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
obius band, 34
obius geometry, 11
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
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