微分流形导论 : 第2版

Foreword

Acknowledgments

CHAPTER I

Differential Calculus

1. Categories

2. Finite Dimensional Vector Spaces

3. Derivatives and Composition of Maps

4. Integration and Tayior'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

l. 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

Oiretions on Vector Fields end 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 of Frobenius

1. 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

CHAPTER VII

Metrics

1. Definition and Functoriality

2. The Metric Group

3. Reduction to the Metric Group

4. Metric Tubular Neighborhoods

5. The Morse Lemma

6. The Riemannian Distance

7. The Canonical Spray

CHAPTER VIII

Integretion of Differential Forms

1. Sets of Measure 0

2. Change of Variables Formula

3. Orientation

4. The Measure Associated with a Differential Form

CHAPTER IX

Stokes' Theorem

1. Stokes' Theorem for a Rectangular Simplex

2. Stokes' Theorem on a Manifold

3. Stokes' Theorem with Singularities

CHAPTER X

Applications of Stokes' Theorem

1. The Maximal de Rham Cohomology

2. Volume forms and the Divergence

3. The Divergence Theorem

4. Cauchy's Theorem

5. The Residue Theorem

Bibliography

Index

《微分流形导论(第2版)(英文版)》是由世界图书出版公司出版的。 《微分流形导论(第2版)(英文版)》内容简介：This book is an outgrowth of my Introduction to Differentiable Manifolds (1962) and Differential Manifolds (1972). Both I and my publishers felt it worth while to keep available a brief introduction to differential manifolds.【作者简介】作者：（美国）朗（Serge Lang）