystem denes asetofcontinuous transition maps g:U\U!GL(m;F); sothateachofthemaps 1 :(U\U)Fm!(U\U)Fmtakes theform (x;v)7!(x;g(x)v): Exercise 2.21. Showthatarealvectorbundle:E!Misorientableif andonlyifitadmits asystem oflocaltrivializations forwhichthetransition mapsg:U\U!GL(m;R)allsatisfy detg(x)>0. Denition 2.22. SupposeMisasmoothmanifold and:E!Misa vector bundle. Asmoothstructur eon:E!Misamaximal system oflocaltrivializations whicharesmoothlycompatible,meaning thatall transition mapsg:U\U!GL(m;F)aresmooth. Thebundle:E!Mtogether withasmoothstructure iscalled a smoothvectorbundle. Thesmoothness ofthetransition mapsg:U\U!GL(m;F)is judged bychoosing localcoordinates onMandtreating GL(m;F)asan opensubset ofthevector space ofF-linear transformations onFm. Remark 2.23.Onecanalsodene vectorbundles ofclassCk,byrequiring thetransition maps tobek-times dieren tiable. Forthistomakesense whenk1,thebaseMmustalsohavethestructure ofaCk-manifold, orbetter. Bythisdenition, allvector bundles arebundles ofclassC0 2.2.SMOOTHNESS 27 (Mneednotevenbeamanifold whenk=0,onlyatopological space). Almost allstatemen tsbelowthatusetheword\smooth"canbeadapted toassume onlynitely manyderivatives. Proposition 2.24. If:E!Misasmoothvectorbundleofrankm anddimM=n,thenthetotalspaceEadmits anaturalsmoothmanifold structur esuchthatisasmoothmap. Thedimension ofEisn+mif F=R,orn+2mifF=C. Exercise 2.25. ProveProposition 2.24.Hint: forx2Uandv2Ex,use thetrivialization andacoordinate chartforsome openneighborhood x2UUtodene acoordinate chartfortheopenneighborhoodv2 EjUE. Thetrivial bundleMRm!Mobviously hasanatural smooth structure ifMisasmoothmanifold. Wecanconstruct asmoothstructure forthetangen tbundleTM!Masfollows.PickanopencoveringM=S Uandacollection ofsmoothcharts':U! Rn.Then dene localtrivializations :TMjU!URnby (X)=(x;d'(x)X); forx2UandX2TxM.Since''1 isalwaysasmoothdieomorphism betweenopensubsets ofRn,thetransition maps g(x)=d(''1 )('(x))2GL(n;R) aresmooth. Exercise 2.26. Verifythatthetensor bundles anddistributions fromEx- amples 2.5,2.6and2.7areallsmoothvector bundles. Inthecontextofsmoothvectorbundles, itmakessense torequire that allsections andbundle maps besmooth.Wethusredene (E)asthe vector space ofallsmoothsections , (E)=fs2C1(M;E)js=IdMg: Notethatthisdenition requires thesmoothmanifold structure ofthetotal spaceE,using Proposition 2.24.Thefollowingexercise demonstrates an alternativ edenition whichisalsoquite revealing, andmakesnoreference tothetotalspace. Exercise 2.27. Showthatthefollowing notion ofasmoothsection is equivalenttothedenition givenabove:asmoothsection isamaps: M!Esuchthatforeverysmoothlocaltrivialization :EjU!UFm, themapsjU:U!UFmtakestheform s(x)=(x;f(x)) forsome smoothmapf:U!Fm.28 CHAPTER 2.BUNDLES Forbundles ofclassCkwithk1,itmakessense tospeakalsoof Cr-sections foranyrk;these canbedened analogously tosmooth sections, byeither ofthetwoapproac hesoutlined above.Thespace of sections ofclassCrisdenoted byCr(E). Similar remarks apply tolinear bundle maps betweendieren tiable vector bundlesE!MandF!Mofrankmand`respectively.In particular, asmoothbundle mapA:E!Fcanbeexpressed inlocal trivializations asamapoftheform (x;v)7!(x;B(x)v) forsome smoothfunctionB(x)withvalues inHom( Fm;F`). Proposition 2.20cannowbeammended tosaythatanysmoothframe (v(1);:::;v(m))overanopensubset orsubmanifoldUMdetermines a smoothtrivialization overU. Wealsohaveanaddendum toDenition 2.14:asubbundleFEis called asmoothsubbund leifjF:F!Madmits asmoothstructure such thattheinclusionF,!Eisasmoothlinear bundle map. Exercise 2.28. IfTMisasmoothdistribution onthemanifoldM, showthat!Misasmoothsubbundle ofTM!M. Exercise 2.29. IfMisasmoothsubmanifold ofRN,showthatTM!M isasmoothsubbundle ofthetrivial bundleMRN!M. Manyofthestatemen tsthatfollowcanbeapplied inobvious waysto either general vectorbundles orsmoothvectorbundles. Since wewillneed toassume smoothness when connections areintroduced inChapter 3,the reader mayaswellassume forsimplicit ythatfromnowonallbundles, sections andbundle maps aresmooth|with theunderstanding thatthis assumption isnotalwaysnecessary . 2.3Operations onvector bundles There aremanywaysinwhichoneortwovectorbundles canbecombined orenhanced tocreate anewbundle. Wenowoutline themostimportant examples; inallcasesitisastraigh tforwardexercise toverifythatthenew objectsadmit (continuousorsmooth)vector bundle structures. Bundles oflinear maps Fortwovector spacesVandW,denote by Hom(V;W) 2.3.OPERA TIONS ONVECTOR BUNDLES 29 thevector space oflinear mapsV!W.Thisnotation assumes thatboth spaces aredened overthesame eldF,inwhichcaseHom(V;W)isalso anF-linear vectorspace. Ifoneorbothspaces arecomplex, onecanalways stilldene aspace ofreallinearmaps Hom R(V;W) byregarding anycomplex space asarealspace oftwicethedimension. NotethatHom R(V;W)isactually acomplex vectorspace ifWiscomplex, sincethescalar multiplication (A)(v)=A(v) thenmakessense for2C,A2Hom R(V;W)andv2V.IfVandWare bothcomplex spaces, wecandistinguish Hom(V;W)fromHom R(V;W) byusing Hom C(V;W)todenote the(complex) space ofcomplex linear maps. Thisnotation willbeusedwhenev erthere ispotentialconfusion. If V=W,weusethenotation End(V)=Hom(V;V); withobvious denitions forEnd R(V)andEnd C(V)whenVisacomplex vector space. Nowfortwovector bundlesE!MandF!Moverthesame base, wedene thebundleoflinearmaps Hom(E;F)!M; whose bersareHom(E;F)x=Hom(Ex;Fx).Thishasanatural smooth bundle structure ifbothE!MandF!Maresmooth.Thesections A2(Hom(E;F))areprecisely thesmoothlinear bundle mapsA:E! F.Wesimilarly denote by End(E)!M thebundle oflinear maps fromEtoitself. Thebundles Hom F(E;F)!M andEnd F(E)forF=RorChavenatural denitions, relating tothe discussion above. WhenVandWarebothcomplex vectorspaces, wecanregard Hom C(V;W) asacomplex subspace ofHom R(V;W);indeed, Hom C(V;W)=fA2Hom R(V;W)jA(iv)=iA(v)forallv2Vg: Another interesting subspace isthespace ofcomplex antiline armaps Hom C(V;W)=fA2Hom R(V;W)jA(iv)=iA(v)forallv2Vg: Thusforcomplex vector bundlesE!MandF!M,wecandene bundles ofcomplex antilinear mapsE!FandE!Erespectively: Hom C(E;F)!M and End C(E)!M: These aresubbundles ofHom R(E;F)!MandEnd R(E)!Mrespec- tively.30 CHAPTER 2.BUNDLES Thedualbundle ThedualspaceVofavector spaceVisHom(V;F),anditselemen tsare called dualvectors.Theextension ofthisconcept tobundles isaspecial caseofthedenition above:wechooseFtobethetrivial linebundle MF!M,anddene thedualbundleofE!Mby E=Hom(E;MF)!M: Foreachx2M,theberE x=Hom(Ex;F)isthedualspace ofEx. Forexample, thedualofatangen tbundleTM!Misthecotangent bundleTM!M,whose sections arethe1-forms onM. Direct sums Thedirectsumofapairofvector spacesVandWisthevector space VW=f(v;w)jv2V;w2Wg withaddition dened by(v;w)+(v0;w0)=(v+v0;w+w0)andscalar multiplication c(v;w)=(cv;cw).Ofcourse, asaset(andasagroup under addition),VWisthesame thing asVW.Thereason forthe redundan tnotation isthatVWgivesrisetoanatural operation ontwo vectorbundlesE!MandF!M,producing anewbundle overMthat really isnottheCartesian productEF.Thedirect sumbundle EF!M isdened tohavebers(EF)x=ExFxforx2M. Exercise 2.30. Fortwocomplex vector bundlesE!MandF!M, showthatHom R(E;F)=Hom C(E;F)Hom C(E;F). It'ssurprisingly complicated tovisualize thetotalspace ofEF:no- tably,it'scertainly notEF.Thelatter consists ofallpairs(v;w)such thatv2Exandw2Fyforanyx;y2M,whereas (v;w)2EFonlyif x=y,i.e.vandwareinbersoverthesamepointofthebase. Onecandene abundle whose totalspace isEF;infact,forthiswe nolonger needassume thatEandFhavethesame base. GivenE!M andF!N,onecandene aCartesian productbundleEF!MN withbers(EF)(x;y)=ExFy.Forourpurposeshowever,thisoperation islessinteresting thanthedirect sum. 2.3.OPERA TIONS ONVECTOR BUNDLES 31 Tensor products Forvector spacesVandWofdimensionmandnrespectively,thetensor productV Wisavectorspace ofdimensionmn.There arevarious ways todene this,details ofwhichweleavetoAppendix A.Fornowletus simply recall themostimportantexample ofatensor productthatarises indieren tialgeometry: Vk `= `O i=1V! kO j=1V! isthenk+`-dimensional vector space ofallmultilinear maps V:::V|{z} `V:::V |{z} k!F: Tensor products extend tobundles intheobvious way:givenE!M andF!M,theirtensor productisthebundle E F!M whose bersare(E F)x=Ex Fx.There ispotentialconfusion hereif bothbundles arecomplex, sowesometimes distinguish thebundles E RF!M andE CF!M: Theformer hasbersthataretensor products ofExandFx,treating both asrealvector spaces. Exercise 2.31. IfVandWarecomplex vectorspaces ofdimensionpand qrespectively,showthatdim C(V CW)=pq,while dim R(V RW)=4pq. Thusthese twospaces cannot beidentied withoneanother. Havingdened dualbundles andtensor products, theideaofthetensor bundlesTk `Monamanifold canbeextended toarbitrary vector bundles bydening Ek `= `O i=1E! kO j=1E! : Theber(Ek `)xforx2Misthenthevectorspace ofallmultilinear maps Ex:::Ex|{z} `E x:::E x|{z} k!F: Alittlemultilinear algebra (seeAppendix A)showsthatthereisacanonical bundle isomorphism Ek `=Hom `i=1E; kj=1E
