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Spivak M. — A Comprehensive Introduction to Differential Geometry. Volume 3
Spivak M. — A Comprehensive Introduction to Differential Geometry. Volume 3



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Название: A Comprehensive Introduction to Differential Geometry. Volume 3

Автор: Spivak M.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2-d edition

Год издания: 1979

Количество страниц: 476

Добавлена в каталог: 18.11.2012

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$C^{\infty}$      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$\ddot{o}$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$\acute{e}$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, M$\ddot{o}$bius 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
M$\ddot{o}$bius      310
M$\ddot{o}$bius strip      216
M$\ddot{o}$bius strip, analytic flat      355
M$\ddot{o}$bius 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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