黎曼几何和几何分析：第4版

1.FoundationalMaterial

1.1ManifoldsandDifferentiableManifolds

1.2TangentSpaces

1.3Submanifolds

1.4RiemannianMetrics

1.5VectorBundles

1.6IntegralCurvesofVectorFields.LieAlgebras

1.7LieGroups

1.8SpinStructures

ExercisesforChapter1

2.DeRhamCohomologyandHarmonicDifferentialForms

2.1TheLaplaceOperator

2.2RepresentingCohomologyClassesbyHarmonicForms

2.3Generalizations

ExercisesforChapter2

3.ParallelTransport,Connections,andCovariantDerivatives

3.1ConnectionsinVectorBundles

3.2MetricConnections.TheYang-MillsFunctional

3.3TheLevi-CivitaConnection

3.4ConnectionsforSpinStructuresandtheDiracOperator

3.5TheBochnerMethod

3.6TheGeometryofSubmanifolds.MinimalSubmanifolds

ExercisesforChapter3

4.GeodesicsandJacobiFields

4.11stand2ndVariationofArcLengthandEnergy

4.2JacobiFields

4.3ConjugatePointsandDistanceMinimizingGeodesics

4.4RiemannianManifoldsofConstantCurvature

4.5TheRauchComparisonTheoremsandOtherJacobiFieldEstimates

4.6GeometricApplicationsofJacobiFieldEstimates

4.7ApproximateFundamentalSolutionsandRepresentationFormulae

4.8TheGeometryofManifoldsofNonpositiveSectionalCurvature

ExercisesforChapter4

AShortSurveyonCurvatureandTopology

5.SymmetricSpacesandKahlerManifolds

5.1ComplexProjectiveSpace

5.2KahlerManifolds

5.3TheGeometryofSymmetricSpaces

5.4SomeResultsabouttheStructureofSymmetricSpaces

5.5TheSpaceSI(n,R)/SO(n,R)

5.6SymmetricSpacesofNoncompactTypeasExamplesofNonpositivelyCurvedRiemannianManifolds

ExercisesforChapter5

6.MorseTheoryandFloerHomology

6.1Preliminaries:AimsofMorseTheory

6.2Compactness:ThePalais-SmaleConditionandtheExistenceofSaddlePoints

6.3LocalAnalysis:NondegeneracyofCriticalPoints,MorseLemma,StableandUnstableManifolds

6.4LimitsofTrajectoriesoftheGradientFlow

6.5TheMorse-Smale-FloerCondition:TransversalityandZ2-Cohomology

6.6OrientationsandZ-homology

6.7Homotopies

6.8Graphflows

6.9Orientations

6.10TheMorseInequalities

6.11ThePalais-SmaleConditionandtheExistenceofClosedGeodesics

ExercisesforChapter6

7.VariationalProblemsfromQuantumFieldTheory

7.1TheGinzburg-LandauFunctional

7.2TheSeiberg-WittenFunctional

ExercisesforChapter7

8.HarmonicMaps

8.1Definitions

8.2TwodimensionalHarmonicMappingsandHolomorphicQuadraticDifferentials

8.3TheExistenceofHarmonicMapsinTwoDimensions

8.4DefinitionandLowerSemicontinuityoftheEnergyIntegral

8.5WeaklyHarmonicMaps.RegularityQuestions

8.6HigherRegularity

8.7FormulaeforHarmonicMaps.TheBochnerTechnique

8.8HarmonicMapsintoManifoldsofNonpositiveSectionalCurvature:Existence

8.9HarmonicMapsintoManifoldsofNonpositiveSectionalCurvature:Regularity

8.10HarmonicMapsintoManifoldsofNonpositiveSectionalCurvature:UniquenessandOtherproperties

ExercisesforChapter8

AppendixA:LinearEllipticPartialDifferentialEquation

A.1SobolevSpaces

A.2ExistenceandRegularityTheoryforSolutionsofLinearEllipticEquations

AppendixB:FundamentalGroupsandCoveringSpaces.

Bibliography

Index

　　本书是一部值得一读的研究生教材(全英文版)，内容主要涉及黎曼几何基本定理的研究，如霍奇定理、Rauch比较定理、Lyusternik和Fet定理调和映射的存在性等，书中还有当代数学研究领域中的最热门论题，有些内容则是首次出现在教科书中。本书各章均附有习题。