本书介绍了微分拓扑、微分几何以及微分方程的基本概念。本书的基本思想源于作者早期的《微分和黎曼流形》，但重点却从流形的一般理论转移到微分几何，增加了不少新的章节。这些新的知识为Banach和Hilbert空间上的无限维流形做准备，但一点都不觉得多余，而优美的证明也让读者受益不浅。在有限维的例子中，讨论了高维微分形式，继而介绍了Stokes定理和一些在微分和黎曼情形下的应用。给出了Laplacian基本公式，展示了其在浸入和浸没中的特征。书中讲述了该领域的一些主要基本理论，如：微分方程的存在定理、唯一性、光滑定理和向量域流，包括子流形管状邻域的存在性的向量丛基本理论，微积分形式，包括经典2-形式的辛流形基本观点，黎曼和伪黎曼流形协变导数以及其在指数映射中的应用，Cartan-Hadamard定理和变分微积分第一基本定理。

Foreword

Acknowledgments

PART I

General Differential Theory

CHAPTER I

Differential Calculus

1.Categories

2.Topological Vector Spaces

3.Derivatives and Composition of Maps

4.Integration and Taylor's Formula

5.The Inverse Mapping Theorem

CHAPTER II

Manifolds

1.Atlases, Charts, Morphisms

2.Submanifolds, Immersions, Submersions

3.Partitions of Unity

4.Manifolds with Boundary

CHAPTER III

Vector Bundles

1.Definition, Pull Backs

2.The Tangent Bundle

3.Exact Sequences of Bundles

4.Operations on Vector Bundles

5.Splitting of Vector Bundles

CHAPTER IV

Vector Fields and Differential Equations

1.Existence Theorem for Differential Equations

2.Vector Fields, Curves, and Flows

3.Sprays

4.The Flow of a Spray and the Exponential Map

5.Existence of Tubular Neighborhoods

6.Uniqueness of Tubular Neighborhoods

CHAPTER V

Operations on Vector Fields and Differential Forms

1.Vector Fields, Differential Operators, Brackets

2.Lie Derivative

3.Exterior Derivative

4.The Poincare Lemma.

5.Contractions and Lie Derivative

6.Vector Fields and l-Forms Under Self Duality

7.The Canonical 2-Form

8.Darboux's Theorem

CHAPTER VI

The Theorem ol Frobenius

I.Statement of the Theorem

2.Differential Equations Depending on a Parameter

3.Proof of the Theorem

4.The Global Formulation

5.Lie Groups and Subgroups

PART II

Metrics, Covariant Derivatives, and Riemannian Geometry

CHAPTER VII

Metrics

1.Definition and Functoriality

2.The Hilbert Group

3.Reduction to the Hiibert Group

4.Hilbertian Tubular Neighborhoods

5.The Morse-Palais Lemma

6.The Riemannian Distance

7.The Canonical Spray

CHAPTER VIII

Covarlent Derivatives and Geodesics

1.Basic Properties

2.Sprays and Covariant Derivatives

3.Derivative Along a Curve and Parallelism

4.The Metric Derivative

5.More Local Results on the Exponential Map

6.Riemannian Geodesic Length and Completeness

CHAPTER IX

Curvature

1.The Riemann Tensor

2.Jacobi Lifts

3.Application of Jacobi Lifts to Texp

4.Convexity Theorems

5.Taylor Expansions

CHAPTER X

Jacobi Lifts and Tensorial Splitting of the Double Tangent Bundle

1.Convexity of Jacobi Lifts

2.Global Tubular Neighborhood of a Totally Geodesic Submanifold

3.More Convexity and Comparison Results

4.Splitting of the Double Tangent Bundle

5.Tensorial Derivative of a Curve in TX and of the Exponential Map

6.The Flow and the Tensorial Derivative

CHAPTER XI

Curvature and the Variation Formula

1.The Index Form, Variations, and the Second Variation Formula

2.Growth of a Jacobi Lift.

3.The Semi Parallelogram Law and Negative Curvature

4.Totally Geodesic Submanifolds

5.Rauch Comparison Theorem

CHAPTER XII

An Example of Seminegative Curvature

1.Pos,,(R) as a Riemannian Manifold

2.The Metric Increasing Property of the Exponential Map

3.Totally Geodesic and Symmetric Submanifolds

CHAPTER XIII

Automorphisms and Symmetries

1.The Tensorial Second Derivative

2.Alternative Definitions of Killing Fields

3.Metric Killing Fields

4.Lie Algebra Properties of Killing Fields

5.Symmetric Spaces

6.Parallelism and the Riemann Tensor

CHAPTER XIV

Immersions and Submersions

1.The Covariant Derivative on a Submanifold.

2.The Hessian and Laplacian on a Submanifold

3.The Covariant Derivative on a Riemannian Submersion

4.The Hessian and Laplacian on a Riemannian Submersion

5.The Riemann Tensor on Submanifolds

6.The Riemann Tensor on a Riemannian Submersion

PART III

Volume Forms and Integration

CHAPTER XV

Volume Forms

1.Volume Forms and the Divergence

2.Covariant Derivatives

3.The Jacobian Determinant of the Exponential Map

4.The Hodge Star on Forms

5.Hodge Decomposition of Differential Forms

6.Volume Forms in a Submersion

7.Volume Forms on Lie Groups and Homogeneous Spaces

8.Homogeneously Fibered Submersions

CHAPTER XVl

Integration of Differential Forms

1.Sets of Measure O

2.Change of Variables Formula

3.Orientation

4.The Measure Associated with a Differential Form

5.Homogeneous Spaces

CHAPTER XVII

Stokes' Theorem

1.Stokes' Theorem for a Rectangular Simplex

2.Stokes' Theorem on a Manifold

3.Stokes' Theorem with Singularities

CHAPTER XVIII

Applications of Stokes' Theorem

1.The Maximal de Rham Cohomology

2.Moser's Theorem

3.The Divergence Theorem

4.The Adjoint of d for Higher Degree Forms

5.Cauchy's Theorem

6.The Residue Theorem

APPENDIX

The Spectral Theorem

1.Hilbert Space

2.Functionals and Operators

3.Hermitian Operators

Bibliography

Index