黎曼几何和几何分析（第6版）

图书标准信息

• 作者 [德]约斯特（Jost J.） 著
• 出版社 世界图书出版公司
• 出版时间 2015-01
• 版次 6
• ISBN 9787510084447
• 定价 99.00元
• 装帧 平装
• 开本 24开
• 纸张 胶版纸
• 页数 611页
• 正文语种 英语

【内容简介】

　　Riemanniangeometryischaracterized,andresearchisorientedtowardsandshapedbyconcepts(geodesics,connections,curvature,...)andobjectives,inparticulartounderstandcertainclassesof(compact)Riemannianmanifoldsdefinedbycurvatureconditions(constantorpositiveornegativecurvature,...).Bywayofcontrast,geometricanalysisisaperhapssomewhatlesssystematiccollectionoftechniques,forsolvingextremalproblemsnaturallyarisingingeometryandforinvestigatingandcharacterizingtheirsolutions.Itturnsoutthatthetwofieldscomplementeachotherverywell;geometricanalysisofferstoolsforsolvingdifficultproblemsingeometry,andRiemanniangeometrystimulatesprogressingeometricanalysisbysettingambitiousgoals.
　　ItistheaimofthisbooktobeasystematicandcomprehensiveintroductiontoRiemanniangeometryandarepresentativeintroductiontothemethodsofgeometricanalysis.ItattemptsasynthesisofgeometricandanalyticmethodsinthestudyofRiemannianmanifolds.
　　ThepresentworkisthesixtheditionofmytextbookonRiemanniangeometryandgeometricanalysis.IthasdevelopedonthebasisofseveralgraduatecoursesItaughtattheRuhr~UniversityBochumandtheUniversityofLeipzig.ThemainnewfeatureofthepresenteditionisasystematicpresentationofthespectrumoftheLaplaceoperatoranditsrelationwiththegeometryoftheunderlyingRiemannianmarufold.Naturally,Ihavealsoincludedseveralsmalleradditionsandminorcorrections(forwhichIamgratefultoseveralreaders).Moreover,theorganizationofthechaptershasbeensystematicallyrearranged.

【目录】

1RiemannianManifolds
1.1ManifoldsandDifferentiableManifolds
1.2TangentSpaces
1.3Submanifolds
1.4RiemannianMetrics
1.5ExistenceofGeodesicsonCompactManifolds
1.6TheHeatFlowandtheExistenceofGeodesics
1.7ExistenceofGeodesicsonCompleteManifolds
ExercisesforChapter1

2LieGroupsandVectorBundles
2.1VectorBundles
2.2IntegralCurvesofVectorFields.LieAlgebras
2.3LieGroups
2.4SpinStructures
ExercisesforChapter2

3TheLaplaceOperatorandHarmonicDifferentialForms
3.1TheLaplaceOperatoronFunctions
3.2TheSpectrumoftheLaplaceOperator
3.3TheLaplaceOperatoronForms
3.4RepresentingCohomologyClassesbyHarmonicForms
3.5Generalizations
3.6TheHeatFlowandHarmonicForms
ExercisesforChapter3

4ConnectionsandCurvature
4.1ConnectionsinVectorBundles
4.2MetricConnections.TheYang—MillsFunctional
4.3TheLevi—CivitaConnection
4.4ConnectionsforSpinStructuresandtheDiracOperator
4.5TheBochnerMethod
4.6EigenvalueEstimatesbytheMethodofLi—Yau
4.7TheGeometryofSubmanifolds
4.8MinimalSubmanifolds
ExercisesforChapter4

5GeodesicsandJacobiFields
5.1FirstandsecondVariationofArcLengthandEnergy
5.2JacobiFields
5.3ConjugatePointsandDistanceMinimizingGeodesics
5.4RiemannianManifoldsofConstantCurvature
5.5TheRauchComparisonTheoremsandOtherJacobiFieldEstimates
5.6GeometricApplicationsofJacobiFieldEstimates
5.7ApproximateFundamentalSolutionsandRepresentationFormulas
5.8TheGeometryofManifoldsofNonpositiveSectionalCurvature
ExercisesforChapter5
AShortSurveyonCurvatureandTopology

6SymmetricSpacesandKahlerManifolds
6.1ComplexProjectiveSpace
6.2KahlerManifolds
6.3TheGeometryofSymmetricSpaces
6.4SomeResultsabouttheStructureofSymmetricSpaces
6.5TheSpaceSl（n，IR）／SO（n，IR）
6.6SymmetricSpacesofNoncompactType
ExercisesforChapter6

7MorseTheoryandFloerHomology
7.1Preliminaries：AimsofMorseTheory
7.2ThePalais—SmaleCondition，ExistenceofSaddlePoints
7.3LocalAnalysis
7.4LimitsofTrajectoriesoftheGradientFlow
7.5FloerCondition，TransversalityandZ2—Cohomology
7.6OrientationsandZ—homology
7.7Homotopies
7.8Graphflows
7.9Orientations
7.10TheMorseInequalities
7.11ThePalais—SmaleConditionandtheExistenceofClosedGeodesics
ExercisesforChapter7

8HarmonicMapsbetweenRiemannianManifolds
8.1Definitions
8.2FormulasforHarmonicMaps.TheBochnerTechnique
8.3TheEnergyIntegralandWeaklyHarmonicMaps
8.4HigherRegularity
8.5ExistenceofHarmonicMapsforNonpositiveCurvature
8.6RegularityofHarmonicMapsforNonpositiveCurvature
8.7HarmonicMapUniquenessandApplications
ExercisesforChapter8

9HarmonicMapsfromRiemannSurfaces
9.1Two—dimensionalHarmonicMappings
9.2TheExistenceofHarmonicMapsinTwoDimensions
9.3RegularityResults
ExercisesforChapter9

10VariationalProblemsfromQuantumFieldTheory
10.1TheGinzburg—LandauFunctional
10.2TheSeiberg—WittenFunctional
10.3Dirac—harmonicMaps
ExercisesforChapter10

ALinearEllipticPartialDifferentialEquations
A.1SobolevSpaces
A.2LinearEllipticEquations
A.3LinearParabolicEquations
BFundamentalGroupsandCoveringSpaces
Bibliography
Index