光滑流形导论

光滑流形导论

(美) 李 (Lee,J.M.) , 著

出版社:世界图书出版公司北京公司

年代:2008

定价:75.0

书籍简介:

本书是一部学习光滑流形的入门教材。旨在让学生熟练的掌握和运用在数学和科学研究中的所需要的流形这个重要的工具。目次: 光滑流形;光滑映射;切向量;向量场;向量丛;余切丛;浸入,嵌入;子流形;李群行为;近似理论;张量;微分形式;定向;流形上的积分;De Rham 同调;de Rham定理;积分曲线与流;积分流形;李群和李代数。

书籍目录:

Preface

SmoothManifolds

TopologicalManifolds

TopologicalPrcpertiescfManifolds

SmoothStructures

ExamplesofSmoothManifolds

ManifoldswithBoundary

Problems

SmoothMaps

SmoothFunctionsandSmoothMaps

LieGroups

SmoothCoveringMaps

ProperMaps

PartitionsofUnity

Problems

3TangentVectors

TangentVectors

Pushforwards

ComputationsinCoordinates

TangentVectorstoCurves

AlternativeDefinitionsoftheTangentSpace

Problems

4VectorFields

TheTangentBundle

VectorFieldsonManifolds

LieBrackets

TheLieAlgebraofaLieGroup

Problems

5VectorBundles

VectorBundles

LocalandGlobalSectionsofVectorBundles

BundleMaps

CategoriesandFunctors

Problems

6TheCotangentBundle

Covectors

TangentCovectorsonManifolds

TheCotangentBundle

TheDifferentialofaFunction

Pullbacks

LineIntegrals

ConservativeCovectorFields

Problems

Submersions,Immersions,andEmbeddings

MapsofConstantRank

TheInverseFunctionTheoremandItsFriends

Constant-RankMapsBetweenManifolds

Submersions

Problems

Submanifolds

EmbeddedSubmanifolds

LevelSets

ImmersedSubmanifolds

RestrictingMapstoSubmanifolds

VectorFieldsandCovectorFieldsonSubmanifolds

LieSubgroups

VectorSubbundles

Problems

LieGroupActions

GroupActions

EquivariantMaps

ProperActions

QuotientsofManifoldsbyGroupActions

CoveringManifolds

HomogeneousSpaces

Applications

Problems

10EmbeddingandApproximationTheorems

SetsofMeasureZeroinManifolds

TheWhitneyEmbeddingTheorem

TheWhitneyApproximationTheorems

Problems

11Tensors

TheAlgebraofTensors

TensorsandTensorFieldsonManifolds

SymmetricTensors

RiemannianMetrics

Problems

12DifferentialForms

TheGeometryofVolumeMeasurement

TheAlgebraofAlternatingTensors

TheWedgeProduct

DifferentialFormsonManifolds

ExteriorDerivatives

SymplecticForms

Problems

13Orientations

OrientationsofVectorSpaces

OrientationsofManifolds

TheOrientationCovering

OrientationsofHypersurfaces

BoundaryOrientations

TheRiemannianVolumeForm

HypersurfacesinRiemannianManifolds

Problems

14IntegrationonManifolds

IntegrationofDifferentialFormsonEuclideanSpace

IntegrationonManifolds

StokessTheorem

ManifoldswithCorners

IntegrationonRiemannianManifolds

IntegrationonLieGroups

Densities

Problems

15DeRhamCohomology

ThedeRhamCohomologyGroups

HomotopyInvariance

TheMayer-VietorisTheorem

Computations

Problems

16ThedeRhamTheorem

SingularHomology

SingularCohomology

SmoothSingularHomology

ThedeRhamTheorem

Problems

17IntegralCurvesandFlows

IntegralCurves

GlobalFlows

TheFundamentalTheoremonFlows

CompleteVectorFields

RegularPointsandSingularPoints

Time-DependentVectorFields

ProofoftheODETheorem

Problems

18LieDerivatives

TheLieDerivative

CommutingVectorFields

LieDerivativesofTensorFields

ApplicationstoGeometry

ApplicationstoSymplecticManifolds

Problems

19IntegralManifoldsandFoliations

TangentDistributions

InvolutivityandDifferentialForms

TheFrobeniusTheorem

ApplicationstoPartialDifferentialEquations

Foliations

Problems

20LieGroupsandTheirLieAlgebras

One-ParameterSubgroups

TheExponentialMap

TheClosedSubgroupTheorem

TheAdjointRepresentation

LieSubalgebrasandLieSubgroups

NormalSubgroups

TheFundamentalCorrespondence

Problems

Appendix:ReviewofPrerequisites

Topology

LinearAlgebra

Calculus

References

Index

内容摘要:

  本书是一部介绍光滑流形的入门教材(全英文版)。该书是针对已经对一般拓扑、基本群、覆盖空间以及基本的线性代数与实分析有较好掌握的本科生和研究生。旨在让学生和相关的工作人员熟练地掌握和运用流形这个重要的数学工具。本书主要介绍了光滑结构,切向量和余向量,向量丛,李导数,浸入和嵌入式子流形,李群和李代数。在讲述上运用图形以及直观的讨论使得内容尽可能的清晰易懂,更重要的是讲述如何用几何的方法思考抽象概念;同时,现代数学方法提供的有力工具得到了充分展示。本书还提供了一些很重要的流形能够提供的几何结构的例子。

书籍规格:

书籍详细信息
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出版地北京出版单位世界图书出版公司北京公司
版次1版印次1
定价(元)75.0语种英文
尺寸14装帧平装
页数印数 1000

书籍信息归属:

光滑流形导论是世界图书出版公司北京公司于2008.03出版的中图分类号为 O189.3 的主题关于 光滑化(数学)-流形-英文 的书籍。