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| Carmo M.P. — Differential geometry of curves and surfaces |
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| Предметный указатель |
Acceleration vector 345
Accumulation point 457
Angle between two surfaces 87
Angle, external 266
Angle, interior 274
Antipodal map 80
ARC 462
Arc length 6
Arc length in polar coordinates 25 (Ex. 11)
Arc length, reparametrization by 22
arc, regular 266
Area 98
Area of a graph 100 (Ex. 5)
Area of surface of revolution 101 (Ex. 11)
Area, geometric definition of 115
Area, oriented 15 (Ex. 10) 166
Area-preserving diffeomorphisms 230 (Ex. 18) 231
Asymptotic curve 148
Asymptotic direction 148
Ball 118
Beltrami — Enneper, theorem of 152 (Ex. 13)
Beltrami’s theorem on geodesic mappings 296 (Ex. 13)
Bertrand curve 26
Bertrand mate 26
Binormal line 19
Binormal vector 18
Bolzano — Weierstrass theorem 112 124 466
Bonnet, O. 535
Bonnet’s theorem 352 424
Boundary of a set 459
Braunmuehl, A. 363
Buck, R.C. 43 97 132
Calabi, E. 354
Catenary 23 (Ex. 8)
Catenoid 221
Catenoid as a minimal surface 202
Catenoid, asymptotic curves of 168 (Ex. 3)
Catenoid, local isometry of, with a helicoid 213 (Ex. 14) 223
Cauchy sequence 461
Cauchy sequence in the intrinsic distance 336 (Ex. 5)
Cauchy — Crofton formula 41
Chain rule 91 (Ex. 24) 126 129
Chern and Lashof, R. 387
Chern, S.S. 318
Christoffel symbols 232
Christoffel symbols for a surface of revolution 232
Christoffel symbols in normal coordinates 295 (Ex. 4)
Cissoid of Diodes 7 (Ex. 3)
Clairaut’s relation 257
Closed plane curve 30
Closed set 458
Closure of a set 458
Compact set 112 465
Comparison theorems 369 (Ex. 3)
Compatibility equations 235
Complete surface 325
Cone 64 65 327
Cone as a ruled surface 189
Cone, geodesies of 307 (Ex. 6)
Cone, local isometry of, with plane 223
Conformal map 226
Conformal map, linear 229 (Ex. 13)
Conformal map, local 226 229
Conformal map, local, of planes 229 (Ex. 15)
Conformal map, local, of spheres into planes 230 (Ex. 16)
Conjugate directions 150
Conjugate locus 363
Conjugate minimal surfaces 213 (Ex. 14)
Conjugate points 362
Conjugate points, Kneser criterion for 370 (Ex. 7)
Connected 462
Connected, arcwise 462
Connected, arcwise, locally 464
Connected, component 469
Connected, simply 382
Connected, simply, locally 383
Connection 306 (Ex. 2) 442
Conoid 210 (Ex.5)
Contact of curves 171 (Ex. 9)
Contact of curves and surfaces 171 (Ex. 10)
Contact of surfaces 91 (Ex.27) 170
Continuous map 120
Continuous map, uniformly 468
Convergence 456
Convergence in the intrinsic distance 336 (Ex.4)
Convex curve 37
Convex hull 48 (Ex. 11)
Convex neighborhood 303
Convex neighborhood, existence of 305
Convex set 48 (Ex. 9)
Convexity and curvature 40 174 387 397
Coordinate curves 53
Coordinate neighborhood 53
Coordinate system 52
Courant, R. 115
Covariant derivative 238
Covariant derivative in terms of parallel transport 305 (Ex. 1)
Covariant derivative, algebraic value of 248
Covariant derivative, along a curve 240
Covariant derivative, expression of 239
Covariant derivative, properties of 306 (Ex. 2)
Covering space 371
Covering space, number of sheets of 377
Covering space, orientable double 443 (Exs. 3 4)
Critical point 58 89
Critical point, nondegenerate 173 (Ex.23)
Critical value 58
Cross product 12
Curvature in arbitrary parameters 25 (Ex. 12)
Curvature of a plane curve 21
Curvature of a space curve 16
Curvature, Gaussian 146 155 see
Curvature, geodesic 248 253
Curvature, lines of 145
Curvature, lines of, differential equations of 161
Curvature, mean 146 156 163
Curvature, mean, vector 201
Curvature, normal 141
Curvature, principal 144
Curvature, radius of 19
Curvature, sectional 442
Curve of class  10 (Ex. 7)
Curve, asymptotic 148
Curve, asymptotic, differential equations for 160
Curve, asymptotic, maximal 410
Curve, closed 30
Curve, closed, continuous 392
Curve, closed, piecewise regular 266
Curve, closed, simple 30
Curve, coordinate 53
Curve, divergent 336 (Ex. 7)
Curve, knotted 402
Curve, level 102 (Ex. 14)
Curve, level, parametrized 3
Curve, level, piecewise differentiable 329
Curve, level, piecewise regular 244
Curve, level, regular 6
Curve, piecewise  35
Curve, simple 10 (Ex.7)
Cut locus 420
Cycloid 7
Cylinder 65 (Ex. 1)
Cylinder, as a ruled surface 188
Cylinder, first fundamental form of 93
Cylinder, isometries of 229 (Ex. 12)
Cylinder, local isometry of, with plane 219
Cylinder, normal sections of 144
Darboux trihedron 261 (Ex. 14)
Degree of a map 390
Developable surface 194 210
| Developable surface as the envelope of a family of tangent planes 195
Developable surface, classification of 194
Developable surface, tangent plane of a 210 (Ex. 6)
Diffeomorphism 74
Diffeomorphism, area-preserving 230 (Exs. 18 19)
Diffeomorphism, local 86
Diffeomorphism, local, orientation-preserving 165
