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P1:JZP 052183774Xpre CUNY374B/Angel 052183774X April9,2006 9:52 P1:JZP 0521853680pre CB980/Chavel February15,2006 11:6 CharCount=0 This page intentionally left blank ii P1:JZP 0521853680pre CB980/Chavel February15,2006 11:6 CharCount=0 RIEMANNIAN GEOMETRY AModernIntroduction SecondEdition ThisbookprovidesanintroductiontoRiemanniangeometry,thegeometryof curvedspaces,foruseinagraduatecourse.Requiringonlyanunderstandingof differentiablemanifolds,thebookcoverstheintroductoryideasofRiemannian geometry, followed by a selection of more specialized topics. Also featured are Notes and Exercises for each chapter to develop and enrich the reader’s appreciationofthesubject.Thissecondeditionhasaclearertreatmentofmany topics from the first edition, with new proofs of some theorems. Also a new chapterontheRiemanniangeometryofsurfaceshasbeenadded. The main themes here are the effect of curvature on the usual notions of classicalEuclideangeometry,andthenewnotionsandideasmotivatedbycur- vatureitself.Amongtheclassicaltopicsshowninanewsettingisisoperimetric inequalities– the interplay of volume of sets and the areas of their bound- aries–incurvedspace.Completelynewthemescreatedbycurvatureinclude theclassicalRauchcomparisontheoremanditsconsequencesingeometryand topology,andtheinteractionofmicroscopicbehaviorofthegeometrywiththe macroscopicstructureofthespace. Isaac Chavel is Professor of Mathematics at The City College of the City UniversityofNewYork.HereceivedhisPh.D.inMathematicsfromYeshiva UniversityunderthedirectionofProfessorHarryE.Rauch.Hehaspublishedin internationaljournalsintheareasofdifferentialgeometryandpartialdifferen- tialequations,especiallytheLaplaceandheatoperatorsonRiemannianmani- folds.HisotherbooksincludeEigenvaluesinRiemannianGeometry(1984)and Isoperimetric Inequalities: Differential Geometric and Analytic Perspectives (2001). He has been teaching at The City College of the City University of NewYorksince1970,andhehasbeenamemberofthedoctoralprogramof theCityUniversityofNewYorksince1976.HeisamemberoftheAmerican MathematicalSociety. i P1:JZP 0521853680pre CB980/Chavel February15,2006 11:6 CharCount=0 ii P1:JZP 0521853680pre CB980/Chavel February15,2006 11:6 CharCount=0 CAMBRIDGESTUDIESINADVANCEDMATHEMATICS EditorialBoard: B.Bolloba´s,W.Fulton,A.Katok,F.Kirwan,P.Sarnak,B.Simon,B.Totaro Alreadypublished 17 W.Dicks&M.DunwoodyGroupsactingongraphs 18 L.J.Corwin&F.P.GreenleafRepresentationsofnilpotentLiegroupsandtheirapplications 19 R.Fritsch&R.PiccininiCellularstructuresintopology 20 H.KlingenIntroductorylecturesonSiegelmodularforms 21 P.KoosisThelogarithmicintegralII 22 M.J.CollinsRepresentationsandcharactersoffinitegroups 24 H.KunitaStochasticflowsandstochasticdifferentialequations 25 P.WojtaszczykBanachspacesforanalysis 26 J.E.Gilbert&M.A.M.MurrayCliffordalgebrasandDiracoperatorsinharmonicanalysis 27 A.Fro¨hlich&M.J.TaylorAlgebraicnumbertheory 28 K.Goebel&W.A.KirkTopicsinmetricfixedpointtheory 29 J.F.HumphreysReflectiongroupsandCoxetergroups 30 D.J.BensonRepresentationsandcohomologyI 31 D.J.BensonRepresentationsandcohomologyII 32 C.Allday&V.PuppeCohomologicalmethodsintransformationgroups 33 C.Souleétal.LecturesonArakelovgeometry 34 A.Ambrosetti&G.ProdiAprimerofnonlinearanalysis 35 J.Palis&F.TakensHyperbolicity,stabilityandchaosathomoclinicbifurcations 37 Y.MeyerWaveletsandoperatorsI 38 C.WeibelAnintroductiontohomologicalalgebra 39 W.Bruns&J.HerzogCohen–Macaulayrings 40 V.SnaithExplicitBrauerinduction 41 G.LaumonCohomologyofDrinfeldmodularvarietiesI 42 E.B.DaviesSpectraltheoryanddifferentialoperators 43 J.Diestel,H.Jarchow,&A.TongeAbsolutelysummingoperators 