微分几何基础

Foreword

Acknowledgments

PART Ⅰ

General Differential Theory，

CHAPTER Ⅰ

Oifferenlial Calculus

Categories

Topological Vector Spaces

Derivatives and Composition of Maps

Integration and Taylors Formula

The Inverse Mapping Theorem

CHAPTER Ⅱ

Manifolds

Atlases， Charts， Morphisms

Submanifolds， Immersions， Submersions

Partitions of Unity

Manifolds with Boundary

CHAPTER Ⅲ

Vector Bundles

Definition， Pull Backs

The Tangent Bundle

Exact Sequences of Bundles

Operations on Vector Bundles

Splitting of Vector Bundles

CHAPTER Ⅳ

Vector Fields and Differential Equations

Existence Theorem for Differential Equations .

Vector Fields， Curves， and Flows

Sprays

The Flow of a Spray and the Exponential Map

Existence of Tubular Neighborhoods

Uniqueness of Tubular Neighborhoods

CHAPTER Ⅴ

Operations on Vector Fields and Differential Forms

Vector Fields， Differential Operators， Brackets

Lie Derivative

Exterior Derivative

The Poincar Lemma

Contractions and Lie Derivative

Vector Fields and l-Forms Under Self Duality

The Canonical 2-Form

Darbouxs Theorem

CHAPTER Ⅵ

The Theorem of Frobenius

Statement of the Theorem

Differential Equations Depending on a Parameter

Proof of the Theorem

The Global Formulation

Lie Groups and Subgroups

PART Ⅱ

Metrics， Covariant Derivatives， and Riemannian Geometry

CHAPTER Ⅶ

Metrics

Definition and Functoriality

The Hilbert Group

Reduction to the Hilbert Group

Hilbertian Tubular Neighborhoods

The Morse-Palais Lemma

The Riemannian Distance

The Canonical Spray

CHAPTER Ⅷ

Covariant Derivatives and Geodesics.

Basic Properties

Sprays and Covariant Derivatives

Derivative Along a Curve and Parallelism

The Metric Derivative

More Local Results on the Exponential Map

Riemannian Geodesic Length and Completeness

CHAPTER Ⅸ

Curvature

The Riemann Tensor

Jacobi Lifts

Application of Jacobi Lifts to Texpx

Convexity Theorems

Taylor Expansions

CHAPTER Ⅹ

Jacobi Lifts and Tensorial Splitting of the Double Tangent Bundle

Convexity of Jacobi Lifts

Global Tubular Neighborhood of a Totally Geodesic Submanifold.

More Convexity and Comparison Results

Splitting of the Double Tangent Bundle

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

The Flow and the Tensorial Derivative

CHAPTER XI

Curvature and the Variation Formula

The Index Form， Variations， and the Second Variation Formula

Growth of a Jacobi Lift

The Semi Parallelogram Law and Negative Curvature

Totally Geodesic Submanifolds

Rauch Comparison Theorem

CHAPTER XII

An Example of Seminegative Curvature

Pos，，(R) as a Riemannian Manifold

The Metric Increasing Property of the Exponential Map

Totally Geodesic and Symmetric Submanifolds

CHAPTER XIII

Automorphisms and Symmetries.，

The Tensorial Second Derivative

Alternative Definitions of Killing Fields

Metric Killing Fields

Lie Algebra Properties of Killing Fields

Symmetric Spaces

Parallelism and the Riemann Tensor

CHAPTER XlV

Immersions and Submersions .

The Covariant Derivative on a Submanifoid

The Hessian and Laplacian on a Submanifold

The Covariant Derivative on a Riemhnnian Submersion .

The Hessian and Laplacian on a Riemannian Submersion

The Riemann Tensor on Submanifolds

The Riemann Tensor on a Riemannian Submersion

PART III

Volume Forms and Integration

CHAPTER XV

Volume Forms

Volume Forms and the Divergence

Covariant Derivatives

The Jacobian Determinant of the Exponential Map

The Hodge Star on Forms

Hodge Decomposition of Differential Forms

Volume Forms in a Submersion

Volume Forms on Lie Groups and Homogeneous Spaces

Homogeneously Fibered Submersions

CHAPTER XVI

Integration of Differential Forms

Sets of Measure 0

Change of Variables Formula

Orientation

The Measure Associated with a Differential Form

Homogeneous Spaces

CHAPTER XVII

Stokes Theorem

Stokes Theorem for a Rectangular Simplex

Stokes Theorem on a Manifold

Stokes Theorem with Singularities

CHAPTER XVIII

Applications of Stokes Theorem

The Maximal de Rham Cohomology

Mosers Theorem

The Divergence Theorem

The Adjoint of d for Higher Degree Forms

Cauchys Theorem

The Residue Theorem

APPENDIX

The Spectral Theorem，

Hilbert Space

Functionals and Operators

Hermitian Operators

Bibliography

Index

《微分几何基础(英文版)》介绍了微分拓扑、微分几何以及微分方程的基本概念。《微分几何基础(英文版)》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。目次：（第一部分）一般微分方程；微积分；流形；向量丛；向量域和微分方程；向量域和微分形式运算；Frobenius定理；（第二部分）矩阵、协变导数和黎曼几何：矩阵；协变导数和测地线；曲率；二重切线丛的张量分裂；曲率和变分公式；半负曲率例子；自同构和对称；浸入和浸没；（第三部分）体积形式和积分：体积形式；微分形式的积分；Stokes定理；Stokes定理的应用；谱理论。