;32 CHAPTER 2.BUNDLES inparticular, E1 0=E,E0 1=EandE1 1=End(E). Thetensor productbundleE0 k= kj=1Ehasanimportantsubbundle kE!M; whose berskE xarespaces ofalternating k-forms onEx.Onecansimi- larlydene analternating tensor productkEEk 0,consisting ofalternat- ingk-forms onE.Asomewhat more abstract (butalsomathematically cleaner) denition ofkEisgiveninAppendix A. Exercise 2.32. Showthatarealvector bundleE!Mofrankmis orientable ifandonlyifthelinebundle mE!Mistrivializable. And thatthisisequivalenttothelinebundle mE!Mbeingtrivializable. Complexication Justasanycomplex vector bundle canbetreated asarealbundle with twicetherank, anyrealbundle determines arelated complex bundle ofthe same rank. Foranindividual realvector spaceV,thecomplexic ation is dened as VC=V RC; where Cisregarded asa2-dimensional realvector space. Thetensor productisarealvector space withtwicethedimension ofV;indeed, any 2VCcanbewritten asasum =(v1 1)+(v2 i) forsome unique pairofvectorsv1;v22V.More importantly,VCcanbe regarded asacomplex vector space bydening scalar multiplication c(v z):=v (cz) forc2C.Then dim CVC=dim RV,andanyrealbasisofVbecomes also acomplex basisofVC. Thesametrickcanbeuseddocomplexify arealvectorbundleE!M, setting EC=E R(MC); whereMC!Misthetrivial complex linebundle, treated asareal vector bundle ofrank2. Pullbac kbundles Suppose:E!Misasmoothvector bundle andf:N!Misa smoothmap, withNasmoothmanifold. Thebundle onMcanthenbe \pulled back"viaftoasmoothvector bundle f:fE!N; 2.4.VECTOR BUNDLES WITH STRUCTURE 33 withbers (fE)x=Ef(x): Thisisoften called aninducedbundle. Exercise 2.33. VerifythatfE!Nadmits asmoothvector bundle structure ifthemapf:N!Missmooth. Exercise 2.34. ForasmoothmanifoldM,dene thediagonal map
: M!MM:x7!(x;x).Showthatthere isanatural isomorphism
T(MM)=TMTM. Example 2.35(Vector elds along amap).Pullbac kbundles appear naturally inmanysituations, e.g.inExercise 2.34.Foranother example, consider asmooth1-parameter family ofsmoothmaps ft:N!M fort2(1;1).Dieren tiating withrespecttotatt=0givesasmooth map:N!TM, (x)=@ @tft(x) t=0: Clearly(x)2Tf0(x)M,thusisasection ofthepullbac kbundlef 0TM! N.Suchasection isalsoknownasavectoreldalongf0. 2.4Vector bundles withextra structure Inthissection wesurveyvarious typesofstructure thatcanbeadded toa vectorbundle, suchasmetrics, volume elemen tsandsymplectic forms. The general pattern isthateachadditional structure allowsustorestrict our attentiontoasystem oftransition maps thattakevalues insome proper subgroupofthegeneral linear group. Thisobserv ation willbeimportant when werelate these structures toprincipal bundles inSection 2.8. 2.4.1 Complex structures Ashasbeenmentioned already ,acomplex vector bundle canalwaysbe though tofasarealbundle withoneadditional pieceofstructure, namely , adenition of\scalar multiplication byi".Toformalize this,consider rstarealvector spaceV,withdimV=2m.Acomplex structur eon Visalinear mapJ2End(V)suchthatJ2= ,where denotes the identitytransformation onV.NotethatnosuchJcanexistifVhasodd dimension. (Proveit!)Butthere aremanysuchstructures onR2m,for example thematrix J0= 0 m m0 ;34 CHAPTER 2.BUNDLES where mdenotes them-by-midentitymatrix. WecallJ0thestandar d complex structure onR2m,forthefollowingreason: ifR2misidentied withCmviathecorresp ondence (x1;:::;xm;y1;:::;ym) !(x1+iy1;:::;xm+iym); thenmultiplication byionCmisequivalenttomultiplication ontheleft byJ0inR2m. Inthesamemanner, achoice ofcomplex structureJonthespaceVof dimension 2mgivesVthestructure ofanm-dimensional complex vector space, withscalar multiplication dened by (a+ib)v:=(a +bJ)v: Exercise 2.36. Showthateverycomplex structureJonR2missimilar tothestandard structureJ0,i.e.there existsS2GL(2m;R)suchthat SJS1=J0.Hint: construct abasis(v1;:::;vm;w1;:::;wm)ofR2mwith wk=Jvkforeachk. Consider nowarealvector bundleE!Mofrank2m.Acomplex structur eonE!Misdened tobeanysmoothsectionJ:M!End(E) suchthatforallx2M,[J(x)]2=IdEx.Thisdenes eachberExasa complex vector space ofdimensionm. Proposition 2.37. TherealbundleE!Mwithcomplex structur eJ admits thestructur eofasmoothcomplex vectorbundleofrankm. Proof.Itsuces ifforeveryx2M,wecanndaneighborhoodx2U Mandasmoothtrivialization :EjU!UCmwhichiscomplex linear oneachber,i.e.fory2Uandv2Ey, (Jv)=i(v): Thiscanbedonebyconstructing asmoothfamily ofcomplex bases forthe bersinaneighborhoodofx,thenapplying Proposition 2.20.Weleave therestasanexercise|the ideaisessentially tosolveExercise 2.36,not justforasingle matrixJbutforasmoothfamily ofthem. Thisresult canberephrased asfollows:since:E!Misasmooth realvectorbundle, thereisalready amaximal system oflocaltrivializations :EjU!UR2mwithsmoothtransition maps g:U\U!GL(2m;R): Identifying CmwithR2msuchthati=J0,wecanidentifyGL(m;C)with thesubgroup GL(m;C)=fA2GL(2m;R)jAJ0=J0Ag: (2.1) Then Proposition 2.37saysthatafterthrowing outenough ofthetriv- ializations inthesystemfU;g,weobtain asystem whichstillcov- ersMandsuchthatalltransition maps takevalues inthesubgroup GL(m;C)GL(2m;R). 2.4.VECTOR BUNDLES WITH STRUCTURE 35 2.4.2 Bundle metrics Ametric onavectorbundle isachoice ofsmoothly varying inner products onthebers.Tobemoreprecise, if:E!Misasmoothvectorbundle, thenabundle metric isasmoothmap h;i:EE!F suchthattherestriction toeachberExExdenes a(realorcomplex) inner productonEx.Intherealcase,h
;
ijExExistherefore abilinear formwhichissymmetric andpositivedenite. Thisiscalled aEuclide an structur e,making thepair(E;h;i)intoaEuclide anvectorbundle. Inthecomplex case,theformh
;
ijExExispositivedenite andsesquilin- ear,whichmeans thatforallv;w2Exandc2C, (i)hv;wi=hw;vi, (ii)hcv;wi=chv;wiandhv;cwi=chv;wi, (iii)hv1+v2;wi=hv1;wi+hv2;wiandhv;w1+w2i=hv;w1i+hv;w2i, (iv)hv;vi>0ifv6=0. Themetrich;iisthencalled aHermitian structur e,and(E;h;i)isa Hermitian vectorbundle. Proposition 2.38. Every smoothvectorbundle:E!Madmits a bundlemetric. Weleavetheproofasanexercise|the keyisthatsince metrics are positivedenite, anysumofmetrics denes another metric. Thuswecan dene metrics inlocaltrivializations andaddthem together viaapartition ofunity.Thisisastandard argumen tinRiemannian geometry ,see[Spi99] fordetails. Example 2.39(Riemannian manifolds ).Themostimportantapplica- tionofbundle metrics istothetangen tbundleTM!Moverasmooth manifold. Ametric onTMisusually denoted byasmoothsymmetric tensor eldg2(T0 2M),andthepair(M;g)iscalled aRiemannian manifold .Themetricgdetermines anotion oflength forsmoothpaths :[t0;t1]!Mbysetting length ( )=Zt1 t0j_ (t)jgdt; where bydenition,jXjg=p g(X;X).Thestudy ofsmoothmanifolds withmetrics andtheirgeometric implications fallsunder theheading of Riemannian geometry .Itisaverylargesubject|w e'lltouchuponafewof thefundamen talprinciples inourlaterstudy ofconnections andcurvature.36 CHAPTER 2.BUNDLES Example 2.40(Quantumwavefunctions ).Anexample ofaHermi- tianvector bundle comes fromquantummechanics, where theprobabilit y distribution ofasingle particle (without spin) inthree-dimensional space canbemodeled byawavefunction :R3!C: Theprobabilit yofnding theparticle within aregionUR3isgivenby PU( )=Z Uh (x); (x)idx; wherehv;wi:=vwdenes thestandard inner productonC.Though onewouldn't phrase things thiswayinanintroductory classonquantum mechanics, onecanthink of asasection ofthetrivial Hermitian line bundle (R3C;h;i).Thisisnotmerely something amathematician would saytointimidate aphysicist|in quantumeldtheory itbecomes quite importanttoconsider elds assections of(possibly nontrivial) Hermitian vector bundles. Where general vector bundles areconcerned, themostimportantthe- oretical result aboutbundle metrics isthefollowing. Recall thatfortwo inner productspaces (V;h;i1)and(W;h;i2),anisometryA:V!Wis anisomorphism suchthat hAv;Awi2=hv;wi1 forallv;w2V. Proposition 2.41. Let:E!Mbeavectorbundlewithametrich;i. Thenforeveryx2M,thereisanopenneighb orhoodx2UManda trivialization :EjU!UFm suchthattheresulting linearmapsEx!fxgFmareallisometries (with respecttoh;iandthestandar dinnerproductonFm). Proof.Using Proposition 2.20,itsuces toconstruct aframe (v1;:::;vm) onsome neighborhoodofxsothattheresulting bases oneachberare orthonormal. Onecandothisbystarting fromanarbitrary frame and applying theGram-Sc hmidt orthogonalization process. Atrivialization thatdenes isometries onthebersiscalled anor- thogonal trivialization, orunitary inthecomplex case. Asaresult of Proposition 2.41,wecanassume without lossofgeneralit ythatasys- temoflocaltrivializationsfU;gforaEuclidean vector bundle con- sistsonlyoforthogonal trivializations, inwhichcasethetransition maps g:U\U!GL(m;R)actually takevalues intheorthogonal group O(m)=fA2GL(m;R)jATA= g: 2.4.VECTOR BUNDLES WITH STRUCTURE 37 ForaHermitian vectorbundle, instead ofO(m)wehavetheunitary group U(m)=fA2GL(m;C)jAyA= g; where bydenitionAyisthecomplex conjugate ofAT.Taking thisdis- cussion astepfurther, wecould formulate analternativ edenition ofa Euclidean orHermitian structure onavectorbundle asamaximal system oflocaltrivializations suchthatalltransition maps takevaluesinthesub- groups O(m)GL(m;R)orU(m)GL(m;C).It'seasytoseethatsuch asystem alwaysdenes aunique bundle metric. These remarks willbeof fundamen talimportance laterwhen westudy structure groups. 2.4.3 Volume forms andvolume elemen ts OnanyrealvectorspaceVofdimensionm,avolume formcanbedened asanonzero elemen toftheone-dimensional vector space mV,called the \top-dimensional exterior product". Forexample, Rmhasanatural volume form =e(1)^:::^e(m); where (e(1);:::;e(m))isthestandard basisofunitvectors inRm.Achoiceof volume form2mVdetermines anotion of\signed volume" forsubsets ofVasfollows:givenanyorderedm-tuple ofvectors (v1;:::;vm),wedene theorientedparallelopip edspanned bythesevectors tohavesigned volume Vol(v1;:::;vm)2R,where v1^:::^vm=Vol(v1;:::;vm)
