Differential Geometry: Connections, Curvature, and Characteristic Classes PDF
Preview Differential Geometry: Connections, Curvature, and Characteristic Classes
Graduate Texts in Mathematics Loring W. Tu Differential Geometry Connections, Curvature, and Characteristic Classes Graduate Texts in Mathematics 275 Graduate Texts in Mathematics SeriesEditors: SheldonAxler SanFranciscoStateUniversity,SanFrancisco,CA,USA KennethRibet UniversityofCalifornia,Berkeley,CA,USA AdvisoryBoard: AlejandroAdem,UniversityofBritishColumbia DavidEisenbud,UniversityofCalifornia,Berkeley&MSRI IreneM.Gamba,TheUniversityofTexasatAustin J.F.Jardine,UniversityofWesternOntario JeffreyC.Lagarias,UniversityofMichigan KenOno,EmoryUniversity JeremyQuastel,UniversityofToronto FadilSantosa,UniversityofMinnesota BarrySimon,CaliforniaInstituteofTechnology Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully written as teaching aids and highlight characteristic features of the theory. Although these books are frequently used as textbooksingraduatecourses,theyarealsosuitableforindividualstudy. Moreinformationaboutthisseriesathttp://www.springer.com/series/136 Loring W. Tu Differential Geometry Connections, Curvature, and Characteristic Classes 123 LoringW.Tu DepartmentofMathematics TuftsUniversity Medford,MA02155,USA ISSN0072-5285 ISSN2197-5612 (electronic) GraduateTextsinMathematics ISBN978-3-319-55082-4 ISBN978-3-319-55084-8 (eBook) DOI10.1007/978-3-319-55084-8 LibraryofCongressControlNumber:2017935362 MathematicsSubjectClassification(2010):53XX;97U20 ©SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright. AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsorthe editorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforanyerrors oromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictionalclaims inpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Differentialgeometryhasalongandglorioushistory. Asitsnameimplies,itisthe study of geometry using differentialcalculus, and as such, itdates back toNewton andLeibnizintheseventeenthcentury. Butitwasnotuntilthenineteenthcentury, with the work of Gauss on surfaces and Riemann on the curvature tensor, that dif- ferentialgeometryflourishedanditsmodernfoundationwaslaid. Overthepastone hundred years, differential geometry has proven indispensable to an understanding ofthephysicalworld,inEinstein’sgeneraltheoryofrelativity,inthetheoryofgravi- tation,ingaugetheory,andnowinstringtheory. Differentialgeometryisalsouseful intopology,severalcomplexvariables,algebraicgeometry,complexmanifolds,and dynamicalsystems,amongotherfields. Ithasevenfoundapplicationstogroupthe- ory as in Gromov’s work and to probability theory as in Diaconis’s work. It is not toofar-fetchedtoarguethatdifferentialgeometryshouldbeineverymathematician’s arsenal. Thebasicobjectsindifferentialgeometryaremanifoldsendowedwithametric, which is essentially a way of measuring the length of vectors. A metric gives rise to notions of distance, angle, area, volume, curvature, straightness, and geodesics. Itisthe presence of a metric thatdistinguishes geometry from topology. However, anotherconceptthatmightcontesttheprimacyofametricindifferentialgeometry is that of a connection. A connection in a vector bundle may be thought of as a way of differentiating sections of the vector bundle. A metric determines a unique connectioncalledaRiemannianconnectionwithcertaindesirableproperties. While a connection is not as intuitive as a metric, it already gives rise to curvature and geodesics. Withthis, theconnectioncanalsolayclaimtobeafundamentalnotion ofdifferentialgeometry. Indeed,in1989,thegreatgeometerS.S.Chernwroteastheeditorofavolume on global differential geometry [5], “The Editor is convinced that the notion of a connectioninavectorbundlewillsoonfinditswayintoaclassonadvancedcalculus, asitisafundamentalnotionanditsapplicationsarewide-spread.” In 1977, the Nobel Prize-winning physicist C. N. Yang wrote in [23], “Gauge fields are deeply related to some profoundly beautiful ideas of contemporary v vi Preface mathematics,ideasthatarethedrivingforcesofpartofthemathematicsofthelast 40years,...,thetheoryoffiberbundles.” Convincedthatgaugefieldsarerelatedto connections on fiber bundles, he tried to learn the fiber-bundle theory from several mathematicalclassicsonthesubject,but“learnednothing. Thelanguageofmodern mathematicsistoocoldandabstractforaphysicist”[24,p.73]. Whilethedefinitionandformalpropertiesofaconnectiononaprincipalbundle canbegiveninafewpages,itisdifficulttounderstanditsmeaningwithoutknowing howitcameintobeing. Thepresentbookisanintroductiontodifferentialgeometry that follows the historical development of the concepts of connection and curva- ture, withthegoalofexplainingtheChern–Weiltheoryofcharacteristicclasseson aprincipalbundle. Thegoal,oncefixed,dictatesthechoiceoftopics. Startingwith directionalderivativesinaEuclideanspace,weintroduceandsuccessivelygeneral- izeconnectionsandcurvaturefromatangentbundletoavectorbundleandfinallyto aprincipalbundle. Alongtheway,thenarrativeprovidesapanoramaofsomeofthe high points in the history of differential geometry, for example, Gauss’ Theorema EgregiumandtheGauss–Bonnettheorem. Initially, the prerequisites are minimal; a passing acquaintance with manifolds suffices. StartingwithSection11,itbecomesnecessarytounderstandandbeableto manipulate differential forms. Beyond Section 22, a knowledge of de Rham coho- mologyisrequired. AllofthisiscontainedinmybookAnIntroductiontoManifolds [21]andcanbelearnedinonesemester. Itismyferventhopethatthepresentbook will be accessible to physicists as well as mathematicians. For the benefit of the reader and to establish common notations, we recall in Appendix A the basics of manifold