Riemannian Geometry and Geometric Analysis

702 Pages·2017·7.62 MB·English

by  Jürgen Jost (auth.)

Universitext
Jürgen Jost
Riemannian
Geometry and
Geometric
Analysis
Seventh Edition
Universitext
Universitext
Serieseditors
SheldonAxler
SanFranciscoStateUniversity
CarlesCasacuberta
UniversitatdeBarcelona
AngusMacIntyre
QueenMary,UniversityofLondon
KennethRibet
UniversityofCalifornia,Berkeley
ClaudeSabbah
Écolepolytechnique,CNRS,UniversitéParis-Saclay,Palaiseau
EndreSüli
UniversityofOxford
WojborA.Woyczyn´ski
CaseWesternReserveUniversity
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JuRrgen Jost
Riemannian Geometry
and Geometric Analysis
Seventh Edition
123
JuRrgenJost
MaxPlanckInstituteforMathematics
intheSciences
MaxPlanckSociety
Leipzig,Germany
ISSN0172-5939 ISSN2191-6675 (electronic)
Universitext
ISBN978-3-319-61859-3 ISBN978-3-319-61860-9 (eBook)
DOI10.1007/978-3-319-61860-9
LibraryofCongressControlNumber:2017950288
MathematicsSubjectClassification(2010):53B21,53L20,32C17,35I60,49-XX,58E20,57R15
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DedicatedtoShing-TungYau,
forsomanydiscussionsabout
mathematicsand Chineseculture
Preface
Riemanniangeometryischaracterized,andresearchisorientedtowardsandshaped
byconcepts(geodesics,connections,curvature,...)andobjectives,inparticularto
understandcertainclassesof(compact)Riemannianmanifoldsdefinedbycurvature
conditions (constant or positive or negative curvature, ...). By way of contrast,
geometricanalysisisaperhapssomewhatlesssystematiccollectionoftechniques,
for solving extremal problems naturally arising in geometry and for investigating
andcharacterizingtheirsolutions.Itturnsoutthatthetwofieldscomplementeach
other very well; geometric analysis offers tools for solving difficult problems in
geometry,and Riemanniangeometrystimulates progressin geometricanalysis by
settingambitiousgoals.
It is theaim ofthis bookto be a systematic and comprehensiveintroductionto
Riemanniangeometryandarepresentativeintroductiontothemethodsofgeometric
analysis.Itattemptsa synthesisofgeometricandanalyticmethodsin thestudyof
Riemannianmanifolds.
ThepresentworkistheseventheditionofmytextbookonRiemanniangeometry
and geometric analysis. It has developed on the basis of several graduate courses
I taught at the Ruhr-University Bochum and the University of Leipzig. The
presenteditioncontainsseveralnewadditions,forinstanceasectiononsymplectic
geometry or a treatment of the Bishop-Gromov volume growth theorem which
elucidates the geometric role of Ricci curvature. Also, many parts of the book
have been systematically reworked and rearranged, and some proofs have been
simplified.Naturally,Ihavealsoincludedseveralsmalleradditionsandcorrections.
Intheend,astheresultofallthesemodificationsandadditions,almostnopageof
thepreviouseditionhasbeenleftunchanged.
Letmenowbrieflydescribethecontents:
In the first chapter, I introduce the basic geometric concepts of Riemannian
geometry.Inparticular,theygiverisetooneofthefundamentalobjectsandtoolsof
Riemanniangeometry,theso-calledgeodesicswhicharedefinedaslocallyshortest
curves.Geodesicswillreappearprominentlyinseverallaterchapters.Here,Itreat
the existence of geodesics with two different methods, both of which are quite
importantingeometricanalysisingeneral.Thus,thereaderhastheopportunityto
vii
viii Preface
understandthebasicideasofthosemethodsinanelementarycontextbeforemoving
on to more difficult versionsin subsequentchapters. The first methodis based on
thelocalexistenceanduniquenessofgeodesicsandwillbeappliedagaininChap.9
for two-dimensionalharmonic maps. The second method is the heat flow method
thatgainedprominencethroughPerelman’ssolutionofthePoincaréconjectureby
theRicciflowmethod.