: Herewe'reusing thefactthatdimmV=1and6=0todeduce that there alwaysissuchanumber.Ifyou'renotusedtothinking aboutwedge products ofvectors butarefamiliar withdieren tialforms, thenyoucan think ofvolume forms alternativ elyasfollows:mVisalso1-dimensional, thusthechoice2mVdenes aunique alternating m-form!2mV suchthat !(v1;:::;vm)=Vol(v1;:::;vm): Avolume formistherefore equivalenttoachoice ofanonzero alternating m-form!onV,andalmost allstatemen tsinthefollowingabout2mV canberephrased asstatemen tsabout!2mV. Exercise 2.42. (a)Showthatif:E!Misany(realorcomplex) linebundle, thena choice ofsmoothnowhere zerosections2(E)determines abundle isomorphism E=Einanatural way.38 CHAPTER 2.BUNDLES (b)Showthatforanyvector bundle:E!Mofrankmandany integerkm,there isacanonical bundle isomorphism k(E)= (kE). (c)Conclude, asinthediscussion above,thatachoiceofvolume formfor arealbundle:E!Mofrankmdenes naturally anisomorphism mE=m(E),andthusacorresp onding volume formforthedual bundleE!M. Youshould takeamomen ttoconvince yourself thatthisdenes some- thing consisten twiththeintuitivenotion ofvolume: inparticular, multi- plying anyofthevectorsvjbyapositivenumberchanges thevolume by thesamefactor, andanylinear dependence among thevectors (v1;:::;vm) produces a\degenerate" parallelopip ed,withzerovolume. Thestandard volume formonRmisdened sothattheunitcubehasvolume one.Some- what lessintuitivelyperhaps, Vol(v1;:::;vm)isnotalwayspositive:it changes signifwereversetheorder ofanytwovectors, sincethischanges theorientation oftheparallelopip ed.Thusonemustgenerally takethe absolute valuejVol(v1;:::;vm)jtondthegeometric volume. These remarks extend easily toacomplex vectorspace ofdimensionm, though inthiscasevolume canbeacomplex number|w ewillforego any attempt tointerpret thisgeometrically .Ineither therealorcomplex case, alinear isomorphism A:(V;1)!(W;2) betweenm-dimensional vector spaces withvolume forms issaidtobevol- umepreserving ifeveryparallelopip edPVhasthesame (signed or complex) volume asA(P)W.Forexample, amatrixA2GL(m;F)is volume preserving withrespecttothenatural volume formonFmifand onlyifdetA=1. Exercise 2.43. Alinear mapA:V!Wdenes acorresp onding linear map A:mV!mW; sendingv1^:::^vmtoAv1^:::^Avm,aswellasapullbac k A:mW!mV; suchthatA!(v1;:::;vm)=!(Av1;:::;Avm).ShowthatA:(V;1)! (W;2)isvolume preserving ifandonlyifA1=2.Equiv alently,den- ingvolume viam-forms!12mVand!22mW,Aisvolume preserv- ingifandonlyifA!2=!1. Foranyvector bundle:E!Mofrankm,thetop-dimensional exterior bundles mE!MandmE!Marelinebundles. Inthe 2.4.VECTOR BUNDLES WITH STRUCTURE 39 realcase,wesawinExercise 2.32thatthese linebundles aretrivial ifand onlyifE!Misorientable, inwhichcasethere existsmoothsections :M!mEand!:M!mEthatarenowher ezero.Sucha section isveryfarfromunique, since wecanalwaysobtain another via multiplication withasmoothpositivefunctionf:M!(0;1). Weshallcallanowhere zerosection2(mE)or!2(mE)a volume formonE,because itdenes asmoothly varying notion ofsigned volume Vol(
;:::;
)oneachber.Thisshould bemostly familiar from thespecialcasewhereEisthetangen tbundle ofasmoothn-manifold M;thenavolume formisdened tobeanowhere zerodieren tialn-form (cf.[Spi65]),i.e.asection ofnTM. Eventhough orientation doesn'tmakesense inthecomplex case,the aboveremarks canstillbeextended almost verbatim toacomplex bundle :E!MofrankmifmEistrivial. Avolume formisagain dened asa smoothnowhere zerosection2(mE),anditnowdenes a\complex volume" Vol(v1;:::;vm)foranyorderedm-tuple ofvectors inthesame ber. Proposition 2.44. Let:E!Mbeavectorbundleofrankmwith avolume form2(mE).Then foreveryx2M,thereisanopen neighb orhoodx2UMandatrivialization :EjU!UFm suchthattheresulting linearmapsEx!fxgFmareallvolume preserv- ing.Exercise 2.45. ProveProposition 2.44.Hint: givenanyframe (v1;:::;vm) overasubsetUM,onecanmultiplyv1byasmoothfunction suchthat Vol(v1;:::;vm)1. Inparticular, amaximal system oflocaltrivializationsfU;gcan bereduced toasystem ofvolume preserving trivializations, forwhichall transition maps takevalues inthespeciallineargroup SL(m;F)=fA2GL(m;F)jdetA=1g: AEuclidean structureh;ionanorientedrealbundle:E!M determines aunique preferred volume formsuchthattheorientedcube spanned byanypositivelyorientedorthonormal basisofExhasvolume one. Thatthisiswelldened followsfromExercise 2.46below.Noworientation preserving orthogonal transformations arealsovolume preserving, sowe cancombinePropositions 2.41and2.44toassume thatalltransition maps takevalues inthespecialorthogonalgroup SO(m)=O(m)\SL(m;R):40 CHAPTER 2.BUNDLES NotethatSO(m)canalternativ elybedened asthesubgroup SO(m)=fA2O(m)jdetA=1g: Thisisequivalenttothefactthateverytransformation ofRmthatpreserv es boththeinner productandtheorientation alsopreserv essigned volume, i.e.ametric andorientation together determine apreferred volume form. Exercise 2.46. Consider Fmwithitsstandard inner product, anddenote thestandard unitbasisvectors bye(1);:::;e(m). (a)Showthatif(v(1);:::;v(m))isanyother ordered basisofFmthen v(1)^:::^v(m)=det(V)
(e(1)^:::^e(m)) whereV2GL(m;F)isthematrix whosejthcolumn isv(j)2Fm. Hint: sincemFmis1-dimensional, theexpression (v(1);:::;v(m))7!v(1)^:::^v(m) e(1)^:::^e(m) denes anantisymmetric m-linear formonFm.Howmanysuchforms arethere? (b)IfF=RandwegiveRmthestandard orientation, showthatfor everypositivelyorientedorthonormal basis(v(1);:::;v(m)), v(1)^:::^v(m)=e(1)^:::^e(m): (c)Conclude thateverym-dimensional realvector space withanori- entation andaninner productadmits avolume formforwhichall positivelyorientedunitcubeshavesigned volume one. Thesituation foracomplex bundle:E!MwithHermitian struc- tureh;iisslightlydieren t.Nowavolume form2(mE)canbecalled compatible withh;iifeveryberhasanorthonormal basiswithvolume one.Notice howeverthatifVol(v1;:::;vm)=1,then(eiv1;v2;:::;vm)is another orthonormal basis andVol(eiv1;v2;:::;vm)=ei,sowecannot require thatallorthonormal bases havevolume one.Relatedly ,there isno welldened notion of\orien tation" thatcould helpuschooseacompatible volume formuniquely; givenanycompatible form,another suchformis dened byfiff:M!Cisanysmoothfunction withjfj=1. Givenh;iandacompatible volume form,aneasyvariation on Proposition 2.44showsthatwecanassume alltransition maps takevalues inthespecialunitary group SU(m)=U(m)\SL(m;C): 2.4.VECTOR BUNDLES WITH STRUCTURE 41 Thenonuniqueness ofthevolume formisrelated tothefactthatSU(m) isasubmanifold ofonedimension lessinU(m);bycontrast, SO(m)and O(m)havethesame dimension, theformer beingaconnected component ofthelatter. Anon-orien table realbundle:E!Mofrankmdoesnotadmit anyvolume form, butonecanstilldene asmoothly varying unsigned volume onthebers.Thisismost easily seeninthecasewhere there is abundle metrich;i:thenitisnatural tosaythatallorthonormal bases spannon-orien tedcubesofvolume one,anddene apositive volume for allother bases andparallelopip edsaccordingly .Theresulting structure iscalled avolume element onthebundle:E!M.Inparticular, everyRiemannian manifold (M;g),orientableornot,hasanatural volume elemen tonitstangen tbundle, whichonecanusetodene integration of real-valued functions onM. More generally ,onecandene avolume element onanyrealbundle :E!Mofrankmtobeachoice oftwononzero volume formsx2 mExoneachberEx,whichvarysmoothly inthefollowingsense: every x02Miscontained inanopenneighborhoodUMwhichadmits a smoothvolume formU:U!mEsuchthat x=U(x) forallx2U.Theexistence ofvolume elemen tsingeneral followsfromthe existence ofbundle metrics, withalittlehelpfromExercise 2.46.Then thevolume ofparallelopip edsisdened by v1^:::^vm=Vol(v1;:::;vm)
(x) foranyunordered basisfv1;:::;vmgofEx,withtheplusorminussign chosen sothatvolume ispositive. Exercise 2.47. Showthateveryrealbundle withavolume elemen tadmits asystem oflocaltrivializations suchthatthetransition maps takevalues inthegroup fA2GL(m;R)jdetA=1g: Example 2.48(Volume elemen tsincoordinates ).It'suseful tohave explicit coordinate expressions forvolume elemen tsonRiemannian mani- folds, particularly ifonewantstounderstand thephysicsliterature, where coordinates areusedinpreference togeometrically invariantobjects.4We beginbynding anexpression forthevolume formonageneral realbundle ofrankmwithmetrich;i,withrespecttoalocalframe (e(1);:::;e(m)) 4Forinstance, physicists tendtodene atensor purely interms ofitscoordinate expression, stipulating thatitmust\transform properly" withrespecttocoordinate changes.42 CHAPTER 2.BUNDLES onsome opensetUM.Since thequestion ispurely local,weloseno generalit yinassuming thatEjU!Uisanorientedbundle. Theframe (e(1);:::;e(m))isthenassumed tobepositivelyoriented,butisotherwise arbitrary; inparticular, wedonotrequire ittobeorthonormal. Themetric nowdenes arealm-by-msymmetric matrixGwithentries Gij=he(i);e(j)i; andusing theframe toexpress vectorsv=vie(i)2Exforx2U,wehave hv;wi=Gijviwj: Nowchooseanorthonormal frame (^e(1);:::;^e(m));thecorresp onding ma- trixbGisthentheidentity,withbGij=ij.Wecanusethisnewframe to identifyEjU!Uwiththetrivial Euclidean vector bundleURm!U, thusconsidering theother frame (e(1);:::;e(m))toconsist ofsmoothRm- valued functions onU.Thevectors (^e(1);:::;^e(m))arenowsimply the standard unitbasisvectors ofRm.ByExercise 2.46,wehave e(1)^:::^e(m)=det(V)^e(1)^:::^^e(m)=det(V) (2.2) whereisthenatural volume formdetermined bythemetric andorienta- tion,andVisthematrix whose columns arethevectorse(1);:::;e(m)2Rm. Wecanrelate thematricesVandGasfollows.Computing intheorthonor- malcoordinates, Gij=he(i);e(j)i=bGk`Vk iV` j=(VT)k ik`V` i=(VT V)ij=(VTV)ij; thusG=VTV,anddetG=(detV)2.Combining thiswith(2.2)givesa formulaforthevolume forminterms ofthemetric andtheframe, =1p detGe(1)^:::^e(m): (2.3) Thisassumes there isanorientation and(e(1);:::;e(m))ispositivelyori- ented,butitclearly applies tothenon-orien tedcaseaswellifweinsert signsintheappropriate places. Letusapply thisformulatothecaseofaRiemannian manifold (M;g) withcoordinates (x1;:::;xn)onsome neighborhoodUM.Thecoordi- nates determine aframing (@1;:::;@n)ofTMoverU,andoneexpresses themetric inthese coordinates bythen-by-nsymmetric matrix gij=g(@i;@j); sothatiftangen tvectors arewrittenX=Xi@i,theng(X;Y)=gijXiYj. Avolume elemen ton(M;g)isactually avolume elemen tonthecotangent 2.4.VECTOR BUNDLES WITH STRUCTURE 43 bundleTM,andwewanttoexpress thenatural volume elemen tdeter- mined byginterms oftheframedx1^:::^dxnonnTMjU.Assign toUtheorientation determined bythecoordinates, andletgdenote the corresp onding volume formonTMjU!U.By(2.3)wehave g=1pdetg@1^:::^@n: Thisisrelated tothedieren tialn-form Vol(
;:::;
