- 现代几何学方法和应用(第2卷英文版)
- - 作者：B.A.Dubrovin//A.T.Fomenko//S.P.Novikov|责编:刘慧//高蓉
  - 出版社：世界图书出版公司
  - ISBN：9787506201339
  - 出版日期：1999/11/01
  - 页数：430
- 售价：39.6

内容大纲
本卷主要介绍流形的几何学和拓扑学，包括同伦群、纤维丛、动态系统和叶状结构以及近年来拓扑方法在现代理论物理中的应用。
流形例子，基本问题、涉及流形函数的基本事实及典型的光滑映射，映射度、子流形的相交指数与应用，流形的定向、基本群覆盖空间，同伦群，光滑纤维丛，动态系统和叶状结构，高维变分问题解的整体结构。
作者介绍
目录
CHAPTER 1 Examples of Manifolds
  1. The concept of a manifold
    1.1. Definition of a manifold
    1.2. Mappings of manifolds; tensors on manifolds
    1.3. Embeddings and immersions of manifolds. Manifolds with boundary
  2. The simplest examples of manifolds
    2.1. Surfaces in Euclidean space. Transformation groups as manifolds
    2.2. Projective spaces
    2.3. Exercises
  3. Essential facts from the theory of Lie groups
    3.1. The structure of a neighbourhood of the identity of a Lie group.The Lie algebra of a Lie group. Semisimplicity
    3.2. The concept of a linear representation. An example of a non-matrix Lie group
  4. Complex manifolds
    4.1. Definitions and examples
    4.2. Riemann surfaces as manifolds
  5. The simplest homogeneous spaces
    5.1. Action of a group on a manifold
    5.2. Examples of homogeneous spaces
    5.3. Exercises
  6. Spaces of constant curvature (symmetric spaces)
    6.1. The concept of a symmetric space
    6.2. The isometry group of a manifold. Properties of its Lie algebra
    6.3. Symmetric spaces of the first and second types
    6.4. Lie groups as symmetric spaces
    6.5. Constructing symmetric spaces. Examples
    6.6. Exercises
  7. Vector bundles on a manifold
    7.1. Constructions involving tangent vectors
    7.2. The normal vector bundle on a submanifold
CHAPTER 2 Foundational Questions. Essential Facts Concerning Functions on a Manifold. Typical Smooth Mappings
  8. Partitions of unity and their applications
    8.1. Partitions of unity
    8.2. The simplest applications of partitions of unity. Integrals over a manifold and the general Stokes formula
    8.3. Invariant metrics
  9. The realization of compact manifolds as surfaces in RN
  10. Various properties of smooth maps of manifolds
    10.1. Approximation of continuous mappings by smooth ones
    10.2. Sard's theorem
    10.3. Transversal regularity
    10.4. Morse functions
  11. Applications of Sard's theorem
    11.1. The existence of embeddings and immersions
    11.2. The construction of Morse functions as height functions
    11.3. Focal points
CHAPTER 3 The Degree of a Mapping. The Intersection Index of Submanifolds.Applications
  12. The concept of homotopy
    12.1. Definition of homotopy. Approximation of continuous maps and homotopies by smooth ones
    12.2. Relative homotopies
  13. The degree of a map
    13.1. Definition of degree

    13.2. Generalizations of the concept of degree
    13.3. Classification of homotopy classes of maps from an arbitrary manifold to a sphere
    13.4. The simplest examples
  14. Applications of the degree of a mapping
    14.1. The relationship between degree and integral
    14.2. The degree of a vector field on a hypersurface
    14.3. The Whitney number. The Gauss-Bonnet formula
    14.4. The index of a singular point of a vector field
    14.5. Transverse surfaces of a vector field. The Poincar6-Bendixson theorem
  15. The intersection index and applications
    15.1. Definition of the intersection index
    15.2. The total index of a vector field
……
CHAPTER 4 Orientability of Manifolds. The Fundamental Group.Covering Spaces (Fibre Bundles with Discrete Fibre)
CHAPTER 5 Homotopy Groups
CHAPTER 6 Smooth Fibre Bundles
CHAPTER 7 Some Examples of Dynamical Systems and Foliations on Manifolds
CHAPTER 8 The Global Structure of Solutions of Higher-Dimensional Variational Problems
Bibliography
Index

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