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。

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