)by X1^:::^Xn=Vol(X1;:::;Xn)1pdetg@1^:::^@n; fromwhichwecompute Vol(X1;:::;Xn)=p detgX1^:::^Xn @1^:::^@n: Thefraction ontherightdenes anantisymmetric n-linear formonthe vectors (X1;:::;Xn),whichweseeisidenticaltodx1^:::^dxn.Weare thusleadtotheformula Vol=p detgdx1^:::^dxn: (2.4) Again, thisassumes thatUMhasanorientation andthecoordinates are positivelyoriented,butwecaneasily removethisassumption. Theproper expression foravolume element ,expressed inarbitrary coordinates, isthen dvolg=p detgdx1:::dxn; where thewedges havebeenremovedtoindicate thatwearenolonger keeping trackofsigns. Thisshowshowtouselocalcoordinate patches forcomputing integralsR Mfdvolgofsmoothfunctions onRiemannian manifolds. 2.4.4 Indenite metrics Itissometimes useful toconsider \metrics" thatarenotpositivedenite. Onarealvector bundleE!M,onecandene anindenite metric to beasmoothmaph;i:EE!Rwhichrestricts oneachbertoa bilinear formthatissymmetric andnondegenerate;thelatter condition means there isnononzerov2Exwithhv;wi=0forallw2Ex.An example ofsuchabilinear formonR4istheMinkowski metric ,dened by hv;wi=vTw,whereisthesymmetric matrix =0BB@1000 0100 0010 00011CCA:44 CHAPTER 2.BUNDLES Aformofthistype,havingthree negativ eeigenvalues andonepositive,is saidtohaveLorentzsignature|these areimportantinEinstein's theory ofrelativit y.Infact,general relativit ydeals exclusiv elywithsmooth4- manifolds thathaveLorentzian metrics.5 Givenarealm-dimensional vectorspaceVwithanindenite (i.e.sym- metric andnondegenerate) inner producth;i,theindex ofh;iisthe largest dimension ofanysubspaceWVonwhichh;ijWisnegativ e denite. Thustheindex isatmostm,andiszeroifandonlyiftheinner productispositivedenite. Choosing abasis toidentifyVwith Rm,the symmetric formh;icanbewritten as hv;wi=vTw forsomesymmetric matrix .Then byastandard result inlinear algebra, =STSforsomeS2O(m)andadiagonal matrix ,thuswecan change thebasisandassume isdiagonal without lossofgeneralit y.Bythe nondegeneracy condition, mustbeinvertible, soallitsdiagonal entries arenonzero. Wecanthenrescale thebasisvectors andassume nally that takestheform m;k:= mk k forsomek2f0;:::;mg,where kdenotes thek-by-kidentitymatrix. Clearlykistheindex ofh;i. Similar remarks apply toanysmoothindenite metrich;ionareal vector bundle:E!Mofrankm.OnanyberEx,thediagonaliza- tionprocedure outlined aboveimplies thatthere isanorthonormal basis (v1;:::;vm)ofEx,whichmeans inthiscasehvi;vji=ij. Exercise 2.49. Givenanorthonormal basisontheberEx,showthatit canbeextended toasmoothorthonormal frame oversomeopenneighbor- hoodofx.Hint: Gram-Sc hmidt! Wementiontwoimportantconsequences ofExercise 2.49: 1.Theindex ofh;iislocallyconstan t|itisthusuniquely dened over everyconnected componentofthebaseM. 2.Theorthonormal frames giverisetoasystem oflocaltrivializations whichareorthogonal inageneralized sense. Wecanthenassume thatthetransition maps takevalues inthesubgroup O(mk;k)=fA2GL(m;R)jATm;kA=m;kg: 5Thephysical signicance ofLoren tzsignature isthatspace-time isa4-manifold inwhichoneofthedimensions isnotliketheother three. Objectswithmass are allowedtomovethrough space-time onlyalong so-called time-like paths x(t),which satisfyh_x(t);_x(t)i>0.Thismeans thatinanycoordinate system, themotion dened byx(t)isslowerthanacertain limiting speed|the speedoflight!Onecanalsomakea distinction betweenspace-time directions thatpoint\forward"or\backward"intime. 2.4.VECTOR BUNDLES WITH STRUCTURE 45 InthecasewhereE!Misoriented,thiscanfurther bereduced to SO(mk;k)=O(mk;k)\SL(m;R): Itshould benoted thatmorerened notions oforientation arepossible ifkandmkarebothnonzero. Forinstance, Minkowskispace (R4;4;3) canbegivenseparate orientations forthetime-lik edirections andspace- likehypersurfaces, andtheassociated group O(1;3)(also knownasthe fullLorentzgroup)hasfourcomponentsrather thantwo.Inparticular, thecomponentsofO(1;3)aredistinguished bywhether their elemen ts preserv eorreversethedirection oftimeandtheorientation ofspace. The twocomponentsthateither preserv eorreversebothmakeupSO(1;3),and thesubgroup SO(1;3)+SO(1;3)thatpreserv esbothiscalled theproper orthochronous Lorentzgroup.(Seeforexample [SU01].) Formetrics onthetangen tbundle ofasmoothmanifold, most ofthe basic theory ofRiemannian geometry (connections, curvature, geodesics etc.) applies justaswelltotheindenite case. Ontheother hand the existence result, Proposition 2.38,doesnotapply tononpositivemetrics ingeneral. Forexample onecanusecovering spaces toshowthatS2does notadmit anymetric ofindex 1(see[Spi99,Problem 9.7]). Exercise 2.50. Adapt theargumen tsofExample 2.48towrite downthe natural volume elemen tdetermined byanindenite metric. Onething to bewareofhereisthatdetgmaybenegativ e;butthiscanbexed by inserting theappropriate signchanges. 2.4.5 Symplectic structures Onarealvector spaceVofevendimension 2m,asymplectic structure is anantisymmetric bilinear form !:VV!R whichisnondegenerate,inthesense that!(v;w)=0forallw2Vonly ifv=0.Note thatbyasimple argumen tinlinear algebra, there isno symplectic structure onanodd-dimensional vector space (Proveit!).The standard symplectic structure onR2misdened as !0(v;w)=vTJ0w; whereJ0isthestandard complex structure. Fortwosymplectic vector spaces (V;!1)and(W;!2),alinear mapA:V!Wiscalled symple cticif !2(Av;Aw)=!1(v;w)46 CHAPTER 2.BUNDLES forallv;w2V.Byconstructing anappropriate basis(asymple cticbasis), onecanshowthateverysymplectic vectorspace (V;!)ofdimension 2mis symplectically isomorphic to(R2m;!0).Wedenote bySp(m)GL(2m;R) thesubgroup consisting ofinvertible symplectic linear maps. Onecannowdene asymple cticstructur eonavector bundle:E! Mofrank2masasmoothfamily ofsymplectic structures onthebers, i.e.asmoothmap !:EE!R thatdenes asymplectic structure oneachber.Thepair(E;!)isthen called asymple cticvectorbundle,andwehavethefollowing statemen t aboutlocaltrivializations: Proposition 2.51. If(E;!)isasymple cticvectorbundleoverM,thenfor everyx2M,thereisanopenneighb orhoodx2UMandasymplectic trivialization, i.e.atrivialization :EjU!UR2m suchthattheresulting linearmapsEx!fxgR2mareallsymple ctic (withrespectto!andthestandar dsymple cticstructur e!0onR2m). Theprooffollowstheusual recipe:oneneeds toconstruct asymple ctic frameinaneighborhoodofx,whichisnomore dicult inprinciple than constructing asymplectic basis onasingle vector space. Weleavethe details asanexercise. Adapting theremarks followingProposition 2.41tothissetting, asym- plectic vectorbundle canalternativ elybedened asabundle withasystem oflocaltrivializations forwhichthetransition maps takevaluesinSp(m). Notethateverysymplectic bundle (E;!)ofrankminherits anatural volume form, dened onthedualbundleEby !m=!^:::^!: Thatthisisnonzero followsfromthenondegeneracy of!.Thussymplectic bundles havenatural orientations andvolume elemen ts;thisisrelated to thefactthateverysymplectic matrixA2Sp(m)hasdeterminan tone, i.e.Sp(m)SL(2m;R). Example 2.52(Hermitian bundles assymplectic bundles ).AHer- mitian bundle (E;h;i)ofrankmcanberegarded asarealbundle ofrank 2m,withacomplex structureJandacomplex-v alued bilinear form hv;wi=g(v;w)+i!(v;w); wheregand!arerealbilinear forms. Itiseasytocheckthatgdenes aEuclidean structure ontherealbundleE,while!denes asymplectic 2.4.VECTOR BUNDLES WITH STRUCTURE 47 structure. Thusaunitary trivialization onsomesubsetUMisalsoboth anorthogonal trivialization andasymplectic trivialization. Thisisrelated totheconvenientfactthatifGL(m;C)isregarded asthesubgroup of matrices inGL(2m;R)thatcomm utewiththestandard complex structure J0,then U(m)=O(2m)\Sp(m): Conversely,everysymplectic bundle (E;!)canbegivenacompatible complex structureJ,suchthatg:=!(
;J
)denes aEuclidean metric andh
;
i:=g+i!becomes aHermitian metric. (See[HZ94]foraproof ofthisfact.) Thusthere isaclosecorresp ondence betweensymplectic and complex bundles. Example 2.53(Symplectic manifolds ).JustasinRiemannian geome- try,themostimportantexamples ofsymplectic vectorbundles aretangen t bundles, though hereitisappropriate toaddoneextra condition. If!isa symplectic structure onTMforsomesmooth2n-dimensional manifoldM, thenthepair(M;!)iscalled analmost symple cticmanifold .Thecondi- tionforremovingtheword\almost" comes fromobserving that!isnowa dieren tial2-form onM;wecallitasymple cticform,andthepair(M;!) asymple cticmanifold ,if!satises the\integrabilit ycondition" d!=0: Theimportance ofthiscondition lieslargely inthefollowingresult, which showsthatallsymplectic manifolds arelocallythesame. Proposition 2.54(Darb oux's theorem) .If(M;!)isasymple cticmani- foldandx2M,thensomeneighb orhoodofxadmits acoordinate system (q1;:::;qn;p1;:::;pn)inwhich!takesthecanonic alform !=nX k=1dpk^dqk: See[Arn89 ]or[HZ94]foraproof.There isnosuchstatemen tforRie- mannian manifolds, where thecurvatur edened byametricgprovides an obstruction fordieren tmanifolds tobelocallyisometric. Thussymplectic geometry isquiteadieren tsubject,andalldierences betweensymplectic manifolds manifest themselv esontheglobalrather thanlocallevel|this isthesubjectoftheeldknownassymple ctictopology. Some motivation forthestudy ofsymplectic manifolds comes fromthe factthattheyarethenatural geometric setting forstudying Hamiltonian systems. Givenasymplectic manifold (M;!)andasmoothfunctionH: M!R,there isaunique smoothvectoreld,theHamiltonian vectoreld XH2Vec(M),dened bythecondition !(XH;
