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| Ïîèñê ïî óêàçàòåëÿì |
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| O'Neill B. — Elementary differential geometry |
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| Ïðåäìåòíûé óêàçàòåëü |
 , m, n 211—213
1-segment 167
2-segment 169
Acceleration 54 68
Acceleration of a curve in a surface 196 324(Ex.
Acceleration, intrinsic 320
Adapted frame field 246—251
Algebraic area 289—290 386(Ex.
All-umbilic surface 257—259
Alternation rule 27 47 153
angle 44
Angle function 50(Ex. 12) 295(Ex.
Angle, coordinate 210
Angle, exterior 374
Angle, interior 374
Angle, oriented 291
Antipodal mapping 165(Ex. 5)
Antipodal points 182
Arc length 51—52 220(Ex.
Arc length function 51
Area 280—286
Area form 283—284 292 309(Ex.
Area-preserving mapping 295(Ex. 16)
Associated frame field 276—277
Asymptotic curves 226—227 230
Asymptotic directions 225—227
Attitude matrix 46 88
Basis formulas 251
Bending 268 275(Ex.
binomial 57 66 69
Boundary of a 2-segment 170
Boundary of a region 386
Bracket operation 81(Ex. 7) 195(Ex.
Bugle surface 241—242 244(Ex. 282—283 291
Canonical isomorphism 7—8 44 59
Canonical parametrization 238
Cartan, E. 42 92 96 303
Cartesian Plane 306
Cartesian product 187(Ex. 9)
Catenoid 236
Catenoid as minimal surface 236—238
Catenoid, Gauss mapping 296(Ex. 20)
Catenoid, Gaussian curvature 236 239
Catenoid, Gaussian curvature, total 287—288
Catenoid, local isometry onto 267—268
Catenoid, principal curvatures 236
Center of curvature 64(Ex. 6)
circle 59 62
Clairut parametrization 330—339
Closed surface in  181(Ex. 10) 263n
Codazzi equations 249 255
Compact surface 176—177 259—260
Complete surface see "Geodesically complete surface"
Composite function 1
Cone 140(Ex. 5) 231(Ex.
Conformal geometric structure 305—306 308(Ex. 312—313
Conformal mapping 268—271 310
Conformal patch 270(Ex. 7) 279(Ex.
Congruence of curves 116—123
Congruence of curves determined by curvature and torsion 117
Congruence of surfaces 189 297—303
Conjugate point 353—359
Connected surface 176
Connection equations on a surface 248 318
Connection equations on Euclidean space 86 248
Connection forms on a surface 248 272 277 306
Connection forms on Euclidean space 85—90
Conoid 233(Ex. 20)
Consistent formula for a mapping 165(Ex. 10)
Coordinate angle 210 220(Ex.
Coordinate expression 143 165(Ex.
Coordinate patch see "Patch"
Coordinate system 158(Ex. 9) 276—277
Covariant derivative formula 91(Ex. 5) 189—190 321
Covariant derivative on a patch 212 324—326
Covariant derivative, Euclidean 77—80 116(Ex. 189—190
Covariant derivative, intrinsic 318—326
Cross product 47—49 107 110
Cross-sectional curve 138
Curvature see also "Gaussian curvature"
Curvature of a curve in  65(Ex. 8) 122—123
Curvature of a curve in 57 66 69
Curve 15
Curve in a surface 144—145
Curve plane 61
Curve segment 56(Ex. 10) 167
Curve, closed 169
Curve, coordinate expression for 144
Curve, one-one 20
Curve, periodic 20
Curve, regular 20
Curve, simple closed 151n
Curve, unparametrized 20—21
Cylinder 129 231(Ex.
Cylinder parametrization 137 141(Ex.
Cylinder, geodesies 275(Ex. 2) 352(Ex.
