光滑流形導論：英文(第2版)

1 Smooth Manifolds
Topological Manifolds
Smooth Structures
Examples of Smooth Manifolds
Manifolds with Boundary
Problems

2 Smooth Maps
Smooth Functions and Smooth Maps
Partitions of Unity
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 Fur ctors
Problems

4 Submersions，Immersions，and Embeddings
Maps of Constant Rank
Embeddings
Submersions
Smooth Covering Maps
Problems

5 Submanifolds
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

7 Lie Groups
Basic Definitions
Lie Group Homomorphisms
Lie Subgroups
Group Actions and Equivariant Maps
Problems

8 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 Tile 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 Symplectic 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
Ccnnectedness and Compactness
Homotopy and the Fundamental Group
Ccvering Maps
Appendix B Review of Linear Algebra
Vector Spaces
Linear Maps
The Determinant
lnner 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 Solutien Techniques
References
Notation Index
Subject Index