光滑流形导论(第2版)(英文版)

李著的《光滑流形导论》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、deRham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。

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