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

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