Index 313

linear reflections, 229

Liouville’s Theorem, 179

longitude, 105

loop, 46

manifold, 28, 59

map, 24

mapping class group, 260, 263

marked loop, 190

marked parallelograms, 251

matrix group, 30

maximal atlas, 136

Maximum Principle, 170

metric, 22

metric completeness, 154

metric space, 22

M¨ obius band, 34

M¨ obius geometry, 11

M¨ obius transformation, 11, 115

modular group, 14, 242, 244, 254

moduli space, 16, 256, 263

negative genus, 258

nonorientable, 35

nonorientable surfaces, 41

norm, 87

octagon, 4

octagon surface, 9, 34

one-to-one, 21

onto, 21

open set, 23

orientable, 35

overlap function, 136

pairs of pants, 263

pants decomposition, 265

parabolic isometry, 130

parallelogram tiling, 68

path connectivity, 48

path homotopic, 70

pathification, 32

Peano curve, 51

periodic billiard paths, 219

permutation group, 43

Pick’s Theorem, 92

Poincar´ e homology sphere, 62

Poincar´ e Recurrence Theorem, 219

Poincar´ e Uniformization Theorem,

201

Polygon Dissection Theorem, 96

polygonal billiards, 213

power series, 172

product space, 51

projective plane, 34, 59, 69

projective space, 59

properly discontinuous action, 224

Pythagorean Theorem, 90

quadratic irrationals, 248

quaternions, 62

real linear transformation, 121

removable singularity, 171

Riemann Mapping Theorem, 188,

199

Riemann surface, 195

Riemann surface covering, 196

Riemannian cover, 152

Riemannian geometries, 9

Riemannian isometry, 135

Riemannian length, 139

Riemannian manifold, 152

Riemannian metric, 134, 139

Riemannian surface, 139

right-angled hexagon, 150

saddle connection, 227

Schwarz–Christoffel

Transformation, 188

scissors congruent, 287

sectors, 207

set, 21

simple connectivity, 70

Sinai Robins, 92

Small Picard Theorem, 202

smooth curve, 133, 137

smooth surface, 136

sphere, 1, 58

spherical isometries, 103

spherical metrics, 103

spherical triangle, 107

square torus, 1

stereographic projection, 111, 181

surface, 21, 27

tangent plane, 103, 137