theory. In an attempt to make the exposition more self-contained, I have alsoincludedsectionsonalgebraicconstructionssuchasthetensorproductandthe exteriorpower. In two decades of teaching from this manuscript, I have generally been able to coverthefirsttwenty-fivesectionsinonesemester,assumingaone-semestercourse on manifolds as the prerequisite. By judiciously leaving some of the sections as independentreadingmaterial,forexample,Sections9,15,and26,Ihavebeenable tocoverthefirstthirtysectionsinonesemester. Everybookreflectsthebiasesandinterestsofitsauthor. Thisbookisnoexcep- tion. For a different perspective, the reader may find it profitable to consult other books. Afterhaving readthisone, itshouldbeeasiertoreadtheothers. Thereare manygoodbooksondifferentialgeometry,eachwithitsparticularemphasis. Some oftheonesIhavelikedincludeBoothby[1],Conlon[6],doCarmo[7],Kobayashi andNomizu[12],Lee[14],MillmanandParker[16],Spivak[19],andTaubes[20]. Forapplicationstophysics,seeFrankel[9]. As a student, I attended many lectures of Phillip A. Griffiths and Raoul Bott on algebraic and differential geometry. It is a pleasure to acknowledge their influ- ence.IwanttothankAndreasArvanitoyeorgos,JeffreyD.Carlson,BenoitCharbon- neau,HanciChi,BrendanFoley,GeorgeLeger,ShiboLiu,IshanMata,StevenScott, andHuaiyuZhangfortheircarefulproofreading,usefulcomments,anderratalists. JeffreyD.Carlsoninparticularshouldbesingledoutforthemanyexcellentpieces ofadvicehehasgivenmeovertheyears. IalsowanttothankBruceBoghosianfor Preface vii helpingmewithMathematicaandforpreparingthefigureoftheFrenet–Serretframe (Figure 2.5). Finally, I am grateful to the Max Planck Institute for Mathematics in Bonn,NationalTaiwanUniversity,andtheNationalCenterforTheoreticalSciences inTaipeiforhostingmeduringthepreparationofthismanuscript. Medford,MA,USA LoringW.Tu April2017 Contents Preface v Chapter1 CurvatureandVectorFields 1 §1 RiemannianManifolds ......................................... 2 1.1 InnerProductsonaVectorSpace ........................... 2 1.2 RepresentationsofInnerProductsby SymmetricMatrices ...................................... 3 1.3 RiemannianMetrics ...................................... 4 1.4 ExistenceofaRiemannianMetric .......................... 6 Problems...................................................... 7 §2 Curves ....................................................... 9 2.1 RegularCurves .......................................... 9 2.2 ArcLengthParametrization................................ 10 2.3 SignedCurvatureofaPlaneCurve.......................... 11 2.4 OrientationandCurvature ................................. 13 Problems...................................................... 14 §3 SurfacesinSpace.............................................. 17 3.1 Principal,Mean,andGaussianCurvatures.................... 17 3.2 Gauss’sTheoremaEgregium............................... 19 3.3 TheGauss–BonnetTheorem ............................... 20 Problems...................................................... 21 §4 DirectionalDerivativesinEuclideanSpace ....................... 22 4.1 DirectionalDerivativesinEuclideanSpace................... 22 4.2 OtherPropertiesoftheDirectionalDerivative................. 24 ix x Contents 4.3 VectorFieldsAlongaCurve ............................... 25 4.4 VectorFieldsAlongaSubmanifold ......................... 26 4.5 DirectionalDerivativesonaSubmanifoldofRn ............... 27 Problems...................................................... 28 §5 TheShapeOperator ........................................... 29 5.1 NormalVectorFields ..................................... 29 5.2 TheShapeOperator ...................................... 30 5.3 CurvatureandtheShapeOperator .......................... 32 5.4 TheFirstandSecondFundamentalForms.................... 35 5.5 TheCatenoidandtheHelicoid ............................. 36 Problems...................................................... 39 §6 AffineConnections ............................................ 43 6.1 AffineConnections....................................... 43 6.2 TorsionandCurvature .................................... 44 6.3 TheRiemannianConnection ............................... 45 6.4 OrthogonalProjectiononaSurfaceinR3 .................... 46 6.5 TheRiemannianConnectiononaSurfaceinR3............... 47 Problems...................................................... 48 §7 VectorBundles ................................................ 49 7.1 DefinitionofaVectorBundle .............................. 49 7.2 TheVectorSpaceofSections .............................. 51 7.3 ExtendingaLocalSectiontoaGlobalSection ................ 52 7.4 LocalOperators.......................................... 53 7.5 RestrictionofaLocalOperatortoanOpenSubset............. 54 7.6 Frames ................................................. 56 7.7 F-LinearityandBundleMaps .............................. 56 7.8 MultilinearMapsoverSmoothFunctions .................... 59 Problems...................................................... 59 §8 Gauss’sTheoremaEgregium.................................... 61 8.1 TheGaussandCodazzi–MainardiEquations ................. 61 8.2 AProofoftheTheoremaEgregium ......................... 63 8.3 TheGaussianCurvatureinTerms ofanArbitraryBasis ..................................... 64 Problems...................................................... 64 §9 GeneralizationstoHypersurfacesinRn+1 ........................ 66 9.1 TheShapeOperatorofaHypersurface....................... 66 9.2 TheRiemannianConnectionofaHypersurface ............... 67 9.3 TheSecondFundamentalForm............................. 68 9.4 TheGaussCurvatureandCodazzi–MainardiEquations ........ 68
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