Lectures on the Geometry of Manifolds

LecturesontheGeometryofManifoldsLiviuI.NicolaescuNovember14,2007IntroductionShapeisafascinatingandintriguingsubjectwhichhasstimulatedtheimaginationofmanypeople.Itsu±cestolookaroundtobecomecurious.Eucliddidjustthatandcameupwiththe¯rstpurecreation.Relyingonthecommonexperience,hecreatedanabstractworldthathadalifeofitsown.Asthehumanknowledgeprogressedsodidtheabilityofformulatingandansweringpenetratingquestions.Inparticular,mathematiciansstartedwonderingwhetherEuclid's\obviousabsolutepostulateswereindeedobviousand/orabsolute.ScientistsrealizedthatShapeandSpacearetwocloselyrelatedconceptsandaskedwhethertheyreallylookthewayoursensestellus.AsFelixKleinpointedoutinhisErlangenProgram,therearemanywaysoflookingatShapeandSpacesothatvariouspointsofviewmayproducedi®erentimages.Inparticular,themostbasicissueof\measuringtheShapecannothaveaclearcutanswer.ThisisabookaboutShape,Spaceandsomeparticularwaysofstudyingthem.Sinceitsinception,thedi®erentialandintegralcalculusprovedtobeaveryversatiletoolindealingwithpreviouslyuntouchableproblems.ItdidnottakelonguntilitfoundusesingeometryinthehandsoftheGreatMasters.Thisisthepathwewanttofollowinthepresentbook.Intheearlydaysofgeometrynobodyworriedaboutthenaturalcontextinwhichthemethodsofcalculus\feelathome.Therewasnoneedtoaddressthisaspectsincefortheparticularproblemsstudiedthiswasanon-issue.Asmathematicsprogressedasawholethe\naturalcontextmentionedabovecrystallizedinthemindsofmathematiciansanditwasanotionsoimportantthatithadtobegivenaname.Thegeometricobjectswhichcanbestudiedusingthemethodsofcalculuswerecalledsmoothmanifolds.Specialcasesofmanifoldsarethecurvesandthesurfacesandthesewerequitewellunderstood.B.Riemannwasthe¯rsttonotethatthelowdimensionalideasofhistimewereparticularaspectsofahigherdimensionalworld.The¯rstchapterofthisbookintroducesthereadertotheconceptofsmoothmanifoldthroughabstractde¯nitionsand,moreimportantly,throughmanywebelieverelevantexamples.Inparticular,weintroduceatthisearlystagethenotionofLiegroup.Themaingeometricandalgebraicpropertiesoftheseobjectswillbegraduallydescribedasweprogresswithourstudyofthegeometryofmanifolds.Besidestheirobvioususefulnessingeometry,theLiegroupsareacademicallyveryfriendly.Theyprovideamarveloustestinggroundforabstractresults.Wehaveconsistentlytakenadvantageofthisfeaturethrough-outthisbook.Asabonus,bytheendoftheselecturesthereaderwillfeelcomfortablemanipulatingbasicLietheoreticconcepts.Toapplythetechniquesofcalculusweneed\thingstoderivateandintegrate.Theseiii\thingsareintroducedinChapter2.Thereasonwhysmoothmanifoldshavemanydi®erentiableobjectsattachedtothemisthattheycanbelocallyverywellapproximatedbylinearspacescalledtangentspaces.Locally,everythinglooksliketraditionalcalculus.Eachpointhasatangentspaceattachedtoitsothatweobtaina\bunchoftangentspacescalledthetangentbundle.Wefounditappropriatetointroduceatthisearlypointthenotionofvectorbundle.Ithelpsinstructuringboththelanguageandthethinking.Oncewehave\thingstoderivateandintegrateweneedtoknowhowtoexplicitlyperformtheseoperations.WedevotetheChapter3tothispurpose.Thisisperhapsoneofthemostunattractiveaspectsofdi®erentialgeometrybutiscrucialforallfurtherdevelopments.Tospiceupthepresentation,wehaveincludedmanyexampleswhichwillfoundapplicationsinlaterchapters.Inparticular,wehaveincludedawholesectiondevotedtotherepresentationtheoryofcompactLiegroupsessentiallydescribingtheequivalencebetweenrepresentationsandtheircharacters.ThestudyofShapebeginsinearnestinChapter4whichdealswithRiemannmanifolds.Weapproachtheseobjectsgradually.The¯rstsectionintroducesthereadertothenotionofgeodesicswhicharede¯nedusingtheLevi-Civitaconnection.Locally,thegeodesicsplaythesameroleasthestraightlinesinanEuclidianspacebutgloballynewphenomenaarise.Weillustratetheseaspectswithmanyconcreteexamples.Inthe¯nalpartofthissectionweshowhowtheEuclidianvectorcalculusgeneralizestoRiemannmanifolds.ThesecondsectionofthischapterinitiatesthelocalstudyofRiemannmanifolds.Upto¯rstorderthesemanifoldslooklikeEuclidianspaces.Thenoveltyariseswhenwestudy\secondorderapproximationsofthesespaces.TheRiemanntensorprovidesthecompletemeasureofhowfarisaRiemannmanifoldfrombeing°at.Thisisaveryinvolvedobjectand,toenhanceitsunderstanding,wecomputeitinseveralinstances:onsurfaces(whichcanbeeasilyvisualized)andonLiegroups(whichcanbeeasilyformalized).WehavealsoincludedCartan'smovingframetechniquewhichisextremelyusefulinconcretecomputations.AsanapplicationofthistechniqueweprovethecelebratedTheoremaEgregiumofGauss.Thissectionconcludeswiththe¯rstglobalresultofthebook,namelytheGauss-Bonnettheorem.Wepresentaproofinspiredfrom[25]relyingonthefactthatallRiemannsurfacesareEinsteinmanifolds.TheGauss-Bonnettheoremwillbearecurringthemeinthisbookandwewillprovideseveralotherproofsandgeneralizations.OneofthemostfascinatingaspectsofRiemanngeometryistheintimatecorrelation\local-global.TheRiemanntensorisalocalobjectwithglobale®ects.Therearecur-rentlymanytechniquesofcapturingthiscorrelation.WehavealreadydescribedoneintheproofofGauss-Bonnettheorem.InChapter5wedescribeanothersuchtechniquew