Differential geometry and integral geometry （英文版）（微分几何与积分几何）

样章

作者：

陈省身

定价：

79.00 元

版面字数：

310千字

开本：

16开

装帧形式：

精装

版次：

最新版次
印刷时间：

2016年

ISBN：

978-7-04-046518-1

物料号：

46518-00

出版时间：

2016-10-24

读者对象：

学术著作

一级分类：

自然科学

二级分类：

数学与统计

三级分类：

几何分析

分析学包括微分学与积分学。在几何中，也有对应的微分几何和积分几何。本书介绍几何的这两个方面，包含四部分。第一部分内容是1971年陈省身在国际数学家大会上所作的1小时报告，向学生和非专家介绍微分几何当时的整体面貌。作者首先简要介绍历史概况，概述了一些基本概念和工具，并介绍了当时微分几何的五个分支：正曲率流形、曲率和欧拉特征、极小子流形、等距映射、全纯映射。第二部分系统地介绍了积分几何。第三部分为微分流形，是作者在1959年微分几何正成为数学的一个主要领域时所写的讲义，该讲义给出了微分流形和微分几何的平稳和快速的引入，给当时的数学界送来一股清新之风。第四部分为积分几何，提供了一个高效但通俗易懂的介绍，并给出了对整个数学的全局的观点。例如，除了介绍在欧氏空间中的积分几何的标准主题，它还讨论了齐次空间的积分几何。

本书不仅对初学者非常有价值，对科研工作者也是很好的补充阅读材料。

目录
- 前辅文
- Part I What is Geometry and Differential Geometry
  - 1 WhatIsGeometry?.
    - 1.1 Geometry as a logical system; Euclid
    - 1.2 Coordinatization of space; Descartes
    - 1.3 Space based on the group concept; Klein’s Erlanger Programm
    - 1.4 Localization of geometry;Gauss and Riemann
    - 1.5 Globalization; topology
    - 1.6 Connections in a fiber bundle; Elie Cartan
    - 1.7 An application to biology
    - 1.8 Conclusion
  - 2 Differential Geometry; Its Past and Its Future
    - 2.1 Introduction
    - 2.2 The development of some fundamental notions and tools
    - 2.3 Formulation of some problems with discussion of related results
      - 2.3.1 Riemannian manifolds whose sectional curvatures keep a
      - 2.3.2 Euler-Poincar´e characteristic.
      - 2.3.3 Minimal submanifolds
      - 2.3.4 Isometricmappings
      - 2.3.5 Holomorphic mappings
- Part II Lectures on Integral Geometry
  - 3 Lectures on Integral Geometry
    - 3.1 Lecture I
      - 3.1.1 Buffon’s needle problem
      - 3.1.2 Bertrand’s parabox
    - 3.2 Lecture II
    - 3.3 Lecture III
    - 3.4 Lecture IV
    - 3.5 LectureV
    - 3.6 LectureVI
    - 3.7 LectureVII
    - 3.8 LectureVIII
- Part III DifferentiableManifolds
  - 4 Multilinear Algebra
    - 4.1 The tensor (or Kronecker) product
    - 4.2 Tensor spaces
    - 4.3 Symmetry and skew-symmetry; Exterior algebra
    - 4.4 Duality in exterior algebra
    - 4.5 Inner product
  - 5 DifferentiableManifolds
    - 5.1 Definition of a differentiable manifold
    - 5.2 Tangent space
    - 5.3 Tensor bundles
    - 5.4 Submanifolds; Imbedding of compactmanifolds
  - 6 Exterior Differential Forms
    - 6.1 Exterior differentiation
    - 6.2 Differential systems; Frobenius’s theorem
    - 6.3 Derivations and anti-derivations
    - 6.4 Infinitesimal transformation
    - 6.5 Integration of differential forms
    - 6.6 Formula of Stokes
  - 7 Affine Connections
    - 7.1 Definition of an affine connection: covariant differential
    - 7.2 The principal bundle
    - 7.3 Groups of holonomy
    - 7.4 Affine normal coordinates
  - 8 Riemannian Manifolds
    - 8.1 The parallelismof Levi-Civita
    - 8.2 Sectional curvature
    - 8.3 Normal coordinates; Existence of convex neighbourhoods
    - 8.4 Gauss-Bonnet formula
    - 8.5 Completeness
    - 8.6 Manifolds of constant curvature
- Part IV Lecture Notes on Differentiable Geometry
  - 9 Review of Surface Theory
    - 9.1 Introduction
    - 9.2 Moving frames.
    - 9.3 The connection form
    - 9.4 The complex structure
  - 10 Minimal Surfaces
    - 10.1 General theorems.
    - 10.2 Examples
    - 10.3 Bernstein -Osserman theorem
    - 10.4 Inequality on Gaussian curvature.
  - 11 Pseudospherical Surface
    - 11.1 General theorems.
    - 11.2 B¨acklund’s theorem.