Index 285

inverted, 79, 83

Hessian, 133, 170

holomorphic, 121

homeomorphism, 29, 51, 249

homology, 65, 92, 148

homothety, 108, 129, 182, 216

homotopy, 65, 87, 146, 154, 256

horocycle, 206, 213

hyperbolic isometry, 207, 216, 226

hyperbolic matrix, 210

hyperbolic plane, 181, 217, 225

hyperboloid, 239, 241

hypercircle, see equidistant curve

ideal boundary, 185, 211, 213

index

Morse, 141

of a continuous map, 267

of a curve, 152, 243, 250, 258

of a vector field, 151, 259

inner product, 160

integral curve, 158

isometry, 31, 34

orientation reversing, 199

isometry group, 44, 183, 211, 224,

227

Jordan Curve Theorem, 51, 249,

256

Klein bottle, 7, 24, 72, 264

lattice, 45

length, 170, 175, 181, 234

lift, 145

limit circle, see horocycle

Liouville’s theorem, 125

local coordinates, 13, 109, 155, 163

manifold, 49, 70

complex, 122

Riemannian, 161, 172

smooth, 103, 109, 251

map, 60, 67, 69, 229

homology of, 100

metric, 17

metric space, 28

metrisable, 49

M¨ obius cap, 80, 82, 91, 250

M¨ obius strip, 6, 18, 77, 80

M¨ obius transformation, see

fractional linear

transformation

modular surface, 129

Morse function, 136

Morse lemma, 134

neutral geometry, see absolute

geometry

norm, 170

normal subgroup, 211

orientability, 77, 81, 84, 88, 99, 113,

252

and homology, 100

osculating circle, 241

parabolic isometry, 206, 216

parabolic matrix, 210

parallel, 2, 212, 214

parallel postulate, 187, 198, 213

parametric representation

of curves, 22

of surfaces, 16, 27

partition of unity, 164, 165, 169,

266

patch, 104, 106, 163

Peano curve, 244

pencil, 206

planar model, 8, 69, 71, 73, 80, 224

notation, 72

smooth structure, 120

Poincar´ e conjecture, 90, 138

Poincar´ e duality, 270

Poincar´ e-Hopf Index Formula, 259

polarisation identity, 40

positive definite, 160, 162

pretzel, see surface of genus two

projective plane, 10, 18, 38, 66, 71,

269

pseudosphere, 186, 190

quotient space, 11, 260

smooth structure, 116, 118, 227

refinement, 61

reflection, 36

regular point, 14, 131