44 P.MattilaGeometryofsetsandmeasuresinEuclideanspaces 45 R.PinskyPositiveharmonicfunctionsanddiffusion 46 G.TenenbaumIntroductiontoanalyticandprobabilisticnumbertheory 47 C.PeskineAnalgebraicintroductiontocomplexprojectivegeometry 48 Y.Meyer&R.CoifmanWavelets 49 R.StanleyEnumerativecombinatoricsI 50 I.PorteousCliffordalgebrasandtheclassicalgroups 51 M.AudinSpinningtops 52 V.JurdjevicGeometriccontroltheory 53 H.VolkleinGroupsasGaloisgroups 54 J.LePotierLecturesonvectorbundles 55 D.BumpAutomorphicformsandrepresentations 56 G.LaumonCohomologyofDrinfeldmodularvarietiesII 57 D.M.Clark&B.A.DaveyNaturaldualitiesfortheworkingalgebraist 58 J.McClearyAuser’sguidetospectralsequencesII 59 P.TaylorPracticalfoundationsofmathematics 60 M.P.Brodmann&R.Y.SharpLocalcohomology 61 J.D.Dixonetal.Analyticpro-pgroups 62 R.StanleyEnumerativecombinatoricsII 63 R.M.DudleyUniformcentrallimittheorems 64 J.Jost&X.Li-JostCalculusofvariations 65 A.J.Berrick&M.E.KeatingAnintroductiontoringsandmodules 66 S.MorosawaHolomorphicdynamics 67 A.J.Berrick&M.E.KeatingCategoriesandmoduleswithK-theoryinview 68 K.SatoLevyprocessesandinfinitelydivisibledistributions 69 H.HidaModularformsandGaloiscohomology 70 R.Iorio&V.IorioFourieranalysisandpartialdifferentialequations 71 R.BleiAnalysisinintegerandfractionaldimensions 72 F.Borceaux&G.JanelidzeGaloistheories 73 B.Bolloba´sRandomgraphs 74 R.M.DudleyRealanalysisandprobability 75 T.Sheil-SmallComplexpolynomials (continuedonoverleaf) iii P1:JZP 0521853680pre CB980/Chavel February15,2006 11:6 CharCount=0 Serieslist(continued) 76 C.VoisinHodgetheoryandcomplexalgebraicgeometry,I 77 C.VoisinHodgetheoryandcomplexalgebraicgeometry,II 78 V.PaulsenCompletelyboundedmapsandoperatoralgebras 79 F.Gesztesy&H.HoldenSolitonequationsandtheiralgebro-geometricsolutions 81 S.MukaiAnIntroductiontoinvariantsandmoduli 82 G.TourlakisLecturesinlogicandsettheoryI 83 G.TourlakisLecturesinlogicandsettheoryII 84 R.BaileyAssociationschemes 85 J.Carlson,S.Mu¨ller-Stach,&C.PetersPeriodmappingsandperioddomains 86 J.Duistermaat&J.KolkMultidimensionalrealanalysisI 87 J.Duistermaat&J.KolkMultidimensionalrealanalysisII 89 M.Golumbic&A.TrenkTolerancegraphs 90 L.HarperGlobalmethodsforcombinatorialisoperimetricproblems 91 I.Moerdijk&J.MrcunIntroductiontofoliationsandliegroupoids 92 J.Kollar,K.Smith&A.CortiRationalandnearlyrationalvarieties 93 D.ApplebaumLe´vyprocessesandstochasticcalculus 95 M.SchechterAnintroductiontononlinearanalysis iv P1:JZP 0521853680pre CB980/Chavel February15,2006 11:6 CharCount=0 RIEMANNIAN GEOMETRY A Modern Introduction Second Edition ISAAC CHAVEL DepartmentofMathematics TheCityCollegeofthe CityUniversityofNewYork v cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press TheEdinburghBuilding,Cambridgecb22ru,UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521853682 ©CambridgeUniversityPress1994,2006 Thispublicationisincopyright.Subjecttostatutoryexceptionandtotheprovisionof relevantcollectivelicensingagreements,noreproductionofanypartmaytakeplace withoutthewrittenpermissionofCambridgeUniversityPress. Firstpublishedinprintformat 2006 isbn-13 978-0-511-21991-7 eBook(EBL) isbn-10 0-511-21991-1 eBook(EBL) isbn-13 978-0-521-85368-2 hardback isbn-10 0-521-85368-0 hardback isbn-13 978-0-521-61954-7 paperback isbn-10 0-521-61954-8 paperback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofurls forexternalorthird-partyinternetwebsitesreferredtointhispublication,anddoesnot guaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate. P1:JZP 0521853680pre CB980/Chavel February15,2006 11:6 CharCount=0 for HARRYERNESTRAUCH (1925–1979) vii P1:JZP 0521853680pre CB980/Chavel February15,2006 11:6 CharCount=0 viii