)=dH:48 CHAPTER 2.BUNDLES Onethenconsiders solutionsx:R!Mtothedieren tialequation _x(t)=XH(x(t)).Thismayseemrather arbitrary atrst,butanyphysicist willrecognize theformthat_x=XH(x)takesinthespecialcoordinates provided byDarboux's theorem: writing x(t)=(q1(t);:::;qn(t);p1(t);:::;pn(t)); wendthatfork2f1;:::;ng, _qk=@H @pkand _pk=@H @qk: These areHamilton's equations fromclassical mechanics, whichdetermine themotion ofasystem withndegrees offreedom inphase space. Thusthe symplectic form!isprecisely thestructure needed fordening Hamilton's equations inageometrically invariantwayonM. WerefertothebookbyArnold [Arn89 ]forfurther details. 2.4.6 Arbitrary G-structures Onething alloftheexamples abovehaveincommon isthateachstructure denes asystem ofpreferred localtrivializations (e.g.theorthogonal, or symplectic trivializations), withtransition maps thattakevalues insome propersubgroupGGL(m;F).Wenowformalize thisidea. Denition 2.55. Let:E!Mbeavector bundle ofrankm,and supposeGisasubgroup ofGL(m;F).AG-structur eonEisamaximal system ofsmoothlocaltrivializations :EjU!UFmwhichcover MandhavesmoothG-valued transition maps: g:U\U!G: Thetrivializations (U;)arecalledG-compatible. Thisdenition israther hardtouseinpractice, because oneseldom wantstoconstruct asystem oflocaltrivializations explicitly .Butwhat itlacksinpracticalit y,itmakesupforinstriking generalit y:allofthe structures we'veconsidered sofararespecialcases ofG-structures. On arealvector bundle forinstance, bundle metrics areequivalenttoO(m)- structures, volume forms areSL(m;R)-structures andsymplectic struc- turesareSp(m)-structures. These particular examples canallbeexpressed interms ofsome chosen tensor eldonE,suchasametric orsymplec- ticform|but ingeneral onecandeneG-structures thathavenothing directly todowithanytensor eld. Indeed, thenotion makessense for anysubgroupGGL(m;F).Oneshould ofcourse beawarethatgivenG andabundleE,itcannot beassumed thatG-structures alwaysexist|this 2.5.INFINITE DIMENSIONAL BUNDLES 49 isn'teventrueformostoftheexamples above,e.g.avolume formexists onlyifEisorientable. Wecangeneralize onestepfurther byconsidering notjustsubgroups of GL(m;F)butarbitrary groups thatadmitm-dimensional representations . Bydenition, anm-dimensional (realorcomplex) represen tation ofGisa group homomorphism :G!GL(m;F); thusassociating witheachelemen tg2Ganm-by-mmatrix(g).The represen tation iscalled faithful ifisaninjectiv ehomomorphism, inwhich caseitidentiesGwithasubgroup ofGL(m;F).More generally ,we cannowmakesense ofthetermG-valuedtransition mapwhenev eran m-dimensional represen tation ofGisgiven,sinceg2Gthenactsonvec- torsv2Fmviathematrix(g).Therepresen tation neednotbefaithful, andGneednotbeasubgroup ofGL(m;F). 2.5Innite dimensional vector bundles With alittlecare,thetheory ofsmoothmanifolds andvector bundles can beextended toinnite dimensional settings, withquite powerfulconse- quences inthestudy ofdieren tialequations. Wewon'tgoverydeeply intothissubjectbecause theanalysis israther involved|but wecanat leastpresen tthemain ideas, andanexample thatshould besucien t motivation forfurther study. Innite dimensional spaces aretypically function spaces, e.g.C1(M;Rn), thespace ofallsmoothmaps fromasmoothcompact manifoldMtoRn. Oneofthemostremark ablefactsaboutdieren tialcalculus isthatmostof itcanbeextended inanelegan twaytoinnite dimensional vectorspaces| however,thespaces forwhichthisworksarenotarbitrary .Wearemost interested inthefollowingspecialclassofobjects. Denition 2.56. ABanach spaceisavector spaceXwithanormkk suchthateveryCauchysequence converges. Recall thataCauchysequencexk2Xisanysequence forwhichthe distanceskxkxjkapproac hzeroasbothkandjgo(indep enden tly) toinnit y.Thenontrivial aspectofaBanac hspace isthatgivenany suchsequence, there exists anelemen tx2Xsuchthatkxxkk!0, i.e.xk!x.Metric spaces withthispropertyarecalled complete . Therequiremen tseems likenothing atrstglance because, infact, allnite dimensional normed vector spaces (realorcomplex) areBanac h spaces. ThespacesCk(M;Rn)withk<1arealsoBanac hspaces when givensuitable norms, e.g.thestandard norm onC1([0;1];Rn)is kfk=sup x2[0;1]jf(x)j+sup x2[0;1]jf0(x)j:50 CHAPTER 2.BUNDLES Similarly ,onanysmoothvector bundle:E!Moveracompact base M,thespacesCk(E)ofk-times dieren tiable sections areBanac hspaces.6 Ontheother hand, spaces consisting ofsmoothmaps, e.g.C1(M;Rn) and(E),generally arenot.Thispresen tsabitofatechnical annoyance, because onewouldgenerally prefer toworkonlywithspaces ofsmooth maps. We'lluseastandard trick(Exercise 2.64)togetaround thisproblem inourexample below. Onething tobewareofinBanac hspaces isthatnotalllinear maps arecontinuous. FortwoBanac hspaces XandY,wedenote thespace of continuous linear mapsX!YbyL(X;Y),withL(X):=L(X;X).An invertible mapA:X!Yisnowcalled anisomorphism ifbothAand A1arecontinuous. ThespaceL(X;Y)admits anatural norm, dened by kAkL(X;Y)=sup x2Xnf0gkAxkY kxkX; andL(X;Y)isthenalsoaBanac hspace. Wenowsummarize somefactsaboutcalculus inBanac hspaces. Thisis nottheplace toexplore thesubjectinanydetail, buttheinterested reader mayconsult thebookbyLang [Lan93 ]tollinthegaps. Tostartwith, thefollowingdenition should lookveryfamiliar. Denition 2.57. LetXandYberealBanac hspaces, withUXan opensubset. Then afunction F:U!Yisdierentiable atx2Uif there exists acontinuouslinear mapdF(x)2L(X;Y)suchthatforall sucien tlysmallh2X, F(x+h)=F(x)+dF(x)h+khkX
(h); where isamapfromaneighborhoodof02XintoYwithlimh!0(h)= 0.Thelinear operator dF(x)iscalled thederivative (oralsolinearization ) ofFatx. ThemapF:U!Yiscontinuously dierentiable onUifdF(x)exists forallx2Uanddenes acontinuousfunction dF:U!L(X;Y). NotethatFisautomatically continuousatx2Uifitisdieren tiable there. Thedenition canbeapplied tocomplex Banac hspaces aswellby treating them asrealBanac hspaces. Thederivativeisthenrequired tobe onlyareallinear map.7 6Technically ,oneshould notcalltheseBanac hspaces butrather Banachable ,because although theydobecome Banac hspaces when givensuitable norms, these norms are notcanonical. Onemustthenshowthatallsuitable choices ofnorms areequivalent , meaning thattheyalldene thesame notion ofconvergence. 7Amapwithacomplex linear derivativeiscalled complex analytic |the study of these isadieren tsubjectaltogether. 2.5.INFINITE DIMENSIONAL BUNDLES 51 ThederivativeofamapfromanopensetintoaBanac hspace hasnow beendened asanother mapfromanopensetintoaBanac hspace. Thus theprocesscanbeiterated, dening higher derivativesandthenotion of smoothmapsU!Y.Standard features ofdieren tialcalculus suchas thesumrule,productruleandchainruleallhaveelegan tgeneralizations tothissetting. Thedenitions sofarmakesense forallnormed vector spaces, not justBanac hspaces. Ontheother hand, thecompleteness condition plays acrucial roleinprovingthefollowinggeneralization ofastandard result fromadvanced calculus. Theorem 2.58 (InverseFunction Theorem) .LetXandYbeBanach spaces,UXanopensubset, andF:U!Yasmoothmapsuchthat forsomex2UwithF(x)=y2Y,thederivative dF(x):X!Y isanisomorphism. Thenthereareopenneighb orhoodsx2UUand y2VYsuchthattherestriction ofFtoUisaninvertible mapU!V. Itsinverse F1isalsosmooth,and d(F1)(y)=(dF(x))1:Y!X: There issimilarly aninnite dimensional version oftheimplicit func- tiontheorem, whichcanbetreated asacorollary oftheinversefunction theorem. Wereferto[Lan93 ]foraprecise statemen t,andtheproofsof bothresults. Wecannowdene smoothmanifolds andvector bundles modeled on Banac hspaces. Herewewillpresen tonlyabriefdiscussion ofthemain denitions|the recommended reference forthismaterial isLang's geome- trybook[Lan99 ]. Denition 2.59. AsmoothBanach manifold isaHausdor topological space MwithanopencoveringM=S Uandasystem ofcharts ':U! ; where each isanopensubset ofsome Banac hspaceX,andthetran- sition maps ''1 aresmoothwherev ertheyaredened. Denition 2.60. GivenasmoothBanac hmanifold M,aBanach space bundle:E!MoverMisasmoothBanac hmanifold Ewithasmooth surjectiv emap,suchthateachberEx=1(x)forx2MisaBanac h space, andthere isasystem oflocaltrivializationsf(U;)g,where the opensetsfUgcoverM,andfgisasetofhomeomorphisms :1(U)!UY:52 CHAPTER 2.BUNDLES HereYisaBanac hspace, andtherestriction oftoeachberisa Banac hspace isomorphism. Furthermore, these trivializations mustbe smoothlycompatible,inthatfor6=,1 takestheform (x;v)7!(x;g(x)v) wherev eritisdened, andthetransition maps g:U\U!L(Y;Y) areallsmooth. Thislastpointaboutthetransition mapsgcontainssomeveryserious analysis intheinnite dimensional case,foritisastronger condition than justrequiring that (U\U)Y!Y:(x;v)7!g(x)v beasmoothmap. Unlikeinnite dimensional calculus, where maps are usually presumed smoothuntilprovennonsmo oth,ittypically takescon- siderable eort toprovethatamapbetweenBanac hspaces issmooth. Thustheconstruction ofasmoothBanac hmanifold orbundle structure onanygivenobjectcanbearather involvedprocess,evenifonecansee intuitivelythatcertain spaces should beBanac hmanifolds. Averygeneral procedure forthese constructions onso-called \manifolds ofmaps" was presen tedinapaperbyH.Eliasson [El67],whose ideascanbeapplied to theexample below. Example 2.61 (Stabilit yofperiodicorbits ).Consider asmoothn- dimensional manifoldMwithasmoothtime-dep enden tvector eldXt whichis1-periodicintime, i.e.Xtisasmoothfamily ofsmoothvector elds suchthatXt+1Xt.Wewishtoconsider theorbitsx:R!Mof thedynamical system _x(t)=Xt(x(t)); (2.5) andparticularly those thatare1-periodic,satisfyingx(t+1)=x(t).Such asolution canberegarded asamapx:S1!M,where thecircleS1is dened asR=Z.Denote byt:M!Mthe owofthissystem fromtime 0totimet,i.e.tisasmoothfamily ofdieomorphisms suchthatforany x02M,x(t):=t(x0)istheunique orbitof(2.5)withx(0)=x0.8 Givenany1-periodicsolutionx(t)withx(0)=x0,wecalltheorbit nondegenerateifthelinear isomorphism d1(x0):Tx0M!Tx0M 8tmaynotbeglobally welldened ifMisnoncompact, butthiswillcause no problem inourdiscussion. 2.5.INFINITE DIMENSIONAL BUNDLES 53 doesnothave1asaneigenvalue. Thisimplies thattheorbitisisolate d, i.e.there isnoother 1-periodicorbitinsome neighborhoodofthisone(a proofisoutlined inExercise 2.63). Nondegeneracy hasanevenstronger consequence: thattheorbit is \stable" under small perturbations ofthesystem. Tostatethisprecisely , addanextra parameter tothevector eldX t,producing asmooth2- parameter family ofvector elds, 1-periodicint,withtherealnumber varying oversome neighborhoodof0suchthat X0 tXt: Thiswillbecalled asmooth1-parameter perturbation ofthetimedependen t vector eldXt.Weaimtoprovethefollowing: Theorem 2.62. Given anynondegenerate1-periodicorbitx(t)ofthesys- tem(2.5)andasmooth1-parameter perturbationX tofXt,thereisa unique smooth1-parameter family ofloops x:S1!M forinasuciently smallneighb orhoodof0,suchthat_x=X t(x)and x0(t)x(t). Exercise 2.63. Thefollowingargumen tprovesbothTheorem 2.62and theprevious statemen tthatanondegenerate periodicorbitisisolated. Let
