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| Carmo M.P. — Differential geometry of curves and surfaces |
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| Предметный указатель |
Hopf, H. and W. Rinow 326 354
Hopf’s theorem on surfaces with H=const. 234 (Ex.4)
Hurewicz, W. 177
Hyperbolic paraboloid (saddle surface) 66 (Ex. 11) Fig.
Hyperbolic paraboloid (saddle surface) as a ruled surface 193
Hyperbolic paraboloid (saddle surface), asymptotic curves of 184
Hyperbolic paraboloid (saddle surface), first fundamental form of 99 (Ex. 1)
Hyperbolic paraboloid (saddle surface), Gauss map of 139
Hyperbolic paraboloid (saddle surface), Parametrization of 66 (Ex. 11)
Hyperbolic plane 431
Hyperboloid of one sheet 88 (Ex. 2) Fig.
Hyperboloid of one sheet as a ruled surface 189 209
Hyperboloid of one sheet, Gauss map of 151 (Ex. 8)
Hyperboloid of two sheets 61
Hyperboloid of two sheets, first fundamental form of 99 (Ex. 1)
Hyperboloid of two sheets, parametrization of 67 (Ex. 13) 99
Immersion 433
Immersion, isometric 433
Index form of a geodesic 422
Index of a vector field 280
Infimum (g.l.b.) 460
Integral curve 178
Intermediate Value Theorem 124
Intrinsic distance 225 329
Intrinsic geometry 217 235 238
Inverse function theorem 131
inversion 121
Isometry 218
Isometry, linear 228 (Ex. 7)
Isometry, local 219
Isometry, local, in local coordinates 220 228
Isometry, local, of tangent surfaces to planes 228 (Ex. 6)
Isoperimetric inequality 33
Isoperimetric inequality for geodesic circles 295 (Ex. 9)
Isothermal coordinates 201 227
Isothermal coordinates for minimal surfaces 213 (Ex. 13(b))
Jacobi equation 357
Jacobi field 357
Jacobi field on a sphere 362
Jacobian determinant 128
Jacobi’s theorem on the normal indicatrix 278
Jacobi’s theorems on conjugate points 419 423
JacQbian matrix 128
Joachimstahl, theorem of 152 (Ex. 15)
Jordan curve theorem 393
Kazdan, J. and F. Warner 446
Klein bottle 427
Klein bottle, embedding of, into  436 437
Klein bottle, non-orientability of 436
Klirigenberg’s lemma 388 (Ex. 8)
Kneser criterion for conjugate points 370 (Ex. 7)
Knotted curve 402
Lashof, R. and S.S. Chern 387
Lebesgue number of a family 113
Levi-Civita connection 442
Lifting of a homotopy 379
Lifting of an arc 376
Lifting, property of, arcs 380
Lima, E. and M. do Carmo 387
Limit of a sequence 456
Limit point 457
Line of curvature 145
Liouville, formula of 253
Liouville, surfaces of 263
Local canonical form of a curve 27
Locally convex 174 (Ex.24) 387
Locally convex, strictly 174 (Ex.24)
Logarithmic spiral 9
Loxodromes of a sphere 96 230
Mainardi — Codazzi equations 235
Mangoldt, H. 363
Map, antipodal 80 (Ex. 1)
Map, conformal 226
Map, conformal, linear 229 (Ex. 13)
Map, continuous 120
Map, covering 371
Map, differentiable 73 126 426
Map, distance-preserving 228 (Ex. 8)
Map, exponential 284
Map, Gauss 136
Map, geodesic 296 (Ex. 12)
Map, self-adjoint linear 214
Massey, W. 408
Mean curvature 146 156 163
Mean curvature vector 201
Mercator projection 230 (Ex. 16) 231
Meridian 76
Meusnier theorem 142
Milnor, T. Klotz 454
Minding’s theorem 288
Minimal surfaces 197
Minimal surfaces as solutions to a variational problem 199
Minimal surfaces of revolution 202
Minimal surfaces, conjugate 213 (Ex. 14)
Minimal surfaces, Gauss map of 212 (Ex. 13)
Minimal surfaces, isothermal parameters on 202 213
Minimal surfaces, ruled 204
Moebius strip 106
Moebius strip, Gaussian curvature of 172 (Ex. 18)
Moebius strip, infinite 443 (Ex. 2)
Moebius strip, nonorientability of 107 109 7)
Moebius strip, parametrization of 106
Monkey saddle 159 171
Morse index theorem 422
Neighborhood 119 123
Neighborhood, convex 303
Neighborhood, coordinate 53
Neighborhood, distinguished 371
Neighborhood, normal 285
Nirenberg, L. and P. Hartman 408
Norm of a vector 4
Normal coordinates 286
Normal curvature 141
Normal indicatrix 278
Normal line 87
Normal plane to a curve 19
Normal principal 19
Normal section 142
Normal vector to a curve 17
Normal vector to a surface 87
Olinde Rodrigues, theorem of 145
Open set 118
Orientation for curves 109 (Ex. 6)
Orientation for surfaces 103 136
Orientation of a vector space 12
Orientation, change of, for curves 6
Orientation, positive, of  12
Oriented area in  15 (Ex. 10)
Oriented positively, boundary of a simple region 267
Oriented positively, simple closed plane curve 31
Oriented surface 103 106
Oriented volume in  16 (Ex. 11)
Orthogonal families of curves 102 (Ex. 15) 181 186
Orthogonal fields of directions 181 185 186
Orthogonal parametrization 95 183
Orthogonal projection 80 (Ex. 2) 121
Orthogonal transformation 23 (Ex. 6) 228
Osculating circle to a curve 30 (Ex. 2(b))
Osculating paraboloid to a surface 170 (Ex. 8(c))
Osculating plane to a curve 17 29 29 30
Osculating sphere to a curve 171 (Ex. 10(c))
