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曲面几何学 英文版PDF|Epub|txt|kindle电子书版本下载
- JohnStillwell著 著
- 出版社： 世界图书北京出版公司
- ISBN：9787510005312
- 出版时间：2010
- 标注页数：216页
- 文件大小：10MB
- 文件页数：230页
- 主题词：曲面论－英文
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图书目录
Chapter 1．The Euclidean Plane1
1.1 Approaches to Euclidean Geometry1
1.2 Isometries2
1.3 Rotations and Reflections5
1.4 The Three Reflections Theorem9
1.5 Orientation-Reversing Isometries11
1.6 Distinctive Features of Euclidean Geometry14
1.7 Discussion18
Chapter 2．Euclidean Surfaces21
2.1 Euclid on Manifolds21
2.2 The Cylinder22
2.3 The Twisted Cylinder25
2.4 The Torus and the Klein Bottle26
2.5 Quotient Surfaces29
2.6 A Nondiscontinuous Group33
2.7 Euclidean Surfaces34
2.8 Covering a Surface by the Plane36
2.9 The Covering Isometry Group39
2.10 Discussion41
Chapter 3．The Sphere45
3.1 The Sphere S2 in R345
3.2 Rotations48
3.3 Stereographic Projection50
3.4 Inversion and the Complex Coordinate on the Sphere52
3.5 Reflections and Rotations as Complex Functions56
3.6 The Antipodal Map and the Elliptic Plane60
3.7 Remarks on Groups,Spheres and Projective Spaces63
3.8 The Area of a Triangle65
3.9 The Regular Polyhedra67
3.10 Discussion69
Chapter 4．The Hyperbolic Plane75
4.1 Negative Curvature and the Half-Plane75
4.2 The Half-Plane Model and the Conformal Disc Model80
4.3 The Three Reflections Theorem85
4.4 Isometries as Complex Functions88
4.5 Geometric Description of Isometries92
4.6 Classification of Isometries96
4.7 The Area of a Triangle99
4.8 The Projective Disc Model101
4.9 Hyperbolic Space105
4.10 Discussion108
Chapter 5．Hyperbolic Surfaces111
5.1 Hyperbolic Surfaces and the Killing-Hopf Theorem111
5.2 The Pseudosphere112
5.3 The Punctured Sphere113
5.4 Dense Lines on the Punctured Sphere118
5.5 General Construction of Hyperbolic Surfaces from Polygons122
5.6 Geometric Realization of Compact Surfaces126
5.7 Completeness of Compact Geometric Surfaces129
5.8 Compact Hyperbolic Surfaces130
5.9 Discussion132
Chapter 6．Paths and Geodesics135
6.1 Topological Classification of Surfaces135
6.2 Geometric Classification of Surfaces138
6.3 Paths and Homotopy140
6.4 Lifting Paths and Lifting Homotopies143
6.5 The Fundamental Group145
6.6 Generators and Relations for the Fundamental Group147
6.7 Fundamental Group and Genus153
6.8 Closed Geodesic Paths154
6.9 Classification of Closed Geodesic Paths156
6.10 Discussion160
Chapter 7．Planar and Spherical Tessellations163
7.1 Symmetric Tessellations163
7.2 Conditions for a Polygon to Be a Fundamental Region167
7.3 The Triangle Tessellations172
7.4 Poincaré's Theorem for Compact Polygons178
7.5 Discussion182
Chapter 8．Tessellations of Compact Surfaces185
8.1 Orbifolds and Desingularizations185
8.2 From Desingularization to Symmetric Tessellation189
8.3 Desingularizations as(Branched) Coverings190
8.4 Some Methods of Desingularization194
8.5 Reduction to a Permutation Problem196
8.6 Solution of the Permutation Problem198
8.7 Discussion201
References203
Index207