Lecture Notes on Symmetric Spaces

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1、LectureNotesonSymmetricSpacesJ.-H.EschenburgToRenato.Contents1.DefinitionandExamples(p.2)2.TransvectionsandHolonomy(p.7)3.KillingFields(p.9)4.CartanInvolutionandCartanDecomposition(p.11)5.LocallySymmetricSpaces(p.15)6.Compact,NoncompactandEuclideanType;Duality(p.16)7.TheIsomet

2、ryGroup(p.17)8.LieSubtriplesandTotallyGeodesicSubspaces(p.19)9.IsotropyRepresentationandRank(p.19)10.TheWeylGroup(p.22)References(p.24)0.IntroductionRiemanniansymmetricspacesarethemostbeautifulandmostimportantRie-mannianmanifolds.Ontheonehand,thisclassofspacescontainsmanyprom

3、inentexampleswhichareofgreatimportanceforvariousbranchesofmathematics,likecom-pactLiegroups,Grassmanniansandboundedsymmetricdomains.Anysymmetricspacehasitsownspecialgeometry;euclidean,ellipticandhyperbolicgeometryareonlytheveryfirstexamples.Ontheotherhand,thesespaceshavemuchin

4、common,andthereexistsarichtheory.Thepurposeofthesenotesistogiveabriefintroductiontothetheoryandtosomeoftheexamples.Symmetricspacescanbeconsideredfrommanydifferentpointsofview.TheycanbeviewedasRiemannianmanifoldswithpointreflectionsorwithparallelcurvaturetensororwithspecialholon

5、omyorasahomogeneousspacewithaspecialisotropyorspecialKillingvectorfields,orasLietriplesystems,orasaLiegroupwithacertaininvolution.Wearetryingtodiscussalltheseaspects.Therefore,thechaptersareonlylooselyconnectedandcanbereadalmostseparately.Prerequisitesarethefundamentalconcepts

6、ofRiemanniangeometryandsomebasicknowledgeofLiegroupswhichcanbefounde.g.inChapter4ofCheeger-Ebin[CE].ThemainreferenceisHelgason[H],butalsoLoos[L]andBesse[B].Recently,P.Eberlein[E]gaveabeautifulapproachtosymmetricspacesofnoncompacttype.WeowethanksalsotoHermannKarcherforhislectu

7、resonsymmetricspaces.11.DefinitionandExamplesA(Riemannian)symmetricspaceisaRiemannianmanifoldSwiththepropertythatthegeodesicreflectionatanypointisanisometryofS.Inotherwords,foranyx∈Sthereissomesx∈G=I(S)(theisometrygroupofS)withthepropertiessx(x)=x,(dsx)x=−I.(∗)Thisisometrysxisc

8、alledsymmetryatx.Asafirstconsequenceofthisdefinition,Sisgeodesicallyco