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O'Neill B. — Elementary differential geometry
O'Neill B. — Elementary differential geometry



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Название: Elementary differential geometry

Автор: O'Neill B.

Язык: en

Рубрика: Математика/Геометрия и топология/Дифференциальная геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1966

Количество страниц: 411

Добавлена в каталог: 20.04.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Minimal surface examples      238
Minimal surface of revolution      236—238
Minimal surface ruled      233(Ex. 22)
Minimal surface, fiat      263(Ex. 1)
Minimal surface, Gauss mapping      294(Ex. 6) 296(Ex.
Minimization of arc length      340
Mobius band      178 180(Ex.
Monge patch      127 219
Monkey saddle      132(Ex. 6) 205 231(Ex.
Monkey saddle, Gaussian curvature      220(Ex. 5)
n-polygon      388—389
Natural coordinate functions      4
Natural frame field      9
Neighborhood      43 125
Neighborhood, normal      341
Norm      43
Normal curvature      195—202 221(Ex.
Normal curvature, sign of      198
Normal section      197
Normal vector field      147
One-to-one      2
Onto      2
Open interval      15
Open set      5 43 151—152
Orientable surface      177—178 185(Ex.
Orientation of a frame      107
Orientation of a patch      285
Orientation of a paving      285
Orientation of a surface      195 284
Orientation of tangent frame fields      291
Orientation, determined by a patch      374
Orientation-preserving (-reversing) isometry      109 296
Orientation-preserving (-reversing) reparametrization      52
Oriented angle      291
Orthogonal coordinates      276—280
Orthogonal coordinates, Gaussian curvature formula      278
Orthogonal matrix      46
Orthogonal transformation      100
Orthogonal vectors      44
Orthonormal expansion      45—46 83
Orthonormal frame      see "Frame"
Osculating circle      64(Ex. 6)
Osculating plane      61
Parallel curves      116
Parallel postulate      336
Parallel surfaces      209—210
Parallel translation      323
Parallel vector field      54—55 322
Parallel vectors      6
Parameter curves      133
Parametrization in a surface      135
Parametrization in a surface, criteria for regularity      136 140(Ex.
Parametrization in a surface, decomposable into patches      165(Ex. 9)
Parametrization of a curve      21
Partial velocities      134 136 146
Patch      124—137
Patch principal      220(Ex. 9) 280(Ex.
Patch, abstract      182 184
Patch, geometric computations      210—216
Patch, Monge      127
Patch, orthogonal      220(Ex. 9) 276—280
Patch, proper      124 152(Ex.
Patchlike 2-segment      281
Paving      283 377
Planar point      205
Plane in $E^{3}$      60 131(Ex. 228 256
Plane in $E^{3}$, identified with $E^{2}$      126
Poincare half-plane      309(Ex. 2)
Poincare half-plane, Gaussian curvature      317(Ex. 1)
Poincare half-plane, geodesies      337(Ex. 6)
Poincare half-plane, isometric to hyperbolic plane      371(Ex. 15)
Poincare half-plane, polar circles      351(Ex. 2)
Point of application      6
Pointwise principle      8
Polar circle      346 359—360
Polar disc      361(Ex. 2)
Polygonal region      386
Pre-geodesic      330
Principal curvatures      199—207
Principal curvatures as characteristic values      200
Principal curvatures, formula for      206
Principal curve      223—225 230—232 263(Ex.
Principal direction      199
Principal frame field      254
Principal normal      57 66 69
Principal vectors      199 220(Ex. 223(Ex.
Principal vectors as characteristic vectors      200
Projective plane      182—183
Projective plane, geodesies      337(Ex. 8)
Projective plane, geometric structure      317(Ex. 6)
Projective plane, homogeneity      371(Ex. 11)
Projective plane, natural mapping (projection)      183
Projective plane, topological properties      186(Ex. 2)
Pseudosphere      see "Bugle surface"
Pullback      163
Quadratic approximation      202—204
Quadric surface      142 294(Ex.
Rectangular decomposition      377
Reflection      109
Regular curve      20
Regular mapping      38 161
Reparametrization      18
Reparametrization, monotone      56(Ex. 10)
Reparametrization, orientation-preserving (-reversing)      52
Reparametrization, unit-speed      51
Riemann      304 336
Riemannian geometry      308 389—390
Riemannian manifold      308
Rigid motion      see "Isometry of Euclidean space"
Rigidity      301(Ex. 1)
Rotation      111(Ex. 4)
Ruled surface      140—143 227 231—233
Ruled surface, noncylindrical      232(Ex. 14)
Ruled surface, total Gaussian curvature      294(Ex. 9)
Ruling      140
Saddle surface      192
Saddle surface, doubly ruled      227
Saddle surface, Euclidean symmetries      303(Ex. 9)
Saddle surface, patch computations      214—216
Saddle surface, principal vectors      221(Ex. 11)
Scalar multiplication      3 8—9
Scale factor      268
Scherk's surface      222(Ex. 21)
Scherk's surface, Gauss mapping      296(Ex. 20)
Scherk's surface, patch in      303(Ex. 11)
Schwartz inequality      44
Serret      81
Shape operator      190—194
Shape operator as derivative of Gauss mapping      289
Shape operator of an immersed surface      368
Shape operator, characteristic polynomial      208(Ex. 4)
