Hyperbolic GeometryThoroughly updated, featuring new material on important topics such as hyperbolic geometry in higher dimensions and generalizations of hyperbolicity Includes full solutions for all exercises Successful first edition sold over 800 copies in North America |
Contents
The Basic Spaces | xi |
12 The Riemann Sphere C | 6 |
13 The Boundary at Infinity of H I | 16 |
The General mőbius Group | 21 |
22 Transitivity Properties of mŐb | 28 |
23 The Cross Ratio | 34 |
24 Classification of MŐbius Transformations | 37 |
25 A Matrix Representation | 40 |
36 Isometries | 101 |
37 Metric Properties of H dH | 106 |
Planar Models of the Hyperbolic Plane | 115 |
42 A General Construction | 128 |
Convexity Area and Trigonometry | 143 |
52 Hyperbolic Polygons | 152 |
53 The Definition of Hyperbolic Area | 162 |
54 Area and the GaussBonnet Formula | 167 |
26 Reflections | 46 |
27 The Conformality of Elements of Mőb | 51 |
28 Preserving H | 54 |
29 Transitivity Properties of MőbH | 60 |
210 The Geometry of the Action of MőbH | 63 |
Length and Distance in H | 71 |
32 The Element of Arclength on H | 78 |
33 Path Metric Spaces | 86 |
34 From Arclength to Metric | 90 |
35 Formulae for Hyperbolic Distance in H | 97 |
55 Applications of the GaussBonnet Formula | 172 |
56 Trigonometry in the Hyperbolic Plane | 179 |
Nonplanar models | 187 |
62 Higher Dimensional Hyperbolic Spaces | 207 |
Solutions to Exercises | 215 |
References | 263 |
List of Notation | 267 |
Index | 271 |
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Common terms and phrases
acts transitively az+b boundary at infinity C¹ path f circle with Euclidean closed half-planes completes the proof consider contained convex set cos(a cz+d defined definition distinct points element of arc-length element of Möb(H endpoints at infinity equation Euclidean centre Euclidean circle Euclidean line Euclidean radius Exercise fixed points function given Hence holomorphic disc holomorphic homeomorphism Homeo(C homeomorphism hyperbolic area hyperbolic centre hyperbolic distance hyperbolic element hyperbolic geometry hyperbolic length hyperbolic line segment hyperbolic polygon hyperbolic rays hyperbolic triangle hyperboloid model Im(z interior angle intersection invariant isometry law of cosines length(ƒ line in H line segment joining locally finite collection loxodromic metric space Möb(D Möb+ Möbius transformation Note perpendicular piecewise C¹ path plane Poincaré disc model positive imaginary axis proof of Proposition quadratic form Re(z Riemann sphere sin(a sinh(u subgroup subset triples of distinct vertices