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| Spivak M. — A Comprehensive Introduction to Differential Geometry. Volume 3 |
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| Предметный указатель |
 see Smooth
Adapted moving frame 24
Affine see also Special affine
Affine, conformal structure 119
Affine, invariants 101
Affine, maps 101
Affine, normal direction 145
Affine, special linear 101
Affine, special orthogonal 101
Ambient Space 1
Analytic flat M bius strip 355
Angle, between two vectors 387
Angle, interior 394
Apolar 141 142
Apolarity condition 142 158
Apolarity condition, geometric interpretation of 170
Approximate a surface up to second order 52
Arclength, special affine 105 107
Asymptote 69
Asymptotic, at t 285
Asymptotic, curve 284
Asymptotic, directions 69
Asymptotic, Tschebyscheff net 373
Asymptotic, vector 199
Authoritarian viewpoint 322
Auto-parallel submanifold 32
Axes, principal 68
Beltrami 291 336
Beltrami — Enneper Theorem 291
Bol 290
Bonnet 79 270 295
Branch points 327 436
Branched covering space 327
Canal surface 421
Canonical parameterization for a catenoid 235
Canonical parameterization for a surface of revolution 230
Carath odory 289
Cartaa’s lemma 26
Cartan, Elie 282
Catenary 233
Catenoid 233 247
Catenoid, canonical parameterization for 235
Cayley — Hamilton theorem 88
Characteristic line 260
Characteristic point 261
Clairaut 315
Classical classification of developable surfaces 353
Classical counterexample 241
Classical flat: surfaces 205
Classical tensor analysis treatment of submanifolds 17
Codazzi — Mainardi equations 15 17 23 30 79 99 113 195 323
Codazzi — Mainardi equations, special affine 192
Codimension 1
Compact surfaces with constant negative curvature 430
Complete analytic function 436
Complete surfaces of constant curvature 347
Cone, generalized 206
conformal 302 435
Conformal structure 119
Connection, forms 24
Connection, induced 34
Constant curvature 16
Constant curvature, compact surfaces with (negative) 430
Constant curvature, complete surfaces of 347
Constant curvature, isometry of simply-connected manifolds with the same 42 ff.
Constant curvature, manifolds 37
Constant curvature, rotation surfaces of 235
Continuation of an isometry 43
Convex 91
Convex, hull 214
Covering space 434
Covering space, branched 327
Cubic 138
Cubic, forms, invariants of 165
Cubic, osculating 165
Curvature see also Constant curvature
Curvature, absolute total 411
Curvature, forms 24
Curvature, functions of a curve 105
Curvature, functions of a curve, special affine 108
Curvature, Gaussian 69 198
Curvature, Gaussian, formula for 322
Curvature, geodesic 273
Curvature, line of (q.v.) 284
Curvature, mean 69 198
Curvature, normal 272
Curvature, positive 90
Curvature, principal 69 198
Curvature, sectional 8
Curvature, special affine (extrinsic) 187
Curvature, special affine mean 187
Curvature, special affine principal 187
Curvature, total 421 459
Curvature, total absolute 411
Curvature, vector, geodesic 5 272
Curvature, vector, normal 272
Cuspidal edge 208
Cuts 436
Cylinder, generalized 206
Cylinder, parabolic 59
Darboux 277 294 302
Darboux, frame 277
Degenerate see Non-degenerate
Degree of the normal map 439
Developable surfaces 287 351
Developable surfaces, classical classification of 353
Developable tangent 207
Development 212
Directions, asymptotic 69
Directions, principal 68 198
Directions, principal, special affine 187
Directrix 214
Discontinuity of the angle 395
Distribution parameter 216
Doubly ruled surfaces 227
Dual one-forms 23
Dupin 300
Dupin, indicatrix 68
Edge, cuspidal 208
Edge, of regression 208
Efimov 385
Ellipsoid 221
Ellipsoid, lines of curvature on 300
Ellipsoid, lines of curvature on, in a neighborhood of an umbilic 288
Elliptic hyperboloid (of one sheet) 222
Elliptic hyperboloid (of two sheets) 225
Elliptic hyperboloid (of two sheets), umbilics on 226
Elliptic paraboloid 56 226
Elliptic paraboloid, umbilics on 226
Elliptic point 56 118
Enneper 291
Enneper’s minimal surface 352
Envelope 255
Envelope, of one-parameter family of planes 259
Equations of structure of SO(3) 101 111
Euclidean motion, proper 73
Euler characteristic 399
Euler’s theorem 274
Fary 428
Fenchel 424 426
First fundamental form 45
First fundamental form of f 46
First fundamental form of f, special affine 124 133
First fundamental form, special affine 124 133
First structural equation 24
Flat 69
Flat, Mbius strip 217 355
Flat, surfaces, classical 205
Flat, torus 87
Frame, adapted moving 24
| Frame, Darboux 277
Frame, Frenet 277
Frenet frame 277
Frobenius 343
Fundamental forms see First second third
Fundamental theorem of spec Lai affine surface theory 192
Fundamental theorem of surface theory 79 111 113
Fundamental theorems of Lie 448 ff.
