- 微分几何基础(英文版)
- - 作者：(美)朗|责编:刘慧//高蓉
  - 出版社：世界图书出版公司
  - ISBN：9787510005404
  - 出版日期：2010/02/01
  - 页数：535
- 售价：47.6

内容大纲
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。
作者介绍
目录
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
    §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
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.Posn(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 ⅩⅥ  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

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