Graduate Texts in Mathematics Peter Petersen Riemannian Geometry Third Edition Graduate Texts in Mathematics 171 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 Peter Petersen Riemannian Geometry Third Edition 123 PeterPetersen DepartmentofMathematics UniversityofCalifornia,LosAngeles LosAngeles,CA,USA ISSN0072-5285 ISSN2197-5612 (electronic) GraduateTextsinMathematics ISBN978-3-319-26652-7 ISBN978-3-319-26654-1 (eBook) DOI10.1007/978-3-319-26654-1 LibraryofCongressControlNumber:2015960754 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerScience+BusinessMediaNewYork1998 ©SpringerScience+BusinessMedia,LLC2006 ©SpringerInternationalPublishingAG2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Printedonacid-freepaper SpringerInternational PublishingAGSwitzerlandispartofSpringerScience+Business Media(www. springer.com) To mywife, Laura Preface ThisbookisintendedasacomprehensiveintroductiontoRiemanniangeometry.The readerisassumedtohavebasicknowledgeofstandardmanifoldtheory,including the theoryof tensors, forms,and Lie groups.At times it is also necessary to have some familiarity with algebraic topology and de Rham cohomology.Specifically, werecommendthatthereaderbefamiliarwithtextssuchas[15,72]or[97,vol.1]. Onmywebpage,therearelinkstolecturenotesonthesetopicsaswellasclassical differentialgeometry(see[90]and[89]).Itisalsohelpfulifthereaderhasanodding acquaintance with ordinary differential equations. For this, a text such as [74] is morethansufficient.Morebasicprerequisitesarerealanalysis,linearalgebra,and some abstract algebra. Differential geometry is and always has been an “applied discipline” within mathematics that uses many other parts of mathematics for its ownpurposes. Most of the material generally taught in basic Riemannian geometry as well as several more advanced topics is presented in this text. The approach we have takenoccasionallydeviatesfromthestandardpath.Alongsidetheusualvariational approach, we have also developed a more function-oriented methodology that likewiseusesstandardcalculustogetherwithtechniquesfromdifferentialequations. Our motivation for this treatment has been that examples become a natural and integral part of the text rather than a separate item that is sometimes minimized. Another desirable by-product has been that one actually gets the feeling that HessiansandLaplaciansareintimatelyrelatedtocurvatures. Thebookisdividedintofourparts: PartI:Tensorgeometry,consistingofchapters1,2,3,and4 PartII:Geodesicanddistancegeometry,consistingofchapters5,6,and7 PartIII:GeometryàlaBochnerandCartan,consistingofchapters8,9,and10 PartIV:Comparisongeometry,consistingofchapters11and12 There are significant structural changes and enhancementsin the third edition, sochaptersnolongercorrespondtothoseofthefirsttwoeditions.Weofferabrief outlineofeachchapterbelow. Chapter 1 introduces Riemannian manifolds, isometries, immersions, and sub- mersions. Homogeneous spaces and covering maps are also briefly mentioned. vii viii Preface There is a discussion on varioustypes of warped products.This allows us to give bothanalyticandgeometricdefinitionsofthebasicconstantcurvaturegeometries. TheHopffibrationasaRiemanniansubmersionisalsodiscussedinseveralplaces. Finally,thereisasectionontensornotation. Chapter 2 discusses both Lie and covariant derivatives and how they can be used to define several basic concepts such as the classical notions of Hessian, Laplacian, and divergence on Riemannian manifolds. Iterated derivatives and abstractderivationsarediscussedtowardtheendandusedlaterinthetext. Chapter 3 develops all of the important curvature concepts and discusses a few simple properties. We also develop several important formulas that relate curvature and the underlying metric. These formulas can be used in many places asareplacementforthesecondvariationformula. Chapter 4 is devoted to calculating curvatures in several concrete situations such as spheres, product spheres, warped products, and doubly warped products. This is used to exhibit several interesting examples. In particular, we explain how the Riemannian analogue of the Schwarzschild metric can be constructed. Thereis a new section that explainswarped productsin generaland howthey are characterized.Thisis an importantsection forlater developmentsas itleadsto an interestingcharacterizationofbothlocalandglobalconstantcurvaturegeometries fromboththewarpedproductandconformalviewpoint.WehaveasectiononLie groups.Heretwoimportantexamplesofleftinvariantmetricsarediscussedaswell asthegeneralformulasforthecurvaturesofbiinvariantmetrics.Itisalsoexplained howsubmersionscanbeusedtocreatenewexampleswithspecialfocusoncomplex projectivespace. Thereare also