微分流形与黎曼几何引论

I.IntroductiontoManifolds

1.PreliminaryCommentsonRn1

2.RnandEuclideanSpace4

3.TopologicalManifolds6

4.FurtherExamplesofManifolds.CuttingandPastingIl

5.AbstractManifolds.SomeExamples14

II.FunctionsofSeveralVariablesandMappings

1.DifferentiabilityforFunctionsofSeveralVariables20

2.DifferentiabilityofMappingsandJacobians25

3.TheSpaceofTangentVectorsataPointofR"29

4.AnotherDefinitionofTa(Rn)32

5.VectorFieldsonOpenSubsetsofRn36

6.TheInverseFunctionTheorem41

7.TheRankofaMapping46

III.OifferentiableManifoldsandSubmanifolds

1.TheDefinitionofaDifferentiableManifold52

2.FurtherExamples59

3.DifferentiableFunctionsandMappings65

4.RankofaMapping,Immersions68

5.Submanifolds74

6.LieGroups80

7.TheActionofaLieGrouponaManifold.TransformationGroups87

8.TheActionofaDiscreteGrouponaManifold93

9.CoveringManifolds98

IV.VectorFieldsonaManifold

1.TheTangentSpaceataPointofaManifold104

2.VectorFields113

3.One-ParameterandLocalOne-ParameterGroupsActingonaManifold119

4.TheExistenceTheoremforOrdinaryDifferentialEquations127

5.SomeExamplesofOne-ParameterGroupsActingonaManifold135

6.One-ParameterSubgroupsofLieGroups142

7.TheLieAlgebraofVectorFieldsonaManifold146

8.FrobeniussTheorem153

9.HomogeneousSpaces160

V.TensorsandTensorFieldsonManifolds

I.TangentCovectors171

CovectorsonManifolds172

CovectorFieldsandMappings174

2.BilinearForms.TheRiemannianMetric177

3.RiemannianManifoldsasMetricSpaces181

4.PartitionsofUnity186

SomeApplicationsofthePartitionofUnity188

5.TensorFields192

TensorsonaVectorSpace192

TensorFields194

MappingsandCovariantTensors195

TheSymmetrizingandAlternatingTransformations196

6.MultiplicationofTensors199

MultiplicationofTensorsonaVectorSpace199

MultiplicationofTensorFields201

ExteriorMultiplicationofAlternatingTensors202

TheExteriorAlgebraonManifolds206

7.OrientationofManifoldsandtheVolumeElement207

8.ExteriorDifferentiation212

AnApplicationtoFrobeniussTheorem216

VI.IntegrationonManifolds

1.IntegrationinR"DomainsofIntegration223

BasicPropertiesoftheRiemannIntegral224

2.AGeneralizationtoManifolds229

IntegrationonRiemannianManifolds232

3.IntegrationonLieGroups237

4.ManifoldswithBoundary243

5.StokessTheoremforManifolds251

6.HomotopyofMappings.TheFundamentalGroup258

HomotopyofPathsandLoops.TheFundamentalGroup259

7.SomeApplicationsofDifferentialForms.ThedeRhamGroups265

TheHomotopyOperator268

8.SomeFurtherApplicationsofdeRhamGroups272

ThedeRhamGroupsofLieGroups276

9.CoveringSpacesandFundamentalGroup280

VII.DifferentiationonRiemannianManifolds

1.DifferentiationofVectorFieldsalongCurvesinR"289

TheGeometryofSpaceCurves292

CurvatureofPlaneCurves296

2.DifferentiationofVectorFieldsonSubmanifoldsofR"298

FormulasforCovariantDerivatives303

Vxp,YandDifferentiationofVectorFields305

3.DifferentiationonRiemannianManifolds308

ConstantVectorFieldsandParallelDisplacement314

4.AddendatotheTheoryofDifferentiationonaManifold316

TheCurvatureTensor316

TheRiemannianConnectionandExteriorDifferentialForms319

5.GeodesicCurvesonRiemannianManifolds321

6.TheTangentBundleandExponentialMapping.NormalCoordinates326

7.SomeFurtherPropertiesofGeodesics332

8.SymmetricRiemannianManifolds340

9.SomeExamples346

VIII.Curvature

1.TheGeometryofSurfacesinE3355

ThePrincipalCurvaturesataPointofaSurface359

2.TheGaussianandMeanCurvaturesofaSurface363

TheTheoremaEgregiumofGauss366

3.BasicPropertiesoftheRiemannCurvatureTensor371

4.CurvatureFormsandtheEquationsofStructure378

5.DifferentiationofCovariantTensorFields384

6.ManifoldsofConstantCurvature391

SpacesofPositiveCurvature394

SpacesofZeroCurvature396

SpacesofConstantNegativeCurvature397

REFERENCES403

INDEX411

　　本书是公认的微分流形最佳入门教材之一，自初版出版30多年以来，一直畅销不衰，被国外各大高校广泛采用为教材。本书对国内的教学也产生了深远的影响，北京大学、复旦大学和四川大学等名校长期将其作为主要参考书。本书的特点是对基础知识要求较少，只需有微积分、线性代数和少量拓扑学背景即可研读；而且在内容安排上力求循序渐进，透彻讲述基本概念，避免不必要的抽象和推广，并配合丰富且有详细解答的例子。非常好地降低了这门课程的学习难度。修订版在第2版色经做出较大调整的基础上，又做了一些小的但很重要的订正和更新。使本书更趋完美。　　这是一本非常好的微分流形入门书。全书从一些基本的微积分知识入手，然后一点点深入介绍，主要内容有：流形介绍、多变量函数和映射、微分流形和子流形、流形上的向量场、张量和流形上的张量场、流形上的积分法、黎曼流形上的微分法以及曲率。书后有难度适中的习题，全书配有很多精美的插图。　　本书非常适合初学者阅读，可作为数学系、物理系、机械系等理工科高年级本科生和研究生的教材。【作者简介】　　华盛顿大学圣路易斯分校数学系荣休教授。于1949年在密歇根大学获得博士学位，师出拓扑学大师、沃尔夫奖得主HasslerWhitney门下。除在华盛顿大学任教40多年外，他还在世界各地讲授微分流形，深受学生爱戴。