出版社:世界图书出版公司北京公司
年代:2010
定价:65.0
本书介绍了微分拓扑、微分几何以及微分方程的基本概念。
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。
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出版地 | 北京 | 出版单位 | 世界图书出版公司北京公司 |
版次 | 修订本 | 印次 | 1 |
定价(元) | 65.0 | 语种 | 英文 |
尺寸 | 23 × 15 | 装帧 | 平装 |
页数 | 印数 | 1000 |
微分几何基础是世界图书出版公司北京公司于2010.2出版的中图分类号为 O186.1 的主题关于 微分几何-英文 的书籍。