MMdenote thediagonal submanifold
=f(x;x)jx2Mg,and consider theembedding F:M!MM:x7!(x;1(x)): ShowthatFintersects
transversely atF(x0)2
ifandonlyifthe orbitx(t)isnondegenerate. Theresult followsfrom thisviathenite dimensional implicit function theorem|ll inthegaps. (See[Hir94]for more ontransv ersalit yanditsimplications.) Alternative proofofTheorem2.62.LeavingasideExercise 2.63forthemo- ment,theresult canalsobeprovedbysetting upthesystem (2.5)asa smoothsection ofaBanac hspace bundle. Assume forsimplicit ythatM isorientable (thisrestriction canberemovedwithalittlecleverness). We rstaskthereader tobelievethatthespace ofmaps B:=C1(S1;M) admits anatural smoothBanac hmanifold structure, suchthatforeachx2 C1(S1;M),thetangen tspaceTxBiscanonically isomorphic toaBanac h space ofsections onthepullbac kbundle (cf.Example 2.35), TxB=C1(xTM):54 CHAPTER 2.BUNDLES ThusTB!BisasmoothBanac hspace bundle. Itiscontained inanother smoothBanac hspace bundleE!B,withbers Ex=C0(xTM): Allofthese statemen tscanbeprovedrigorously using theformalism of Eliasson [El67].Wenowconstruct asmoothsection s:B!Easfollows. ForaC1-loopx:S1!M,s(x)willbetheC0-vector eldalongxgiven by s(x)(t)=_x(t)Xt(x(t)): Thefactthatthissection issmoothalsofollowsfrom[El67].Clearly , theC1-solutions ofthesystem (2.5)areprecisely thezeros ofthesection s:B!E. Exercise 2.64. ShowthatanyCk-solution to(2.5)isactually ofclass Ck+1.Byinduction then,allC1-solutions aresmooth!(This isthesimplest example ofanellipticbootstrapping argumen t.) WecannowattackTheorem 2.62viatheinnite dimensional implicit function theorem. Letx2Bbeanorbit of(2.5),hences(x)=0.By choosing appropriate charts andlocaltrivializations, wecanidentifythe section sinaneighborhoodofx2Bwithsome smoothmap F:U!Y; where XandYareBanac hspaces, UXisanopenneighborhoodof0 andF(0)=0.Wecanalsoassume without lossofgeneralit ythatthezero sets1(0)inaneighborhoodofxcorresp ondstothesetF1(0). Lemma 2.65. Thelinearization dF(0):X!Yisanisomorphism. WepostponetheproofofthisuntilChapter 3,where wewilldevelop aconvenientmeans ofcomputing dF(0)viacovariantderivatives|this is where theassumption thatMisorientable willbeused. Toseetheimpli- cations forstabilit yofperiodicorbits, consider nowasmooth1-parameter perturbation X tofthevectoreldXt.Thiscanbeviewedasa1-parameter perturbation softhesection s,orequivalently,asmoothmap eF:(;)U!Y:(;u)7!F(u) where (;)Risasucien tlysmall intervalandF0F.Thelin- earization deF(0;0):RX!Ytakestheform deF(0;0)(h;v)=h@eF @(0;0)+dF(0)v; anditfollowseasily from Lemma 2.65thatthisoperator issurjectiv e, withaone-dimensional kernel. Thusbytheimplicit function theorem, 2.5.INFINITE DIMENSIONAL BUNDLES 55 some neighborhoodof0intheseteF1(0)isasmoothone-dimensional submanifold of(;)U.Itfollowsthatthesame istruefortheset f(;y)2(;)Bjs(y)=0g; andmoduloafewanalytical details, thisimplies Theorem 2.62. Thepreceding mayseem likeanunwarrantedamoun tofeort given thatwealready sawamore elemen taryproofinExercise 2.63.Thereal reason toconsider thisapproac histhatasimilar argumen tcanbeapplied tothestudy ofcertain systems ofpartial dieren tialequations, where no suchnite dimensional metho disavailable. Wenowlookbrie y atan example thatisimportantinsymplectic topology. Example 2.66(Pseudoholomorphic curves).GivenamanifoldWof dimension 2n,analmost complex structur eonWisacomplex structure J2(End(TW))onitstangen tbundle, i.e.foreachx2W,Jdenes a linear mapTxW!TxWwithJ2=Id.Thisgivesthetangen tbundle TW!Wthestructure ofacomplex vector bundle withrankn(cf. Section 2.4),andthepair(W;J)iscalled analmost complex manifold .For n=1,analmost complex manifold (;j)isalsocalled aRiemann surface GivenaRiemann surface (;j)andanalmost complex manifold (W;J) ofdimension 2n,amapu:!Wiscalled apseudoholomorphic (or J-holomorphic )curve ifitsatises Tuj=JTu: (2.6) Working inlocalcoordinates onbothmanifolds, (2.6)becomes asys- temof2nrstorder partial dieren tialequations intwovariables (since dim=2),called thenonline arCauchy-R iemann equations .Thenext exercise explains thereason forthisterminology . Exercise 2.67. If(;j)=(W;J)=(C;i),showthatamapf:C!C satises (2.6)ifandonlyifitisananalytic function. Theinnite dimensional implicit function theorem canbeapplied to thestudy ofJ-holomorphic curvesjustasinExample 2.61. Exercise 2.68. Without worrying abouttheanalytical details, writedown aBanac hspace bundleEovertheBanac hmanifoldB:=C1(;W)and asectionB!Ewhose zeros areprecisely theJ-holomorphic curvesu: (;j)!(W;J)ofclassC1.(AsinExercise 2.64,itturns outthatanysuch solution isactually asmoothmap, duetoellipticregularity theory.Note thattechnically ,mustbecompact inorder toapply Eliasson's formalism, butthere arevarious waystoworkaround thisforcertain noncompact domains.) MuchmoreonJ-holomorphic curvescanbefound inthebookbyMc- DuandSalamon [MS04].56 CHAPTER 2.BUNDLES 2.6General berbundles Bynowitshould beapparen tthatvector bundles areworthyofstudy. Wenextexamine someobjectsthatclearly deserv etobecalled \bundles," though theirbersarenotvector spaces. Example 2.69(Framebundles ).Recall thatif:E!Misasmooth vector bundle ofrankm,aframeoversome subsetUMisanordered set(v1;:::;vm)ofsmoothsectionsvj2(EjU)whichspaneachberEx forx2U.Restricting toasingle ber,aframe onExissimply anordered basisofEx.Denote byFExthesetofallframes onEx,andlet FE=[ x2MFEx: Thisspace hasanatural topology andsmoothmanifold structure suchthat anyframe overanopensubsetUMdenes asmoothmapU!FE. Wecanalsodene asmoothprojection mapF:FE!Msuchthat 1 F(x)=FExforallx2M.Itseems natural tocallthesets1 F(x) bers,andF:FE!Miscalled theframebundleofE.ThebersFEx arenotvector spaces, butrather smoothmanifolds. Exercise 2.70. Showthatforeachx2M,FExcanbeidentied with thegeneral linear group GL(m;F),though notinacanonical way. Asection ofthebundleFE!MoversomesubsetUMisdened| predictably|as amaps:U!FEsuchthats=IdU.Thusasmooth sections:U!FEisprecisely aframe overUforthevector bundleE, andwedenote thespace ofsuchsections by(FEjU).Twothings must beobserv ed: (i)(FEjU)isnotavectorspace, though itdoeshaveanatural topology, i.e.onecandene what itmeans forasequence ofsmoothsections sk2(FEjU)toconverge(uniformly withallderivatives)toanother smoothsections2(FEjU). (ii)Sectionss:U!FEcanalwaysbedened oversucien tlysmall opensubsetsUM,butthere mightnotbeanyglobalsections s:M!FE.Indeed, suchasection exists ifandonlyifthevector bundleE!Mistrivializable. Example 2.71(Unitsphere bundles anddiskbundles ).Givenareal vector bundle:E!Mofrankmwithasmoothbundle metrich;i, dene Sm1E=fv2Ejhv;vi=1g: Then therestriction oftoSm1Eisanother surjectiv emapp:Sm1E! M,whose bersp1(x)areeachdieomorphic tothesphereSm1. 2.6.GENERAL FIBER BUNDLES 57 Asections:M!Sm1Eissimply asection ofthevector bundle Ewhose values areunitvectors. Unitsphere bundles cansometimes be useful asalternativ edescriptions ofvectorbundles: theycarry allthesame information, buthavethenicepropertythatifMiscompact, soisthetotal spaceSm1E.Inparticular, if(M;g)isacompact Riemannian n-manifold, thenS1TMisacompact (2n1)-manifold, andthegeodesicequation on Mdenes asmoothvector eldonS1TM. Aclosely related objectisthediskbundle DmE=fv2Ejhv;vi1g; withbersthataresmoothcompact manifolds withboundary .Suchob- jectsarisenaturally intheconstruction oftubular neighb orhoods:givena manifoldMandaclosed submanifold NM,itturns outthataneigh- borhoodofNinMcanalwaysbeidentied withadiskbundle overN, withbersofdimension dimMdimN(cf.[Bre93 ]). Clearly itisuseful toconsider bundles whose bersarealldieomorphic manifolds, varying overthebaseinsomesmoothway.Eachoftheexamples sofarhasbeendened interms ofagivenvector bundle, butthisneedn't bethecaseingeneral, asthenextexample shows. Example 2.72(Mapping tori).SupposeNisasmoothn-manifold and ':N!Nisadieomorphism. Then wedene themapping torus of' tobethesmooth(n+1)-manifold M'=(RN)=; withtheequivalence relation dened by(t;x)(t+1;'(x)).Theprojec- tionRN!Rthendescends toasmoothsurjectiv emap :M'!S1:=R=Z; withthe\bers"(M')[t]:=1([t])alldieomorphic toN.Toseethat thebersvary\smoothly" withrespecttot2S1,wecanconstruct \local trivializations" :1(U)!UNoveropensubsetsUS1.For example, regardingU1:=(0;1)asanopensubset ofS1,thereistheobvious dieomorphism 1:1(U1)!U1Ndened by 1([(t;x)])=(t;x) fort2(0;1): Thesame expression witht21 2;3 2
givesawelldened dieomorphism 2:1(U2)!U2NfortheopensetU2:=1 2;3 2
