Riemannian Geometry

512 Pages·2016·2.64 MB·English

by  Peter Petersen

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
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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.