光滑流形導論(英文)

Preface
Smooth Manifolds
Topological Manifolds
Topological Prcperties cf Manifolds
Smooth Structures
Examples of Smooth Manifolds
Manifolds with Boundary
Problems
Smooth Maps
Smooth Functions and Smooth Maps
Lie Groups
Smooth Covering Maps
Proper Maps
Partitions of Unity
Problems
3 Tangent Vectors
Tangent Vectors
Pushforwards
Computations in Coordinates
Tangent Vectors to Curves
Alternative Definitions of the Tangent Space
Problems
4 Vector Fields
The Tangent Bundle
Vector Fields on Manifolds
Lie Brackets
The Lie Algebra of a Lie Group
Problems
5 Vector Bundles
Vector Bundles
Local and Global Sections of Vector Bundles
Bundle Maps
Categories and Functors
Problems
6 The Cotangent Bundle
Covectors
Tangent Covectors on Manifolds
The Cotangent Bundle
The Differential of a Function
Pullbacks
Line Integrals
Conservative Covector Fields
Problems
Submersions, Immersions, and Embeddings
Maps of Constant Rank
The Inverse Function Theorem and Its Friends
Constant-Rank Maps Between Manifolds
Submersions
Problems
Submanifolds
Embedded Submanifolds
Level Sets
Immersed Submanifolds
Restricting Maps to Submanifolds
Vector Fields and Covector Fields on Submanifolds
Lie Subgroups
Vector Subbundles
Problems
Lie Group Actions
Group Actions
Equivariant Maps
Proper Actions
Quotients of Manifolds by Group Actions
Covering Manifolds
Homogeneous Spaces
Applications
Problems
10 Embedding and Approximation Theorems
Sets of Measure Zero in Manifolds
The Whitney Embedding Theorem
The Whitney Approximation Theorems
Problems
11 Tensors
The Algebra of Tensors
Tensors and Tensor Fields on Manifolds
Symmetric Tensors
Riemannian Metrics
Problems
12 Differential Forms
The Geometry of Volume Measurement
The Algebra of Alternating Tensors
The Wedge Product
Differential Forms on Manifolds
Exterior Derivatives
Symplectic Forms
Problems
13 Orientations
Orientations of Vector Spaces
Orientations of Manifolds
The Orientation Covering
Orientations of Hypersurfaces
Boundary Orientations
The Riemannian Volume Form
Hypersurfaces in Riemannian Manifolds
Problems
14 Integration on Manifolds
Integration of Differential Forms on Euclidean Space
Integration on Manifolds
Stokes’’s Theorem
Manifolds with Corners
Integration on Riemannian Manifolds
Integration on Lie Groups
Densities
Problems
15 De Rham Cohomology
The de Rham Cohomology Groups
Homotopy Invariance
The Mayer-Vietoris Theorem
Computations
Problems
16 The de Rham Theorem
Singular Homology
Singular Cohomology
Smooth Singular Homology
The de Rham Theorem
Problems
17 Integral Curves and Flows
Integral Curves
Global Flows
The Fundamental Theorem on Flows
Complete Vector Fields
Regular Points and Singular Points
Time-Dependent Vector Fields
Proof of the ODE Theorem
Problems
18 Lie Derivatives
The Lie Derivative
Commuting Vector Fields
Lie Derivatives of Tensor Fields
Applications to Geometry
Applications to Symplectic Manifolds
Problems
19 Integral Manifolds and Foliations
Tangent Distributions
Involutivity and Differential Forms
The Frobenius Theorem
Applications to Partial Differential Equations
Foliations
Problems
20 Lie Groups and Their Lie Algebras
One-Parameter Subgroups
The Exponential Map
The Closed Subgroup Theorem
The Adjoint Representation
Lie Subalgebras and Lie Subgroups
Normal Subgroups
The Fundamental Correspondence
Problems
Appendix: Review of Prerequisites
Topology
Linear Algebra
Calculus
References
Index