S1.Observ ethat U1[U2coversS1,thusthetwotrivializationsfUj;jgj2f1;2gdetermine what wewillmomen tarily dene asa\smoothberbundle structure" for :M'!S1.58 CHAPTER 2.BUNDLES Notice thatthecoordinate vectoreldeV(t;x)=@ @tonRNdescends toawelldened smoothvector eldVonthetotalspaceM',whose 1- periodicorbits canbeidentied withthexedpointsof'.Inthiswaythe mapping torus canbeusedtoanalyze xedpointsbychanging adiscrete dynamical system intoacontinuous dynamical system. Thisnextexample similarly hasnodirect relation toanyvectorbundle. Example 2.73(TheHopfbration ).RegardS3astheunitsphere in C2,anddene amapwithvalues intheextended complex plane by :S3!C[f1g:(z1;z2)7!z1 z2: Using thestereographic projection toidentifyC[f1gwithS2,themap :S3!S2turns outtobesmooth,anditsbers1(z)aresetsofthe formfei(z1;z2)j2Rgforxed (z1;z2)2S3C2.Thuswehavea berbundle:S3!S2withbersdieomorphic toS1.Suchobjects arecalled circlebundles,andthisoneinparticular isknownastheHopf bration.Itplaysanimportantroleinhomotop ytheory|in particular,  isthesimplest (andinsome sense thearchetypal) example ofamapfrom S3toS2thatcannot bedeformed continuously totheidentity.Theproof ofthisfactusesageneral result thatrelates thehigher homotop ygroups ofthetotalspace, berandbaseofanyberbundle. Foratopologist, this givesmore thansucien tmotivation forthestudy ofberbundles. Herenowarethedenitions|w eonceagain begininapurely topolog- icalcontext,andsubsequen tlyincorp orate thenotion ofsmoothness. Denition 2.74. Aberbundleconsists ofthree topological spacesE,M andF,called thetotalspace,baseandstandar dberrespectively,together withacontinuoussurjectiv emap:E!Msuchthatforeverypointx2 Mthere exists anopenneighborhoodx2UMandalocaltrivialization :1(U)!UF: Hereisahomeomorphism thattakes1(x)tofxgFforeachx2U. ThebersEx:=1(x)ofaberbundle arethushomeomorphic to some xedspaceF,byhomeomorphisms thatvarycontinuously overthe base.Remark 2.75.Geometers often usetheterms bration andberbundle synonymously ,though theformer hasaslightlymore general meaning in topology,cf.[ Hat02 ]. Denition 2.76. Suppose:E!Misaberbundle withstandard ber F,suchthatMandFarebothsmoothmanifolds. Asmoothstructur eon :E!Misamaximal setoflocaltrivializationsfU;gsuchthatthe opensetsfUgcoverMandthemapsfgaredieomorphisms. 2.7.STRUCTURE GROUPS 59 Weseefromthisdenition thatasmoothberbundle haseachberdif- feomorphic tothestandard berF,bydieomorphisms thatvarysmoothly overM.It'snothardtoshowfromthisthatthetotalspaceEisthena smoothmanifold ofdimension dimM+dimF. Theterms section andberpreserving maphavethesamedenition in thiscontextasforavector bundle. 2.7Structure groups Theversion ofaberbundle dened aboveissomewhat weakerthanthe denition giveninsometexts, notably theclassic bookbySteenro d[Ste51]. Thestricter denition includes anextra pieceofstructure, called thestruc- turegroupofabundle. Toexplain this,wewillneedtoassume some knowledge oftopological groups andLiegroups, whicharediscussed in Appendix B. ThemostimportantLiegroups areofcourse thelinear groups: GL(n;F), SL(n;F),O(n),SU(n),Sp(n)andsoforth, allofwhicharisenaturally in thestudy ofvectorbundles withextra structure suchasbundle metrics or volume forms. Inthestudy ofgeneral berbundles, onealsoencoun ters various \innite dimensional groups," suchasthefollowingexamples. For anytopological spaceF,there isthegroup Homeo (F)=f':F!Fj'isbijectiv e,'and'1bothcontinuousg: Onanorientedtopological manifold, thishasanimportantsubgroup Homeo+(F)=f'2Homeo(F)j'preserv esorientationg; wereferto[Hat02 ]forthedenition of\orien tation preserving" onatopo- logical manifold. IfFisasmoothmanifold, wehavethecorresp onding subgroups Di(F)=f'2Homeo(F)j'and'1aresmoothg; andintheorientedcase, Di+(F)=f'2Di(F)j'preserv esorientationg: There arestillother groups corresp onding toextra structures onasmooth manifoldF,suchasaRiemannian metricgorasymplectic form!.These twoinparticular giverisetothegroups ofisometries andsymple ctomor- phisms respectively, Isom(F;g)=f'2Di(F)j'ggg; Symp(F;!)=f'2Di(F)j'!!g:60 CHAPTER 2.BUNDLES Mostoftheseexamples areinnite dimensional,9whichintroduces some complications fromananalytical pointofview. They cannot beconsidered smoothLiegroups, though theyclearly aretopological groups. Givenatopological groupGandatopological spaceF,wesaythatG actscontinuously fromtheleftonFifthere isacontinuousmap GF!F:(a;x)7!ax satisfying ex=x and (ab)x=a(bx) foralla;b2Gandx2F,wheree2Gistheidentityelemen t.The simplest example isthenatural action ofanysubgroupGHomeo(F)on F,dened by(';x)7!'(x). Similarly ,arightaction FG!F:(x;a)7!xa mustsatisfy xe=x andx(ab)=(xa)b: Forexample, asubgroupGHomeo (F)denes arightaction onFby (x;')7!'1(x).Foranyleftorrightaction, theoperation ofanysingle group elemen ta2GonFdenes ahomeomorphism. Youshould takeamomen ttoconvince yourself thatthedenitions of \leftaction" and\rightaction" arenotfullyequivalent:thekeyisthe associativit ycondition. Forinstance, themap(x;')7!'1(x)mentioned abovedenes onlyarightaction, notaleftaction. Ontheother hand, anyleftaction (a;x)7!axdenes acorresp onding rightaction by(x;a)7!a1x,andviceversa. Inthiswayonecanswitch backandforth betweenleftandrightactions forthesakeofnotational convenience; thedistinction isessentially amatter ofbookkeeping. Remark 2.77.Thefollowingexample showswhyitcanbeconvenientto consider bothleftandrightactions inthesame discussion. Forsome topological groupG,letF=Gandconsider thenatural actionsG F!FandFG!Fdened bygroup multiplication. These dene homeomorphisms ofFbyLa(h)=ahandRb(h)=hbforeverya;b2G, andduetotheassociativelaw,LaRb=RbLa.Thiswillbeuseful when wedene thegroup action onframe bundles inSection 2.8. Theterms smoothleft/right action nowhaveobvious denitions ifGis aLiegroup andFisasmoothmanifold. Inthiscase,eachelemen ta2G denes adieomorphism onF. 9Theexception isIsom(F;g),whichisgenerated innitessimally bythenite dimen- sional space ofKillingvectorelds;seee.g.[GHL04 ]. 2.7.STRUCTURE GROUPS 61 Wearenowready torelate allofthistoberbundles. Foragen- eralbundle:E!M,it'sclear thatanytwolocaltrivializations :1(U)!UFand:1(U)!UFcanberelated by 1 (x;p)=(x;g(x)p) (2.7) whereg:U\U!Homeo(F)isacontinuous\transition map," dened interms ofthenatural leftaction ofHomeo(F)onF.Atthemostbasic levelthen, wecansaythatthebundle:E!M\hasstructure group Homeo(F)". Denition 2.78. Suppose:E!Misaberbundle, withstandard berF,andGisatopological group. Areduction ofthestructur egroup ofthisbundle toGisasystem oflocaltrivializationsfU;gsuchthat fUgisanopencoveringofM,andthemaps arerelated toeachother asin(2.7)byafamily ofcontinuoustransition maps g:U\U!G; acting onthetrivializations viasome continuousleftactionGF!F. When:E!Misasmoothberbundle andGisaLiegroup acting smoothly onthestandard berF,weadditionally require thetransition maps tobesmooth. Ifsuchareduction exists, wesaythatthebundle:E!Mhas structur egroupG.Aberbundle withstructure groupGiscalled aG- bundle. Foraberbundle withextra structure, itisappropriate tointroduce asomewhat restricted notion ofbundle isomorphism. Denition 2.79. Suppose1:E1!Mand2:E2!Maretwo G-bundles withthesame standard berFandthesameG-action onF. Then aG-bundleisomorphism isaberpreserving homeomorphism (or dieomorphism) f:E1!E2suchthatforeverypairoflocaltrivializations :1 1(U)!UFand :1 2(U)!UF,there isacontinuous (orsmooth)mapf:U\U!Gwith f1 (x;p)=(x;f(x)p): Steenro d[Ste51]regards thestructure group aspartoftheintrinsic structure ofaberbundle, included inthedenition. Inprinciple, our denition isequivalenttohissimply bynaming Homeo(F)asthestructure group|but thisisbesidethepoint,because manyofthemostinteresting cases arethose where thestructure group canbereduced toasimpler groupG,particularly ifGisanite dimensional Liegroup. It'stimeforsome examples.62 CHAPTER 2.BUNDLES Example 2.80 (Vector bundles withstructure ).Asmoothvector bundle:E!Mofrankmcannowbedenedasasmoothberbundle withstandard berFmandstructure group GL(m;F),whichactsonFm inthenatural way.Thenotion ofaGL(m;F)-bundle isomorphism isthen equivalenttoourprevious denition ofvector bundle isomorphism. Thegroup reduces further ifthebundle isgivenextra structure: forin- stance, aEuclidean orHermitian bundle metric reduces GL(m;F)toO(m) orU(m)respectively,while avolume formyields SL(m;F).Thecombina- tionofaHermitian metric andcompatible complex volume formreduces thestructure toSU(m).There arecorresp onding statemen tsaboutsym- plectic structures (Sp(m)),indenite metrics (O(mk;k))andsoforth. Indeed, avector bundle withaG-structure foranyLiegroupGcanbe dened asaberbundle withstandard berFmandstructure groupG, whereGactsonFmviasomegivengroup represen tation. Eachsuchstruc- turecomes withitsowndenition ofabundle isomorphism; forinstance, asmoothvector bundle isomorphism onaEuclidean bundle isanO(m)- bundle isomorphism ifandonlyifitsrestriction toeachpairofbersisan isometry . Example 2.81(Framebundles withstructure ).Inaddition toreduc- ingthestructure group toasubgroup ofGL(m;F),theaforemen tioned structures onvector bundles eachdene specialframe bundles thathave thesame structure group. Forexample, givenaEuclidean vector bundle (E;h;i)overM,wecandene theorthogonalframebundleFO(m)E!M, asubbundle ofFEwhichcontainsonlyorthonormal frames. Thestandard berofFO(m)EistheLiegroup O(m),whichadmits anatural action of O(m),dened bythegroup multiplication. Clearly anytwotrivializations canthenberelated byatransition mapwithvalues inO(m). Thesame remarks apply toanyreduction ofthestructure group ofa vectorbundle: there isalwaysacorresp onding frame bundle, consisting of theframes thatare\preferred" bythestructure inquestion, e.g.symplectic frames, unitary frames, frames ofunitvolume, etc.. Theframe bundle alwayshasthesame structure group asthevector bundle, anditsbers arealsodieomorphic tothisgroup. We'llhavemore tosayaboutthisin thenextsection, onprincipal bundles. Example 2.82(Unitsphere bundles ).ForthebundleSm1E!Mde- nedinExample 2.71withrespecttoaEuclidean vectorbundle (E;h;i), there isanatural reduction ofthestructure group toO(m).Namely ,we restrict toorthonormal frames onEandusethese todene localtrivial- izations ofSm1E,sothattransition maps inO(m)actontheunitsphere Sm1Rm. Example 2.83(Symplectic brations ).Wenowgiveanexample where thestructure group isaninnite dimensional subgroup ofHomeo (F).A 2.8.PRINCIP ALBUNDLES 63 symple cticbration isasmoothberbundle:E!Mwhose standard berisasymplectic manifold (F;!),withstructure group Symp(F;!). Thislastcondition giveseachberExthestructure ofasymplectic man- ifold, suchthatthelocaltrivializations restrict tosymplectomorphisms Ex!fxgF.ThusEcanbethough tofasabundle ofsymplectic man- ifolds thatareallrelated bysmoothly varying symplectomorphisms. A berpreserving dieomorphism f:E1!E2betweentwosuchbundles is aSymp(F;!)