Osserman’s theorem 208 337
Ovaloid 322 387
Paraboloid of revolution 80 (Ex. 3)
Paraboloid of revolution, conjugate points on 368 (Ex. 2)
Paraboloid of revolution, Gauss map of 140
Paraboloid of revolution, geodesies of 258
Parallel curves 47 (Ex. 6)
Parallel surfaces 212 (Ex. 11)
Parallel transport 242
| Parallel transport, existence and uniqueness of 242 253
Parallel transport, geometric construction of 244
Parallel vector field 241
Parallels of a surface of revolution 76
Parallels, geodesic 306 (Ex. 3(d))
Parameter distribution 192
Parameter of a curve 3
Parameters, change of, for curves 82 (Ex. 15)
Parameters, change of, for surfaces 70
Parameters, isothermal 227
Parameters, isothermal, existence of 227
Parameters, isothermal, existence of, for minimal surfaces 213 (Ex. 13(b))
Parametrization of a surface 52
Parametrization of a surface by asymptotic curves 184
Parametrization of a surface by lines of curvature 185
Parametrization of a surface, orthogonal 95
Parametrization of a surface, orthogonal, existence of 183
Partition 10 (Ex. 8) 114
Plane, hyperbolic 431
Plane, normal 19
Plane, osculating 17 29 29 30
Plane, real projective 427
Plane, rectifying 19
Plane, tangent 84
Planes, one-parameter family of tangent 212 (Ex. 10) 308
Plateau’s problem 200
Poincare half-plane 431
Poincare half-plane, completeness of 444 (Ex. 7)
Poincare half-plane, geodesies of 432 444
Poincare’s theorem on indices of a vector field 282
Point, accumulation 457
Point, central 191
Point, conjugate 362
Point, critical 58 89 364
Point, elliptic 146
Point, hyperbolic 146
Point, isolated 461
Point, limit 457
Point, parabolic 146
Point, umbilical 147
Pole 390 (Ex. 11)
Principal curvature 144
Principal direction 144
Principal normal 19
Product, cross 12
Product, dot 4
Product, inner 4
Product, vector 12
Projection 80 (Ex.2) 121
Projection, Mercator 230 (Ex.16) 231
Projection, stereographic 67 (Ex.16) 228
Projective plane 427
Projective plane, embedding of, into 437
Projective plane, nonorientability of 436
Projective plane, orientable double covering of 443 (Ex. 3)
Pseudo-sphere 168 (Ex. 6)
Radius of curvature 19
Ray 336 (Ex. 6)
Rectifying plane 19
Rectifying plane, envelope of 308 (Ex. 7(b))
Region 97
Region, bounded 97
Region, regular 271
Region, simple 267
Regular curve 68 (Ex. 17) 75
Regular parametrized curve 6
Regular parametrized surface 78
Regular surface 52
Regular value 58 92
Regular value, inverse image of 59 92
Reparametrization by arc length 22
Riemannian manifold 441
Riemannian manifold, covariant derivative on 442
Riemannian metric 441
Riemannian metric on abstract surfaces 430
Riemannian structure 442
Rigid motion 23 (Ex. 6) 42
Rigidity of the sphere 317
Rinow, W. and H. Hopf 326 354
Rokhlin, V.A. and M.L. Gromov 454
Rotation 74 86
Rotation axis 76
Rotation index of a curve 37 393
Ruled surface 188
Ruled surface, central points of 191
Ruled surface, directrix of 188
Ruled surface, distribution parameter of 192
Ruled surface, Gaussian curvature of 192
Ruled surface, line of striction of 191
Ruled surface, noncylindrical 190
Ruled surface, rulings of 188
Ruling 188
Samelson, H. 114
Santalo, L. 45
Scherk’s minimal surface 207
Schneider R. 54
Schur’s theorem for plane curves 406 (Ex. 8)
Second fundamental form 141
Segre, B. 408
Set, arcwise connected 462
Set, bounded 112
Set, closed 458
Set, compact 112 466
Set, connected 462
Set, convex 48 (Ex. 9)
Set, locally simply connected 383
Set, open 118
Set, simply connected 382
Similarity 296 (Ex. 12)
Similitude 187 (Ex. 9) 229
Simple region 267
Singular point of a parametrized curve 6
Singular point of a parametrized surface 78
Singular point of a vector field 279
Smooth function 2
Soap films 199
Sphere 55
Sphere as double covering of projective plane 443 (Ex. 2)
Sphere, conjugate locus on 362—363
Sphere, first fundamental form of 95
Sphere, Gauss map of 137
Sphere, geodesies of 246
Sphere, isometries of 229 (Ex. 11) 264
Sphere, isothermal parameters on 228 (Ex. 4)
Sphere, Jacobi field on 362
Sphere, orientability of 104
Sphere, parametrizations of 55—58 67
Sphere, rigidity of 317
Sphere, stereographic projection of 67 (Ex. 16)
Spherical image 152 (Ex.9) 279
Stereographic projection 67 (Ex. 16) 228
Stoker, J.J. 387 408
Stoker’s remark on Efimov’s theorem 454 (Ex. 1)
Stoker’s theorem for plane curves 406 (Ex. 8)
Striction, line of 191
Sturm’s oscillation theorem 370 (Ex. 6)
Supremum (l.u.b.) 460
Surface of Liouville 263
Surface, abstract 425
Surface, complete 325
Surface, connected 61
Surface, developable 194 210
Surface, focal 210 (Ex. 9)
Surface, geometric 430
Surface, minimal 197
Surface, parametrized 78
Surface, parametrized, regular 78
Surface, regular 52
Surface, rigid 317
Surface, tangent 78
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