光滑流形导论（第2版）
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作者（美）John M. Lee（J.M.李）
出版社世界图书出版公司
ISBN9787510098833
出版时间2016-05
装帧平装
开本16开
定价115元
货号23955287
上书时间2024-10-20
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导语摘要
该书是一本关于光滑流形理论的导论性研究生教材，旨在让学生们熟悉掌握将流形用在数学和科研工作中需要的工具，比如光滑结构、切向量和余向量、向量丛、陷入和嵌入的子流形、张量、微分形式、de Rham上同调、向量场、流量、叶状结构、李导数、李群、李代数等。充分利用现代数学提供的强大的工具的同时，书中采用尽可能具体的研究方法, 选取了各种图像，并对用几何思维考虑抽象概念进行了直观的讨论。
作者简介
John M. Lee（J.M.李,美国）是国际知名学者，在数学界享有盛誉。本书凝聚了作者多年科研和教学成果，适用于科研工作者、高校教师和研究生。
目录
1 Smooth Manifolds
Topological Manifolds
Smcoth Structures
Examples of Smeoth Manifolds
Manifolds with Boundary
Problems
2 Smooth Maps
Smooth Functions and Smooth Maps
Partitions of Unily
Problems
3 Tangent Vectors
Tangent Vectors
The Differential of a Smooth Map
Computations in Coordinates
The Tangent Bundle
Velocity Vectors of Curves
Alternative Definitions of the Tangent Space
Categories and Functors
Problems
4 Submersions, Immersions, and Embeddings
Maps of Constant Rank
Embeddings
Submersicns
Smooth Covering Maps
Problems
5 Submanifoids
Embedded Submanifolds
Immersed Submanifolds
Restricting Maps to Submanifolds
The Tangent Space to a Submanifold
Submanifolds with Boundary
Problems
6 Sard's Theorem
Sets of Measure Zero
Sard's Theorem
The Whitney Embedding Theorem
The Whitney Approximation Theorems
Transversality
Problems
Lie Groups
Basic Definitions
Lie Group Homomorphisms
Lie Subgroups
Group Actions and Equivariant Maps
Problems
Vector Fields
Vector Fields on Manifolds
Vector Fields and Smooth Maps
Lie Brackets
The Lie Algebra of a Lie Group
Problems
9 Integral Curves and Flows
Integral Curves
Flows
Flowouts
Flows and Flowouts on Manifolds with Boundary
Lie Derivatives
Commuting Vector Fields
Time-Dependent Vector Fields
First-Order Partial Differential Equations
Problems
10 Vector Bundles
Vector Bundles
Local and Global Sections of Vector Bundles
Bundle Homomorphisms
Subbundles
Fiber Bundles
Problems
11 The Cotangent Bundle
Covectors
The Differential of a Function
Pullbacks of Covector Fields
Line Integrals
Conservative Covector Fields
Problems
12 Tensors
Multilinear Algebra
Symmetric and Alternating Tensors
Tensors and Tensor Fields on Manifolds
Problems
13 Riemannian Metrics
Riemannian Manifolds
The Riemannian Distance Function
The Tangent-Cotangent Isomorphism
Pseudo-Riemannian Metrics
Problems
14 Differential Forms
The Algebra of Alternating Tensors
Differential Forms on Manifolds
Exterior Derivatives
Problems
15 Orientations
Orientations of Vector Spaces
Orientations of Manifolds
The Riemannian Volume Form
Orientations and Covering Maps
Problems
16 Integration on Manifolds
The Geometry of Volume Measurement
Integration of Differential Forms
Stokes's Theorem
Manifolds with Corners
Integration on Riemannian Manifolds
Densities
Problems
17 De Rham Cohomology
The de Rham Cohomology Groups
Homotopy Invariance
The Mayer-Vietoris Theorem
Degree Theory
Proof of the Mayer-Vietoris Theorem
Problems
18 The de Rham Theorem
Singular Homology
Singular Cohomology
Smooth Singular Homology
The de Rham Theorem
Problems
19 Distributions and Foliations
Distributions and Involutivity
The Frobenius Theorem
Foliations
Lie Subalgebras and Lie Subgroups
Overdetermined Systems of Partial Differential Equations
Problems
20 The Exponential Map
One-Parameter Subgroups and the Exponential Map
The Closed Subgroup Theorem
Infinitesimal Generators of Group Actions
The Lie Correspondence
Normal Subgroups
Problems
21 Quotient Manifolds
Quotients of Manifolds by Group Actions
Covering Manifolds
Homogeneous Spaces
Applications to Lie Theory
Problems
22 Symplectie Manifolds
Symplectic Tensors
Symplectic Structures on Manifolds
The Darboux Theorem
Hamiltonian Vector Fields
Contact Structures
Nonlinear First-Order PDEs
Problems
Appendix A Review of Topology
Topological Spaces
Subspaces, Products, Disjoint Unions, and Quotients
Connectedness and Compactness
Homotopy and the Fundamental Group
Covering Maps
Appendix B Review of Linear Algebra
Vector Spaces
Linear Maps
The Determinant
Inner Products and Norms
Direct Products and Direct Sums
Appendix C Review of Calculus
Total and Partial Derivatives
Multiple Integrals
Sequences and Series of Functions
The Inverse and Implicit Function Theorems
Appendix D Review of Differential Equations
Existence, Uniqueness, and Smoothness
Simple Solution Techniques
References
Notation Index
Subject Index
内容摘要
该书是一本关于光滑流形理论的导论性研究生教材，旨在让学生们熟悉掌握将流形用在数学和科研工作中需要的工具，比如光滑结构、切向量和余向量、向量丛、陷入和嵌入的子流形、张量、微分形式、de Rham上同调、向量场、流量、叶状结构、李导数、李群、李代数等。充分利用现代数学提供的强大的工具的同时，书中采用尽可能具体的研究方法, 选取了各种图像，并对用几何思维考虑抽象概念进行了直观的讨论。
主编推荐
John M. Lee（J.M.李,美国）是国际知名学者，在数学界享有盛誉。本书凝聚了作者多年科研和教学成果，适用于科研工作者、高校教师和研究生。
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