-bundle isomorphism ifandonlyifitsrestriction toE1 x!E2 x foreachx2Misasymplectomorphism. Foraverysimple example, onecantakeanysymplectic manifold (F;!) withasymplectomorphism '2Symp(F;!)anddene themapping torus M'asinExample 2.72.Thisisnowasymplectic bration overS1. Wecannowgivethesimplest possible denition of\orien tation" fora berbundle. Tosimplify matters, werestrict tothesmoothcase. Denition 2.84. Forasmoothberbundle:E!Mwithstandard berF,anorientation ofthebundle isareduction ofitsstructure group fromDi(F)toDi+(F). Takeamomen ttoconvince yourself thatthismatchestheolddenition inthecaseofarealvector bundle. Thenextstepisofcourse togivean example ofanon-orien tableberbundle|it justsohappensthatthemost popular example isalsothebestknownnon-orien table manifold inthe world. Example 2.85(TheMobius strip).Recall thenon-orien tablelinebun- dle`!S1fromExample 2.12,whereS1isidentied withtheunitcircle inCandthebersarelinesinR2,specically `ei=Rcos(=2) sin(=2) R2: Wecandene aEuclidean structure onthisbundle using thenatural inner productofR2,andthenconsider thecorresp onding \unit diskbundle" D1E=fv2`jhv;vi1g; asmoothberbundle overS1withstandard ber[1;1].Thetotalspace ofD1EistheMobius strip. 2.8Principal bundles We'vealready seensome examples ofprincipal berbundles, namely the framebundlesFGE!Mcorresp onding toanyvector bundleE!M64 CHAPTER 2.BUNDLES withstructure groupG.These bundles havethespecialpropertythat theirstandard berandstructure group arethesame LiegroupG,which actsfromtheleftonitselfbygroup multiplication. There isalsoanatural berpreserving action ofGonFGE:toseethis,consider theexample oftheorthonormal frame bundleFO(m)Eforarealvector bundleEwith Euclidean metrich;i.Givenanyorthonormal framep=(v1;:::;vm)2 (FO(m)E)xforExandamatrixA2O(m)withentriesAi j,another such framep
A2(FO(m)E)xisobtained byrightmultiplication, p
A= v1


vm
0B@A 1 1


A1 m......... A m 1


Am m1CA=(v iAi 1;:::;viAi m): Similar remarks apply tobundles ofunitary frames, orientedframes and soforth. Since frame bundles playahelpful roleinunderstanding vectorbundles withextra structure, wegiveaspecialname totheclassofberbundles thatshare thesame specialproperties. Thefollowingistherstoftwo equivalentdenitions wewillgive|it applies tosmoothbundles, though itshould beclearhowtogeneralize ittotopological bundles andgroups. Denition 2.86. Asmoothprincip alberbundle:E!Misasmooth berbundle whose standard berisaLiegroupG,suchthatthestructure group reduces toG,acting onitselfbyleftmultiplication. These aresometimes simply called princip albundles,orprincip alG- bundlesifwewanttospecifythestructure group. Thedenition aboveisanalogous tothenotion ofavector bundle as aberbundle withstandard berFmandstructure group GL(m;F)(see Example 2.80).Thisis,ofcourse, notthemostintuitivedenition possible, whichiswhyweoriginally characterized vector bundles interms ofthe intrinsic linear structure onthebers.Ananalogous denition ispossible forprincipal bundles:E!M,assoonasweunderstand whatintrinsic structure theberspossess. Foreachx2M,Exisdieomorphic to aLiegroupG,butnotinacanonic alway|inparticular, there isno specialelemen te2Exthatwecancalltheidentity.Ontheother hand, havingreduced thestructure group toG,we'vechosen aspecialsetoflocal trivializations :1(U)!UGthatarerelated toeachother on eachberbytheleftaction ofG.Thishastheeect ofinducing anatural berpreserving rightaction EG!E:(p;g)7!pg; dened interms ofanylocaltrivialization (p)=((p);f(p))2U Gbyf(pg)=f(p)g.Thisisindependen tofthechoice because the 2.8.PRINCIP ALBUNDLES 65 rightaction ofGonitselfcomm uteswiththeleftaction ofthetransition functions (cf.Remark 2.77);indeed ifp2Exforx2U\U, f(pg)=g(x)f(pg)=g(x)f(p)g=f(p)g: Clearly theaction restricts toarightaction ofGoneachberEx,and oneeasily seesthatthisisidenticaltothegroup action onframes thatwe dened aboveforthespecialcasewhenEisaframe bundle. Twoother importantproperties areworthnoting: (i)Gactsfreely,i.e.without xed points,meaningpg=pforsome p2Eandg2Gifandonlyifgistheidentity.Equiv alently,Gacts freely oneachberEx. (ii)Gactstransitively oneachber:foranyp;q2Ex,thereexistsg2G suchthatq=pg. Conversely,supposewehaveasmoothberbundle:E!Manda LiegroupGwithaberpreserving smoothrightactionEG!Ethat restricts toafreeandtransitiv eaction oneachber.These twoproperties oftheaction imply thateachberisdieomorphic toG.Infact,forany x2M,wecanchooseanopenneighborhoodx2UMandasmooth sections:U!E,anduseittodene theinverseofalocaltrivialization, 1:UG!1(U):(x;g)7!s(x)g: NowchooseanopencoverM=S Uwithsmoothsectionss:U!E, andusethese todene asystem oflocaltrivializationsf;Uginthe same manner. Forx2U\Uthere isaunique elemen tg(x)2Gsuch thats(x)=s(x)g(x),andonecould useanimplicit function theorem argumen ttoshowthatthemapg:U\U!Gissmooth.Then we have 1 (x;g)=(s(x)g)=(s(x)g(x)g)=(x;g(x)g); sogareindeed thetransition maps forthesystemfU;g.We've provedthatthefollowingisequivalenttoDenition 2.86. Denition 2.87. Asmoothprincip alberbundleisasmoothberbundle :E!Mtogether withaLiegroupGandaberpreserving rightaction EG!Ewhichrestricts toeachberfreely andtransitiv ely. Thuseachberhastheintrinsic structure ofasmoothmanifold with afreeandtransitiv eG-action; asaresult, thebermustbedieomorphic toG.Thisistheclosest wecancome togiving thebersanactual group structure without requiring thebundle tobetrivial|indeed, bythesame trickweusedtoconstruct localtrivializations above,wehave:66 CHAPTER 2.BUNDLES Proposition 2.88. Asmoothprincip albundle:E!Mistrivial ifand onlyifitadmits aglobalsmoothsections:M!E. Thisisofcourse quite dieren tfromthesituation inavector bundle, whichalwaysadmits aninnite dimensional vectorspace ofglobal smooth sections. Forprincipal bundles there isaone-to-one corresp ondence be- tweensections andtrivializations. Aconsequence isthatanysmoothly varying group structure ontheberswouldtrivialize thebundle, since there wouldthenbeaglobal sections:M!E,taking everypointtothe identityelemen tinthecorresp onding ber. Denition 2.89. GiventwoprincipalG-bundlesE1andE2overthesame baseM,aprincip albundleisomorphism isasmoothmap :E1!E2 whichisberpreserving andG-equiv ariant,i.e. (pg)= (p)g. Exercise 2.90. Notice thatthedenition abovedoesnotexplicitly require tobeaninvertible map. Showhoweverthateveryprincipal bundle isomorphism isautomatic allyinvertible, anditsinverseisanother principal bundle isomorphism. Exercise 2.91. ShowthatasmoothmapbetweenprincipalG-bundles is aprincipal bundle isomorphism ifandonlyifitisaG-bundle isomorphism (seeDenition 2.79). Clearly ,aprincipal bundle istrivializable ifandonlyifitisisomorphic tothetrivial principal bundle :MG!M; where therightaction is(x;h)g:=(x;hg). Aswe'veseenalready ,anyvector bundle:E!Mwithstructure groupGyields aprincipalG-bundleFGE!M,theframe bundle. Prin- cipalbundles arisenaturally inother contextsaswell. Example 2.92 (TheHopf bration ).Recall thebundle:S3!S2 fromExample 2.73,withstandard berS1,whereS3isregarded asthe unitsphere inC2.Therightaction S3U(1)!S3:((z1;z2);ei)7!ei(z1;z2) givestheHopfbration thestructure ofaprincipal U(1)-bundle. Example 2.93(AnSO(n)-bundle overSn).RegardingSnastheunit sphere inRn+1,thenatural action ofSO(n+1)onRn+1restricts toatran- sitiveaction onSn.Letx0=(1;0;:::;0)2Sn,anddene thesurjectiv e map :SO(n+1)!Sn:A7!Ax0: 2.8.PRINCIP ALBUNDLES 67 Using theinjectiv ehomomorphism SO(n),!SO(n+1):B7! 10 0B ; wecanregard SO(n)asasubgroup ofSO(n+1)anddene thenatural rightaction SO(n+1)SO(n)!SO(n+1):(A;B)7!AB: Nowobserv ethatSO(n)isprecisely thesubgroup ofmatrices inSO(n+1) thatxthepointx0.ThusifA21(x)foranyx2Sn,wehaveAx0=x andABx0=Ax0=x,sotherightaction isberpreserving. Itisnow simple toverifythattheaction isfreeandtransitiv eoneachber,thus: SO(n+1)!Snwiththisaction isaprincipal SO(n)-bundle. Aswiththe Hopfbration, thisconstruction isuseful forunderstanding thetopology ofthespaces involved,particularly thespecialorthogonal groups. Similar constructions canbemade fortheunitary andspecialunitary groups. Exercise 2.94. Construct aprincipal U(n)-bundle overS2n+1withtotal space U(n+1). References [Arn89 ]V.I.Arnol0d,Mathematic almethodsofclassic almechanics ,Springer-V erlag, NewYork,1989. [Bre93 ]G.E.Bredon, Topologyandgeometry ,Springer-V erlag, NewYork,1993. [El67]H.I.Elasson, Geometry ofmanifolds ofmaps,J.Dieren tialGeometry 1 (1967), 169{194. [GHL04 ]S.Gallot, D.Hulin, andJ.Lafontaine, Riemannian geometry ,3rded., Springer-V erlag, Berlin, 2004. [Hat02 ]A.Hatcher,Algebraictopology,Cambridge UniversityPress, Cambridge, 2002. [Hir94]M.W.Hirsch,Dierential topology,Springer-V erlag, NewYork,1994. [HZ94]H.Hofer andE.Zehnder, Symple cticinvariants andHamiltonian dynamics , Birkhauser Verlag, Basel, 1994. [Lan93 ]S.Lang, Realandfunctional analysis ,3rded.,Springer-V erlag, NewYork, 1993. [Lan99 ] ,Fundamentals ofdierential geometry ,Springer-V erlag, NewYork, 1999. [MS04]D.McDu andD.Salamon, J-holomorphic curves andsymple ctictopology, American Mathematical Society,Providence, RI,2004. [SU01]R.U.SexlandH.K.Urban tke,Relativity, groups, particles :Specialrelativ- ityandrelativistic symmetry ineldandparticle physics ,Springer-V erlag, Vienna, 2001.68 CHAPTER 2.BUNDLES [Spi65]M.Spivak,Calculus onmanifolds :Amodernapproachtoclassic altheorems ofadvanc edcalculus ,W.A.Benjamin, Inc.,NewYork-Amsterdam, 1965. [Spi99] ,Acomprehensive introduction todierential geometry ,3rded.,Vol.1, Publish orPerishInc.,Houston, TX,1999. [Ste51]N.Steenro d,TheTopologyofFibreBundles,Princeton UniversityPress, Princeton, N.J.,1951.