Cylindrical frame field 82—83
Cylindrical frame field, connection forms 89—90
Cylindrical frame field, dual 1-forms 96(Ex. 4)
Cylindrical helix 72—75
Darboux 81
Degree of a form 27 152
Degree of a mapping 386(Ex. 6)
Derivative map 35—40
Derivative map of a mapping of surfaces 160—161 166
Derivative map of a patch 149(Ex. 4)
Derivative map of an isometry 104
Derivative of Euclidean vector field 113
Diffeomorphism 38 40(Ex. 161
Differentiability 4 10 33 143 145—146
Differential 23—26
Differential form on 156
Differential form on 22—31
Differential form on a surface 152—157
Differential form, closed or exact 157(Ex. 2) 173—175
Differential form, pullback 163
Direction 196
Directional derivative 11—15 149
Directional derivative, computation of 12 25
Director curve 140
Distribution parameter 232(Ex. 14)
Domain 1
Dot product 42—44 46 82 210
Dot product, preserved by isometries 105
Dual 1-forms 91—92 248 306
Dupin curves 208(Ex. 5)
E, F, G 140(Ex. 2) 210 317(Ex.
Edge (curve) 170 373 377
Ellipsoid 142(Ex. 10)
Ellipsoid isometry group 365
Ellipsoid, Euclidean symmetries 303(Ex. 10)
Ellipsoid, Gaussian curvature 217—219 222(Ex.
Ellipsoid, umbilics 223(Ex. 23)
Elliptic hyperboloid 142(Ex. 10) 221—222
Elliptic paraboloid 142(Ex. 11) 220(Ex.
Enneper's surface 221(Ex. 12)
Euclid 335—336
Euclidean coordinate functions 9 15 24 33 53
Euclidean distance 43 49(Ex.
Euclidean geometry 112 304—305 308
Euclidean plane 5 305 335—336
Euclidean plane, local isometries 363—364
Euclidean space 3 5
Euclidean symmetry 302—303 365
Euclidean vector field 147
Euler — Poincare characteristic 378—380
Euler's formula 201
Evolute 75(Ex. 15)
Exterior angle 374
Exterior derivative 28—31 31—32 154—155
| Faces 377
Flat surface 207 231(Ex. 263(Ex.
Flat torus 316 317(Ex. 370(Ex.
Flat torus imbedding in  369
Focal point 362
FORM see "Differential form"
Frame 44
Frame field on 82
Frame field on a curve 120
Frame field on surface 251 306
Frame field, adapted 246
Frame field, natural 9
Frame field, principal 254
Frame field, transferred 272—273
Frame-homogeneous surface 366
frenet 81
Frenet apparatus 63(Ex. 1) 71
Frenet approximation 60—61 65(Ex.
Frenet formulas 58 67
Frenet frame field 57
Function 1—2
Function, one-to-one 2
Function, onto 2
Gauss 245 274 303
Gauss equation 249
Gauss mapping 194—195 289—291
Gauss — Bonnet formula 375—377 383—385 388(Ex.
Gauss — Bonnet theorem 380—383 387(Ex.
Gaussian curvature 203—207 310—312 see
Gaussian curvature in Jacobi equation 355
Gaussian curvature sign 204—205
Gaussian curvature, differentiability 206
Gaussian curvature, formulas, explicit 206 212 217 253 278
Gaussian curvature, formulas, implicit 205 252
Gaussian curvature, Gauss mapping and 289
Gaussian curvature, geodesic curvature an 339(Ex. 19)
Gaussian curvature, geodesies and 355—358
Gaussian curvature, holonomy and 325(Ex. 5)
Gaussian curvature, interval 275(Ex. 3)
Gaussian curvature, isometric invariance 273—275
Gaussian curvature, polar circles and 359—360
Gaussian curvature, polar discs and 361(Ex. 2)
Gaussian curvature, principal curvatures and 203
Gaussian curvature, shape operator and 203
Geodesic curvature 230(Ex. 7) 329—330 337(Ex.
Geodesic curvature, total 372—373
Geodesic polar mapping 340—341
Geodesic polar parametrization 341—344
Geodesically complete surface 263n 328—329
Geodesically complete surface, shortest geodesic segments in 348
Geodesies 228—232 326—363 see
Geodesies coordinate formulas 327 333
Geodesies on surfaces of nonpositive curvature 358 388(EX.