Gadamard 94
Gauss 75
Gauss formulas 6 20 28 75
Gauss formulas, special affine 156
Gauss — Bonnet formula 396
Gauss — Bonnet theorem 400
Gaussian curvature 69 198
Gaussian curvature, formula for 322
Gauss’ equation 7 17 23 29 78 113
Gauss’ Theorema Egregium 7 78 98
General affine invariants 101
General uniformization theorem 437
Generalized cone 206
Generalized cylinder 206
Geodesic 4 285
Geodesic, at p 35
Geodesic, curvature 273
Geodesic, curvature vector 5 272
Geodesic, on a torus 342
Geodesic, on surface of revolution 317
Geodesic, torsion 278
Geodesic, totally 35
Geometric interpretation of apolarity 170
Hamburger 290
Hazzidaki’s formula 379
Helicoid 247
Helicoid, (right) 218
hilbert 347 367
Holmgren 367
Hopf Umlaufsatz 393
Hopf Umlaufsatz, generalization of 397
Horn 426
Hull, convex 414
Hyperbolic paraboloid 58 227
Hyperbolic point 58 118
Hyperboloid, elliptic (of one sheet) 222
Hyperboloid, elliptic (of two sheets) 225
Hyperboloid, elliptic (of two sheets), umbilics on 226
Hyperboloid, of revolution 222
Hyperboloid, of revolution, striction curve of 266
Hypersurface 10
Idiot, any 153
Immersion, isometric 1
Index of a singularity of a 1-dimensional distribution 325
Indicatrix of Dupin 68
Induced connection 34
Interior angle 394
Invariant affine 101
Invariant meaning 84 85
Invariant under orientation preserving change of parameter 128
Invariant under proper Euclidean motions 73
Invariants for cubic forms 165
inversion 303
Inward 118 131
Isometric immersion 1
Isometric, locally 247
Isometry, continuation of 43
Isometry, continuation of, of simply-connected manifolds of the same curvature 42 ff.
jacobi 407
Joachimsthal 296
K, formula for 322
Klotz 290 385
Knotted 428
Laguerre 282
Levi — Civita 262
Lie’s fundamental theorems 448 ff.
Linear affine maps, special 101
Lines of curvature 284
Lines of curvature, on an ellipsoid 300
Lines of curvature, on an ellipsoid, in a neighborhood of an umbilic 288
Liouville 310
Locally isometric 247
Mbius 310
Mbius strip 216
Mbius strip, analytic flat 355
Mbius strip, smooth flat 217
Malnardl see Codazzi — Mainardi
Malz, R. 362
Mean curvature 69 198
Mean curvature, special affine 187
Meridian 229
Meusnier’s theorem 276
Milnor 428 439
Minimal surface 242
Minimal surface, Enneper’s 252
Minimal surface, Scherk’s 249
Monkey saddle 59
Morse theory 421
Motion, proper Euclidean 73
Moving frame, adapted 24
Navel point 71
Negative (constant) curvature, compact surfaces with 430
Neighborhood, tubular 439
Non — Euclidean plane 432
Non-degenerate 139 441
Normal curvature 272
Normal curvature vector 272
Normal direction, affine 145
Normal field, unit 10
Normal map, degree of 439
Normal projection 1
Normal special affine 150
One-dimensional distribution, index of a singularity of 325
Orthogonal affine maps, special 101
Orthogonal systems of surfaces, triply 297
Osculating circle 453
Osculating cubic 165
Osculating paraboloid 61
Parabolic cylinder 59
Parabolic point 58 118
Paraboloid, elliptic 56 226
Paraboloid, elliptic, umbllics on 226
Paraboloid, hyperbolic 58 227
Paraboloid, osculating 61
Parallel (of a surface of revolution) 229
Parallel along c 4 262
Parallel surface 270
Parameter curve 320
Perpendicular projection 1
Pick invariant 171 345
Planar point 59 118
Plane 205
Plane, support 91
Point inward 118 131
Polygon 394
Positive curvature 90
Principal at t 286
Principal axes 68
Principal curvatures 69 198
Principal curvatures, special affine 187
Principal curve 284
Principal directions 68 198
Principal directions, special affine 187
Principal vectors 68
Profile curve 228
Projection, normal 1
Projection, perpendicular 1
Projection, tangential 1
Proper Euclidean motion 73
Pseudosphere 238
Quadratic 137
Quadratic, (or quadric) surface 174 220
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