黎曼几何和几何分析

图书标准信息

• 作者 [德]约斯特 著
• 出版社 世界图书出版公司
• 出版时间 2008-03
• 版次 1
• ISBN 9787506291927
• 定价 68.00元
• 装帧 平装
• 开本 24开
• 纸张 胶版纸
• 页数 566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.

【目录】

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
Appendix
Bibliography
Index