284 Index bump function, 167 chain, 93, 96 chain complex, 92 for a triangulation, 94 chart, 104, 106, 109, 132 projection, 117 Christoffel symbol, 242 circle map, 145 classification, 29, 48, 53, 111 complex tori, 129 finitely generated abelian groups, 93 isometries of H2, 204 matrices, 210 non-orientable surfaces, 91 orientable surfaces, 85 surfaces, 253 coarsening, 57, 60, 70 compact, 30 complete, 30 cone, 2, 22 conformal, 122, 128, 173, 181 conjugacy, 209 conjugate points, 177 connected, 30, 49 covering space, 83, 84 critical point, 14, 131, 150, 169, 259 degenerate, 141, 154 cross cap, see M¨ obius cap cross-ratio, 193 crystallographic restriction, 45 curvature, 169, 182, 219, 233, 241 and angular excess, 238 Gaussian, 178, 241 principal, 241 cycle, 93, 96 cylinder, 2, 22 degree, 145, 151, 244, 267 in higher dimensions, 148 diffeomorphism, 51, 111, 249 and holomorphic equivalence, 125 differential equation, 158, 174, 234, 259, 266 dimension, 31, 32, 50 direct product, 47, 85, 160 disc model, 185, 197, 217, 232 ellipsoid, 3, 177 elliptic isometry, 205, 215 elliptic matrix, 210 elliptic plane, see projective plane equidistant curve, 208, 214, 226, 252 Euler characteristic, 65, 73, 254, 269 and Betti numbers, 98 and continuous maps, 269 and curvature, 228 and Morse functions, 139 and vector fields, 259 combinatorial definition, 56 in higher dimensions, 270 modifying, 76, 82, 140 of covering space, 84 Euler-Lagrange equation, 170, 174, 176 exact sequence, 92 factor space, see quotient space Fermi geodesic coordinates, 252 flow, 158 fractional linear transformation, 124, 183, 201 preserves cross-ratio, 193 fundamental domain, 10, 224, 227 fundamental group, 101, 256 Fundamental Theorem of Algebra, 246 Gauss-Bonnet Theorem, 229, 235, 239 genus, 99, 254 geodesic, 19, 20, 33, 170, 175, 186, 195, 206, 207, 226, 252, 268 geodesic flip, 45, 178 geodesic polar coordinates, 231, 236, 240 geographic coordinates, 16, 163 geometry, 2, 27, 127, 172, 259 glide reflection, 37 gradient, 152, 155 graph, 13 handle, 4, 74, 89, 250

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