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。
- 前辅文
- Part I What is Geometry and Differential Geometry
- 1 WhatIsGeometry?.
- 1.1 Geometry as a logical system
- 1.2 Coordinatization of space
- 1.3 Space based on the group concept
- 1.4 Localization of geometry
- 1.5 Globalization
- 1.6 Connections in a fiber bundle
- 1.7 An application to biology
- 1.8 Conclusion
- 2 Differential Geometry
- 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
- 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
- 6 Exterior Differential Forms
- 6.1 Exterior differentiation
- 6.2 Differential systems
- 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
- 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.