Shape operator, covariant derivatives and      324(Ex. 3)
Shape operator, frame fields, in terms of      248
Shape operator, Gaussian and mean curvature, and      203
Shape operator, normal curvature and      196
Shape operator, preserved by Euclidean isometries      297—298
Shape operator, principal curvatures and vectors, and      200
Shape operator, proof of symmetry      212—213 251(Ex.
Shortest curve segment      340
Sign of an isometry      108
Simple region      294—295
Simply connected surface      176 363
Slant of a geodesic      331 338
Smooth overlap      145 182
Speed      51
Sphere      128
Sphere with handles      379
Sphere, conjugate points      354 357—358
Sphere, Euclidean symmetries      302(Ex. 8)
Sphere, frame-homogeneity      370(Ex. 6)
Sphere, Gaussian curvature      207 219(Ex. 253 360 361(Ex.
Sphere, geodesies      228—229 346—347
Sphere, geographical patch      134—135 277—278
Sphere, geometric characterizations      258 259 262
Sphere, geometric structures      383
Sphere, holonomy      323—324 337(Ex.
Sphere, local isometries      363—365
Sphere, rigidity      301(Ex. 1)
Sphere, shape operator      191—192
Sphere, topological properties      176—178 378
Spherical curve      63 65(Ex.
Spherical frame field      83
Spherical frame field, adapted to sphere      248—249 277—278
Spherical frame field, dual and connection forms      94—95
Spherical image of a curve      71 75(Ex.
Spherical image of a surface      see "Gauss mapping"
Standard geometric surface      363—366
Stereographic plane      314
Stereographic projection      160 162
Stereographic projection as conformal mapping      271(Ex. 14)
Stereographic sphere      314 337(Ex.
Stokes' theorem      170—172 387(Ex.
Straight line      15 18 55 229(Ex.
Straight line, length-minimizing propertie      56(Ex. 1)
Striction curve      232(Ex. 14)
Structural equations on $E^{3}$      92—95
Structural equations on a surface      249 252 292 311—312
Subset      1
Support function      218 222 256
Surface in $E^{3}$      125 306 367
Surface in $E^{3}$, implicit definition      127—128
Surface of revolution      129—130 234—244
Surface of revolution, area      292(Ex. 2)
Surface of revolution, augmented      133(Ex. 12)
Surface of revolution, diffeomorphism types      187(Ex. 8)
Surface of revolution, Gaussian curvature      235 238 242 243(Ex.
Surface of revolution, geodesies      338(Ex. 13)
Surface of revolution, local characterization      270(Ex. 12)
Surface of revolution, meridians and parallels      130
Surface of revolution, natural frame field      279
Surface of revolution, of constant curvature      239—241 244(Ex. 294(Ex.
Surface of revolution, parametrization, canonical      238
Surface of revolution, parametrization, special      143(Ex. 13)
Surface of revolution, parametrization, usual      138—139
Surface of revolution, principal curvatures      235
Surface of revolution, principal curves      225 235
Surface, abstract      182—184
Surface, geometric      305
Surface, immersed      368
Surface, topological properties      182(Ex. 14)
Surface, total curvature      293(Ex. 5)
Symmetry equation      249
Tangent bundle      185
Tangent line      22(Ex. 9)
Tangent plane      146 150(Ex.
Tangent space      7
Tangent surface      231(Ex. 11)
Tangent surface, isometries of      270(Ex. 5) 301(Ex.
Tangent vector to $E^{3}$      6 14
Tangent vector to a surface      146 183—184
Theorema egregium      273—275
Topological properties      176—182 380n
Toroidal frame field      84(Ex. 4) 2) 250—251
Torsion      58 66
Torsion formula      69
Torsion sign      114—115
Torus of revolution      139
Torus of revolution, Euler — Poincare characteristic      379 387(Ex.
Torus of revolution, Gauss mapping      194(Ex. 5) 290—291
Torus of revolution, Gaussian curvature      204—205 235—236
Torus of revolution, patch computations      235—236
Torus of revolution, total Gaussian curvature      287 291
Torus of revolution, usual parametrization      139
Total curvature of a curve      76(Ex. 16)
Total Gaussian curvature      286—291 380—385
Total Gaussian curvature of a patch      325(Ex. 5)
Total Gaussian curvature, Euler — Poincare characteristic, and      380
Total Gaussian curvature, Gauss mapping, and      290
Total Gaussian curvature, holonomy and      325(Ex. 5)
Total geodesic curvature      372—375 386(Ex. 389(Ex.
Transferred frame field      272—273
Translation      98—100 109
triangle      383—385
Triangle inequality      347
Triple scalar product      48—49 108
Tube      221(Ex. 16)
Umbilic point      200 221(Ex. see
Unit normal function      211 368
Unit normal vector field      180(Ex. 5) 190
Unit points      34
Unit speed curve      51
Unit sphere      125
Unit tangent      56 66 69
Unit vector      44
Vector      see "Tangent vector"
Vector analysis      31(Ex. 8)
Vector field on a curve      52—54 320
Vector field on a surface in $E^{3}$      147—149 151(Ex.
Vector field on a surface in $E^{3}$, normal      147 149
Vector field on a surface in $E^{3}$, tangent      147 149
Vector field on an abstract surface      183
Vector field on Euclidean space      8
Vector part      6
Velocity      17—18 183—184
Vertices      373—374
Wedge product      27—28 153
Winding number      174(Ex. 5)
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