The second chapter introduces another fundamental concept, that of a vector
bundle.Besidesthemostbasicone,thetangentbundleofaRiemannianmanifold,
manyothervectorbundleswillappearinthisbook.Thestructuregroupofavector
bundleisaLiegroup,andIshallthereforeusethisopportunitytoalsodiscussLie
groupsandtheir infinitesimalversions,the Lie algebras.I also discuss symplectic
andspinstructures.
Thethirdchapterthenintroducesbasicconceptsandmethodsfromanalysis.In
particular, the Laplace-Beltrami operator is a fundamental object in Riemannian
geometry.I show theessential propertiesofits spectrumanddiscussrelationships
with the underlying geometry. The discussion then turns to the operation of the
Laplace operator on differential forms. The de Rham cohomology groups are
introduced together with the essential tools from elliptic PDE for treating these
groups.Theexistenceofharmonicformsrepresentingcohomologyclassesisproved
bothbya variationalmethod,therebyintroducinganotherofthebasic schemesof
geometricanalysis,andbytheheatflowmethod.Thelinearsettingofcohomology
classes allows us to understand some key ideas without the technical difficulties
ofnonlinearproblems.I alsodiscussthespectrumoftheLaplacianondifferential
forms.
The fourth chapter begins with fundamental geometric concepts. It treats the
generaltheoryofconnectionsandcurvature.Ialsointroduceimportantfunctionals
liketheYang-Millsfunctionalanditsproperties.TheBochnermethodisappliedto
thefirsteigenvalueoftheLaplacianandharmonic1-formsonmanifoldsofpositive
Ricci curvature, as an example of the interplay between geometry and analysis. I
alsodescribethemethodofLiandYauforobtainingeigenvalueestimatesthrough
gradientboundsforeigenfunctions.
The fifth chapter is devoted to the geometry of submanifolds, and it also
introducesaparticularlyimportantclassofsubmanifolds,theminimalones.
In the sixth chapter, I introduce Jacobi fields, prove the Rauch comparison
theorems for Jacobi fields and apply these results to geodesics. Then the global
geometryofspacesofnonpositivecurvatureisdeveloped.
Thesefirstsixchapterstreatthemoreelementaryandbasicaspectsofthesubject.
Theirresultswillbeusedintheremaining,moreadvancedchapters.
TheseventhchaptertreatsKählermanifoldsandsymmetricspacesasimportant
examplesofRiemannianmanifoldsindetail.
TheeigthchapterisdevotedtoMorsetheoryandFloerhomology.Thebasicidea
of Morse theory is that global invariants of a compact Riemannian manifold can
be recoveredfromthe localanalysis at the criticalpointsof a smooth functionon
thatmanifold.Floertheorythensubstantiallyrefinesthismethodbylookingatthe
Preface ix
relationsbetweencriticalpointsencodedin thegradientflow linesofthe function
thatconnectthem.
Intheninthchapter,harmonicmapsbetweenRiemannianmanifoldsaretreated.
I prove several existence theorems and apply them to Riemannian geometry. The
treatment uses an abstract approach based on convexity that should bring out the
fundamentalstructures.Inthischapter,arepresentativesampleoftechniquesfrom
geometricanalysisisdisplayed.
In the tenth chapter,harmonic maps from Riemann surfaces are discussed. We
encounter here the phenomenon of conformal invariance which makes this two-
dimensionalcasedistinctivelydifferentfromthehigherdimensionalone.
Riemannian geometry has become the mathematical language of theoretical
physics,whereastherigorousdemonstrationofmanyresultsintheoreticalphysics
requiresdeeptoolsfromnonlinearanalysis.Therefore,theeleventhchapterexplores
someconnectionsbetweenphysics,geometryandanalysis.Ittreatsvariationalprob-
lems from quantum field theory, in particular the Ginzburg–Landauand Seiberg–
Witten equations, and a mathematical version of the nonlinear supersymmetric
sigmamodel.Inmathematicalterms,the two-dimensionalharmonicmapproblem
is coupled with a Dirac field. The background material on spin geometry and
Dirac operatorsis alreadydevelopedin earlier chapters.The connectionsbetween
geometryandphysicsaredevelopedinmoregeneralityinmymonograph[236].
A guiding principle for this textbook was that the material in the main body
should be self-contained. The essential exception is that we use material about
Sobolev spaces and linear elliptic and parabolic PDEs without giving proofs.
