內容簡介
《曲面幾何學》揭示了幾何和拓撲之間的相互關係，為廣大讀者介紹了現代幾何的基本概況。書的開始介紹了三種簡單的面，歐幾里得面、球面和雙曲平面。運用等距同構群的有效機理，並且將這些原理延伸到常曲率的所有可以用合適的同構方法獲得的曲面。緊接著主要是從拓撲和群論的觀點出發，講述一些歐幾里得曲面和球面的分類，較為詳細地討論了一些有雙曲曲面。由於常曲率曲面理論和現代數學有很大的聯繫，該書是一本理想的學習幾何的入門教程，用最簡單易行的方法介紹了曲率、群作用和覆蓋面。這些理論融合了許多經典的概念，如，複分析、微分幾何、拓撲、組合群論和比較熱門的分形幾何和弦理論。《曲面幾何學》內容自成體系，在預備知識部分包括一些線性代數、微積分、基本群論和基本拓撲。
作者簡介
作者：（澳大利亞）史迪威（JohnStillwell）
目錄
Preface
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 CoveringisometryGroup
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
5.9 Discussion
Chapter 6.Paths andgeodesics
6.1 Topological Classification of Surfaces
6.2 Geometric Classification of Surfaces
6.3 Paths and Homotopy
6.4 Lifting Paths and Lifting Homotopies
6.5 The Fundamental Group
6.6 Generators and Relations for the Fundamental Group
6.7 Fundamental Group and Genus
6.8 Closed Geodesic Paths
6.9 Classification of Closed Geodesic Paths
6.10 Discussion
Chapter 7.Planar and Spherical TesseUations
7.1 Symmetrictessellations
7.2 Conditions for a Polygon to Be a Fundamental Region
7.3 The Triangle Tessellations
7.4 Poincarr's Theorem for Compact Polygons
7.5 Discussion
Chapter 8.Tessellations of Compact Surfaces
8.1 Orbifolds and Desingularizations
8.2 From Desingularization to Symmetric Tessellation
8.3 Desingularizations as (Branched) Coverings
8.4 Some Methods of Desingularization
8.5 Reduction to a Permutation Problem
8.6 Solution of the Permutation Problem
8.7 Discussion
References
Index
