曲面几何学(英文版)

本书揭示了几何和拓扑 之间的相互关系，为广大 读者介绍了现代几何的基 本概况。书的开始介绍了 三种简单的面，欧几里得 面、球面和双曲平面。运 用等距同构群的有效机理 ，并且将这些原理延伸到 常曲率的所有可以用合适 的同构方法获得的曲面。 紧接着主要是从拓扑和群 论的观点出发，讲述一些 欧几里得曲面和球面的分 类，较为详细地讨论了一 些有双曲曲面。由于常曲 率曲面理论和现代数学有 很大的联系，该书是一本 理想的学习几何的入门教 程，用简单易行的方法 介绍了曲率、群作用和覆 盖面。这些理论融合了许 多经典的概念，如，复分 析、微分几何、拓扑、组 合群论和比较热门的分形 几何和弦理论。本书内容 自成体系，在预备知识部 分包括一些线性代数、微 积分、基本群论和基本拓 扑。书中不少例子和每章 节后面的练习可以帮助读 者好地理解书中知识。

Chapter 1.The Euclidean Plane 1.1 Approaches to Euclidean Geometry 1.2 Isometries 1.3 Rotations and Reflections 1.4 The Three Reflections Theorem 1.5 Orientation-Reversing Isometries 1.6 Distinctive Features of Euclidean Geometry 1.7 Discussion Chapter 2.Euclidean Surfaces 2.1 Euclid on Manifolds 2.2 The Cylinder 2.3 The Twisted Cylinder 2.4 The Torus and the Klein Bottle 2.5 Quotient Surfaces 2.6 A Nondiscontinuous Group 2.7 Euclidean Surfaces 2.8 Covering a Surface by the Plane 2.9 The Covering Isometry Group 2.10 Discussion Chapter 3.The Sphere 3.1 The Sphere S2 in R3 3.2 Rotations 3.3 Stereographic Projection 3.4 Inversion and the Complex Coordinate on the Sphere 3.5 Reflections and Rotations as Complex Functions 3.6 The Antipodal Map and the Elliptic Plane 3.7 Remarks on Groups, Spheres and Projective Spaces 3.8 The Area of a Triangle 3.9 The Regular Polyhedra 3.10 Discussion Chapter 4.The Hyperbolic Plane 4.1 Negative Curvature and the Half-Plane 4.2 The Half-Plane Model and the Conformal Disc Model 4.3 The Three Reflections Theorem 4.4 Isometries as Complex Fnctions 4.5 Geometric Description of Isometries 4.6 Classification of Isometries 4.7 The Area of a Triangle 4.8 The Projective Disc Model 4.9 Hyperbolic Space 4.10 Discussion Chapter 5.Hyperbolic Surfaces 5.1 Hyperbolic Surfaces and the Killing-Hopf Theorem 5.2 The Pseudosphere 5.3 The Punctured Sphere 5.4 Dense Lines on the Punctured Sphere 5.5 General Construction of Hyperbolic Surfaces from Polygons 5.6 Geometric Realization of Compact Surfaces 5.7 Completeness of Compact Geometric Surfaces 5.8 Compact Hyperbolic Surfaces