This material is collected in Appendix A. Appendix B collects some elementary
topologicalresultsaboutfundamentalgroupsandcoveringspaces.
Also,incertainplacesinChap.8,Idonotpresentalltechnicaldetails,butrather
explain some points in a more informal manner, in order to keep the size of that
chapterwithinreasonablelimitsandnottolosethepatienceofthereaders.
Iemploybothcoordinate-freeintrinsicnotationsandtensornotationsdepending
onlocalcoordinates.Iusuallydevelopaconceptinbothnotationswhilesometimes
alternating in the proofs. Besides the fact that I am not a methodological purist,
reasons for often preferring the tensor calculus to the more elegant and concise
intrinsic one are the following. For the analytic aspects, one often has to employ
results about(elliptic) partial differentialequations(PDEs), and in order to check
thattherelevantassumptionslikeellipticityholdandinordertomakecontactwith
thenotationsusuallyemployedinPDEtheory,onehastowritedownthedifferential
equationinlocalcoordinates.Also,manifoldandimportantconnectionshavebeen
established between theoreticalphysics and our subject. In the physicalliterature,
usuallythetensornotationisemployed,andtherefore,familiaritywiththatnotation
isnecessaryforexploringthoseconnectionsthathavebeenfoundtobestimulating
forthedevelopmentofmathematics,orpromisetobesointhefuture.
As appendices to most of the sections, I have written paragraphswith the title
“Perspectives”. The aim of those paragraphs is to place the material in a broader
context and explain further results and directions without detailed proofs. The
materialofthesePerspectiveswillnotbeusedinthemainbodyofthetext.Similarly,
x Preface
after Chap.6, I have inserted a section entitled “A short survey on curvature and
topology”thatpresentsanaccountofmanyglobalresultsofRiemanniangeometry
not covered in the main text. At the end of each chapter, some exercises for the
readeraregiven.Afirstsetofbasicexercisessystematicallyexploresthegeometry
of the most basic examples, the spheres, the Euclidean and the hyperbolicspaces
and their quotients. They represent standard material, and therefore, instructors
should have an easy job going through them with their students. In any case,
every reader should go through those exercises very carefully, to test her or his
basic understandingand to develop computationalskills. Some of those exercises
are simply formulated as statements which the reader then is invited to verify.
These basic exercises are then followed by others, treating various aspects of the
material presented in the corresponding chapter. These exercises are of varying
difficulty,butmanyofthemarerelativelyeasy.Itrustthereadertobeofsufficient
perspicacity to understand my system of numbering and cross-references without
furtherexplanation.
Overtheyears,Ihavediscussedthetopicspresentedinthisbookwithsomany
friends, colleagues and students that if I tried to compile a list of people whom
I should thank I would surely miss and thereby offend many of them. Therefore,
I refrain from listing all the people who have helped me to develop, clarify or
understandthematerialpresentedinthisbook.ButIshouldcertainlymentionthat
AndreasSchäfersuppliedme withaverydetailedlistoftyposandcorrectionsfor
thepreviousedition,andIamverygratefultohimforhisvaluablehelpinimproving
my book. I also thank Wilderich Tuschmann for supplying several references on
the topologyof spaces of positive or nonnegativecurvature.I thank the European
ResearchCouncilforsupportingmyworkwiththeAdvancedGrantFP7-267087.
Sincethelasteditionofthisbookappeared,sadly,severaloftheprotagonistsof
the field have passed away. For me personally, the most severe losses are that of
myacademicteacher,StefanHildebrandt,andmycolleaguefor20years,Eberhard
Zeidler. Stefan Hildebrandtwas one of the pioneersof geometric analysis, due to
his fundamentalwork on minimalsurfacesandharmonicmappings(see [239] for
an obituary). Eberhard Zeidler had enthusiastically explored the deep connection
betweengeometryandtheoreticalhigh-energyphysics(see[240]foranobituary).
Also,thetwomathematiciansthatpioneeredglobalRiemanniangeometry,Marcel
Berger and Wilhelm Klingenberg (see [237]), whom I also counted among my
friends,arenolongerwithus.
This book, like its previous editions, is dedicated to Shing-Tung Yau, for his
profound contributions to geometric analysis and his enormous influence on the
developmentofthefield.
JürgenJost