Geodesies, closed 232(Ex. 13)
Geodesies, existence and uniqueness 328
Geodesies, frame fields and 250(Ex. 1)
Geodesies, length-minimizing properties 339—359
Geodesies, preserved by (local) isometries 275(Ex. 1) 326 362—363
Geodesies, spreading of 353—354
Geographical patch 134—135
Geometric surface 305 308
Gradient 32(Ex. 8) 50(Ex.
Gradient as normal vector field 148 216
Group 103
Group, euclidean 103
Group, Euclidean symmetry 302(Ex. 7)
Group, isometry 365
Group, orthogonal 104
Halmos symbol 9
Handle 379 382
Hausdorff axiom 186(Ex. 5) 345n
Helicoid 141(Ex. 7)
Helicoid as ruled minimal surface 227 233(Ex.
Helicoid, Gauss mapping 296(Ex. 20)
Helicoid, local isometries 267 275—276
Helicoid, patch computations for 213—214
Helix 15—16 58—59 119
Hilbert's Lemma 261
Hilbert's theorem 263
Holonomy 323—325
Holonomy angle 323
Holonomy, Gaussian curvature and 325(Ex. 5)
Homogeneous surface 366—368 370
Homotopic to constant 175
Hopf 386 389
Hyperbolic paraboloid 143(Ex. 12) 220(Ex.
Hyperbolic plane 315—316
Hyperbolic plane of pseudo-radius r 317(Ex. 4)
Hyperbolic plane, geodesic completeness 351(Ex. 1)
Hyperbolic plane, geodesies 334—336
Hyperbolic plane, isometries 371(Ex. 14)
Hyperbolic plane, local isometries 364
Identity mapping 99
Image 1
Image curve 33
Immersed surface 187(Ex. 10) 219 368
Improper integral 285—286
Induced inner product 308 313
Initial velocity 21(Ex. 6)
Inner product 304 316—317
Integral curve 186—187
Integral of function 286 292(Ex.
Integration of differential forms 167—176 283—297
Integration of differential forms, 1-forms over 1-segments 167—169 172—173
Integration of differential forms, 1-forms over oriented regions 285 292(Ex.
Integration of differential forms, 2-forms over 2-segments 169—172 174(Ex. 297(Ex.
Interior angle 374
Intrinsic distance 264—265 269(Ex. 351
Intrinsic geometry 271 304
Inverse function 2
Inverse function theorem 39 161—162
Isometric imbedding 367—369
Isometric immersion 367—369
Isometric invariant 271 304
Isometric surfaces 265
Isometry group 365
Isometry of Euclidean space 98—111
Isometry of Euclidean space, decomposition theorem 101
Isometry of Euclidean space, derivative map 104
Isometry of Euclidean space, determined by frames 105
Isometry of surfaces 263—266 270—275 306
Isometry of surfaces, Euclidean isometries and 297—299
Isometry of surfaces, isometric immersions and 367
Isothermal coordinates 279(Ex. 2)
J (rotation operator) 309(Ex. 5)
Jacobi equation 355—357 361(Ex.
Jacobian 288 294(Ex. 296(Ex.
Jacobian matrix 37
Kronecker delta 23 45
Lagrange identity 209(Ex. 6)
Law of Cosines 370(Ex. 10)
Leibnizian property 13
Length of a curve segment see "Arc length"
Length of a vector 44
Levi Civita 322
Liebmann's Theorem 262
Line of curvature see "Principal curve"
Liouville parametrization 339(Ex. 18)
Local isometry 265—270 362—365
Local isometry determined by frames 363
Local isometry of constant curvature surfaces 364
Local isometry, criteria for 266 269(Ex. 270(Ex.
Local minimization of arc length 352—358
Manifold 184—187
Mapping of Euclidean spaces 32—41
Mapping of surfaces 158—166
Mean curvature 203 205—208 212 217 252
Mercator projection 271(Ex. 13)
Metric tensor 305
Minimal surface 207 275(Ex.
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