Diffeomorphism, local, orientation-reversing 166
Differentiable function 72 80 82 125
Differentiable manifold 438
Differentiable map 73 126 426
Differentiable structure 425 439
Differential of a map 86 127 430
Direction, asymptotic 148
Direction, principal 144
Directions, conjugate 150
Directions, field of 178
Directrix of a ruled surface 188
Distance on a surface 329
Distribution parameter 192
Do Carmo, M. and E. Lima 387
Domain 97
Dot product 4
Dupin indicatrix 148
Dupin indicatrix, geometric interpretation of 164
Dupin’s theorem on triply orthogonal systems 153
Edges of a triangulation 271
Efimov, N.V. 453
Eigenvalue 216
Eigenvector 216
Ellipsoid 61 80 90
Ellipsoid, conjugate locus of 263
Ellipsoid, first fundamental form of 99 (Ex. 1)
Ellipsoid, Gaussian curvature of 173 (Ex. 21)
Ellipsoid, parametrization of 66 (Ex. 12)
Ellipsoid, umbilical points of 172 (Ex. 20)
Embedding 435
Embedding of the Klein bottle into  436
Embedding of the projective plane into 437
Embedding of the torus into 435
Energy of a curve 307 (Ex. 4)
Enneper’s surface 168 (Ex. 5)
Enneper’s surface as a minimal surface 205
Envelope of a family of tangent planes 195 210 212 244 308
Euclid’s fifth axiom 279 431 432
Euler formula 145
Euler — Lagrange equation 365
Euler — Poincare characteristic 272
Evolute 23 (Ex. 7)
Exponential map 284
Exponential map, differentiability of 285
Faces of a triangulation 271
Fary — Milnor Theorem 402
Fenchel’s theorem 399
Fermi coordinates 306 (Ex. 3)
Field of directions 178
Field of directions, differential equation of 179
Field of directions, integral curves of 178
Field of unit normal vectors 104
First fundamental form 92
Flat torus 435
Focal surfaces 210 (Ex. 9)
Folium of Descartes 8 (Ex. 5)
Frenet formulas 19
Frenet trihedron 19
Function, analytic 207
Function, component 120
Function, continuous 119
Function, differentiable 72 125
Function, harmonic 201
Function, height 72
Function, Morse 173 (Ex. 23)
Fundamental theorem for the local theory of curves 19 309
Fundamental theorem for the local theory of surfaces 236 311
Gauss formula 234
Gauss formula in orthogonal coordinates 237 (Ex. 1)
Gauss lemma 288
Gauss map 136
Gauss theorem egregium 234
Gauss — Bonnet theorem (global) 274
Gauss — Bonnet theorem (global), application of 276
Gauss — Bonnet theorem (local) 268
Gaussian curvature 146 155
Gaussian curvature for graphs of differentiable functions 163
Gaussian curvature in terms of parallel transport 270 271
Gaussian curvature, geometric interpretation of 167
Genus of a surface 273
Geodesic circles 287
Geodesic coordinates 306 (Ex. 3)
Geodesic curvature 248 253
Geodesic flow 440
Geodesic mapping 296 (Ex. 12)
Geodesic parallels 306 (Ex. 3)
Geodesic polar coordinates 286
Geodesic polar coordinates, first fundamental form in 287
Geodesic polar coordinates, Gaussian curvature in 288
Geodesic polar coordinates, geodesies in 295 (Ex. 7)
Geodesic torsion 153 (Ex. 19) 261
Geodesics 307
Geodesics as solutions to a variational problem 345
Geodesics of a cone 306 (Ex. 6)
Geodesics of a cylinder 246 247
Geodesics of a paraboloid of revolution 258—260
Geodesics of a sphere 246
Geodesics of the Poincare half-plane 431 432 444
Geodesics, differential equations of 254
Geodesics, existence of 255
Geodesics, minimal 303 332
Geodesics, minimizing properties of 292
Geodesics, of surfaces of revolution 255-258 356
Geodesics, radial 287
Geppert, H. 407
Gluck, H. 41
Gradient on surfaces 101 (Ex. 14)
Graph of a differentiable function 58
Graph of a differentiable function, area of 100 (Ex. 5)
Graph of a differentiable function, Gaussian curvature of 163
Graph of a differentiable function, mean curvature of 163
Graph of a differentiable function, second fundamental form of 163 164
Graph of a differentiable function, tangent plane of 88 (Ex. 3)
Green, L. 363
Gromov, M.L. and V.A. Rokhlin 454
Group of isometries 229 (Ex. 9)
Hadamard’s theorem on complete surfaces with  387 390
Hadamard’s theorem on ovaloids 387
Hartman, P. and L. Nirenberg 408
Heine — Borel theorem 112 124
Helicoid 94
Helicoid as a minimal surface 204
Helicoid as the only minimal ruled surface 204
Helicoid, asymptotic curves of 168 (Ex. 2)
Helicoid, distribution parameter of 209 (Ex. 1)
Helicoid, generalized 101 (Ex. 13) 186
Helicoid, line of striction of 209 (Ex. 1)
Helicoid, lines of curvature of 168 (Ex. 2)
Helicoid, local isometry of, with a catenoid 213 (Ex. 14) 223
Helicoid, tangent plane of 89 (Ex. 9)
Helix 3 22
Helix, generalized 26 (Ex. 17)
hessian 164 173
Hilbert, D. 318 446
Hilbert’s Theorem 446
Holmgren, E. 446
Holonomy group 297 (Ex. 14)
Homeomorphism 123
Homotopic arcs 378
Homotopy of arcs 378
Homotopy of arcs, free 390 (Ex. 10)
Homotopy of arcs, lifting of 379
Hopf — Rinow’s theorem 333
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