some generalcommentson how submersionscan beconstructedusingisometricgroupactions. Chapter 5 further develops the foundational topics for Riemannian manifolds. Theseincludethefirstvariationformula,geodesics,Riemannianmanifoldsasmet- ric spaces, exponentialmaps, geodesic completeness versus metric completeness, andmaximaldomainsonwhichtheexponentialmapisanembedding.Thechapter includesa detailed discussion of the propertiesof isometries. This naturally leads to the classification of simply connected space forms. At a more basic level, we obtain metric characterizationsof Riemannian isometries and submersions. These areused to showthatthe isometrygroupis a Lie groupandto givea proofofthe slicetheoremforisometricgroupactions. Chapter 6 contains three more foundational topics: parallel translation, Jacobi fields, and the second variation formula. Some of the classical results we prove here are the Hadamard-Cartan theorem, Cartan’s center of mass construction in nonpositivecurvatureandwhyitshowsthatthefundamentalgroupofsuchspacesis torsion-free,Preissman’stheorem,Bonnet’sdiameterestimate,andSynge’slemma. Attheendofthechapter,wecovertheingredientsneededfortheclassicalquarter pinched sphere theorem including Klingenberg’s injectivity radius estimates and Berger’sproofofthistheorem.Spheretheoremsarerevisitedinchapter12. Chapter7focusesonmanifoldswithlowerRiccicurvaturebounds.We discuss volumecomparisonanditsuses.TheseincludeproofsofhowPoincaréandSobolev constants can be bounded and theorems about restrictions on fundamentalgroups Preface ix for manifolds with lower Ricci curvature bounds. The strong maximum principle forcontinuousfunctionsisdeveloped.Thisresultisfirstusedinawarm-upexercise toproveCheng’smaximaldiametertheorem.WethenproceedtocovertheCheeger- Gromoll splitting theorem and its consequences for manifolds with nonnegative Riccicurvature. Chapter8coversvariousaspectsofsymmetriesonmanifoldswithemphasison Killingfields.HerethereisafurtherdiscussiononwhytheisometrygroupisaLie group.TheBochnerformulasforKillingfieldsarecoveredaswellasadiscussion on how the presence of Killing fields in positive sectional curvature can lead to topologicalrestrictions.ThelatterisafairlynewareainRiemanniangeometry. Chapter 9 explains both the classical and more recent results that arise from the Bochnertechnique.We startwith harmonic1-formsasBochnerdidand move on to general forms and other tensors such as the curvature tensor. We use an approachthatconsiderablysimplifiesmanyofthetensorcalculationsinthissubject (see, e.g., the first and second editions of this book). The idea is to consistently use how derivations act on tensors instead of using Clifford representations. The Bochnertechniquegivesmanyoptimalboundsonthetopologyofclosedmanifolds with nonnegative curvature. In the spirit of comparison geometry, we show how Betti numbers of nonnegatively curved spaces are bounded by the prototypical compact flat manifold: the torus. More generally, we also show how the Bochner techniquecanbeusedtocontrolthetopologywithmoregeneralcurvaturebounds. Thisrequiresalittle moreanalysis,butisafascinatingapproachthathasnotbeen presentedinbookformyet. The importance of the Bochner technique in Riemannian geometry cannot be sufficientlyemphasized.Itseemsthattime andagain,whenpeopleleastexpectit, newimportantdevelopmentscomeoutofthisphilosophy. Chapter 10 develops part of the theory of symmetric spaces and holonomy. The standard representations of symmetric spaces as homogeneous spaces or via Lie algebras are explained. There are several concrete calculations both specific and more general examples to get a feel for how curvatures behave. Having done this, we define holonomy for general manifolds and discuss the de Rham decomposition theorem and several corollaries of it. In particular, we show that holonomyirreduciblesymmetricspacesareEinsteinandthattheircurvatureshave thesamesignastheEinsteinconstant.Thistheoremandtheexamplesareusedto indicatehowonecanclassifysymmetricspaces.Finally,wepresentabriefoverview ofhowholonomyandsymmetricspacesarerelatedtotheclassificationofholonomy groups.Thisisused,togetherwithmostofwhathasbeenlearneduptothispoint, togivetheGallotandMeyerclassificationofcompactmanifoldswithnonnegative curvatureoperator. Chapter 11 focuseson the convergencetheoryof metric spaces and manifolds. First,weintroducethemostgeneralformofconvergence:Gromov-Hausdorffcon- vergence.Thisconceptisoftenusefulinmanycontextsasawayofgettingaweak formofconvergence.Therealobjecthere isto figureoutwhatweakconvergence impliesinthepresenceofstrongersideconditions.Thereisasectionwithaquick overviewofHölderspaces,Schauder’sellipticestimates,andharmoniccoordinates.