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

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Ќазвание: Elementary differential geometry

јвтор: O'Neill B.

язык: en

–убрика: ћатематика/√еометри€ и топологи€/ƒифференциальна€ геометри€/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1966

 оличество страниц: 411

ƒобавлена в каталог: 20.04.2005

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный указатель
$\ell$, m, n      211Ч213
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      246Ч251
Algebraic area      289Ч290 386(Ex.
All-umbilic surface      257Ч259
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      51Ч52 220(Ex.
Arc length function      51
Area      280Ч286
Area form      283Ч284 292 309(Ex.
Area-preserving mapping      295(Ex. 16)
Associated frame field      276Ч277
Asymptotic curves      226Ч227 230
Asymptotic directions      225Ч227
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      241Ч242 244(Ex. 282Ч283 291
Canonical isomorphism      7Ч8 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      236Ч238
Catenoid, Gauss mapping      296(Ex. 20)
Catenoid, Gaussian curvature      236 239
Catenoid, Gaussian curvature, total      287Ч288
Catenoid, local isometry onto      267Ч268
Catenoid, principal curvatures      236
Center of curvature      64(Ex. 6)
circle      59 62
Clairut parametrization      330Ч339
Closed surface in $E^{3}$      181(Ex. 10) 263n
Codazzi equations      249 255
Compact surface      176Ч177 259Ч260
Complete surface      see "Geodesically complete surface"
Composite function      1
Cone      140(Ex. 5) 231(Ex.
Conformal geometric structure      305Ч306 308(Ex. 312Ч313
Conformal mapping      268Ч271 310
Conformal patch      270(Ex. 7) 279(Ex.
Congruence of curves      116Ч123
Congruence of curves determined by curvature and torsion      117
Congruence of surfaces      189 297Ч303
Conjugate point      353Ч359
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      85Ч90
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) 276Ч277
Covariant derivative formula      91(Ex. 5) 189Ч190 321
Covariant derivative on a patch      212 324Ч326
Covariant derivative, Euclidean      77Ч80 116(Ex. 189Ч190
Covariant derivative, intrinsic      318Ч326
Cross product      47Ч49 107 110
Cross-sectional curve      138
Curvature      see also "Gaussian curvature"
Curvature of a curve in $E^{2}$      65(Ex. 8) 122Ч123
Curvature of a curve in $E^{3}$      57 66 69
Curve      15
Curve in a surface      144Ч145
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      20Ч21
Cylinder      129 231(Ex.
Cylinder parametrization      137 141(Ex.
Cylinder, geodesies      275(Ex. 2) 352(Ex.
Cylindrical frame field      82Ч83
Cylindrical frame field, connection forms      89Ч90
Cylindrical frame field, dual 1-forms      96(Ex. 4)
Cylindrical helix      72Ч75
Darboux      81
Degree of a form      27 152
Degree of a mapping      386(Ex. 6)
Derivative map      35Ч40
Derivative map of a mapping of surfaces      160Ч161 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 145Ч146
Differential      23Ч26
Differential form on $E^{2}$      156
Differential form on $E^{3}$      22Ч31
Differential form on a surface      152Ч157
Differential form, closed or exact      157(Ex. 2) 173Ч175
Differential form, pullback      163
Direction      196
Directional derivative      11Ч15 149
Directional derivative, computation of      12 25
Director curve      140
Distribution parameter      232(Ex. 14)
Domain      1
Dot product      42Ч44 46 82 210
Dot product, preserved by isometries      105
Dual 1-forms      91Ч92 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      217Ч219 222(Ex.
Ellipsoid, umbilics      223(Ex. 23)
Elliptic hyperboloid      142(Ex. 10) 221Ч222
Elliptic paraboloid      142(Ex. 11) 220(Ex.
Enneper's surface      221(Ex. 12)
Euclid      335Ч336
Euclidean coordinate functions      9 15 24 33 53
Euclidean distance      43 49(Ex.
Euclidean geometry      112 304Ч305 308
Euclidean plane      5 305 335Ч336
Euclidean plane, local isometries      363Ч364
Euclidean space      3 5
Euclidean symmetry      302Ч303 365
Euclidean vector field      147
Euler Ч Poincare characteristic      378Ч380
Euler's formula      201
Evolute      75(Ex. 15)
Exterior angle      374
Exterior derivative      28Ч31 31Ч32 154Ч155
Faces      377
Flat surface      207 231(Ex. 263(Ex.
Flat torus      316 317(Ex. 370(Ex.
Flat torus imbedding in $E^{4}$      369
Focal point      362
FORM      see "Differential form"
Frame      44
Frame field on $E^{3}$      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      272Ч273
Frame-homogeneous surface      366
frenet      81
Frenet apparatus      63(Ex. 1) 71
Frenet approximation      60Ч61 65(Ex.
Frenet formulas      58 67
Frenet frame field      57
Function      1Ч2
Function, one-to-one      2
Function, onto      2
Gauss      245 274 303
Gauss equation      249
Gauss mapping      194Ч195 289Ч291
Gauss Ч Bonnet formula      375Ч377 383Ч385 388(Ex.
Gauss Ч Bonnet theorem      380Ч383 387(Ex.
Gaussian curvature      203Ч207 310Ч312 see
Gaussian curvature in Jacobi equation      355
Gaussian curvature sign      204Ч205
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      355Ч358
Gaussian curvature, holonomy and      325(Ex. 5)
Gaussian curvature, interval      275(Ex. 3)
Gaussian curvature, isometric invariance      273Ч275
Gaussian curvature, polar circles and      359Ч360
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) 329Ч330 337(Ex.
Geodesic curvature, total      372Ч373
Geodesic polar mapping      340Ч341
Geodesic polar parametrization      341Ч344
Geodesically complete surface      263n 328Ч329
Geodesically complete surface, shortest geodesic segments in      348
Geodesies      228Ч232 326Ч363 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      339Ч359
Geodesies, preserved by (local) isometries      275(Ex. 1) 326 362Ч363
Geodesies, spreading of      353Ч354
Geographical patch      134Ч135
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 275Ч276
Helicoid, patch computations for      213Ч214
Helix      15Ч16 58Ч59 119
Hilbert's Lemma      261
Hilbert's theorem      263
Holonomy      323Ч325
Holonomy angle      323
Holonomy, Gaussian curvature and      325(Ex. 5)
Homogeneous surface      366Ч368 370
Homotopic to constant      175
Hopf      386 389
Hyperbolic paraboloid      143(Ex. 12) 220(Ex.
Hyperbolic plane      315Ч316
Hyperbolic plane of pseudo-radius r      317(Ex. 4)
Hyperbolic plane, geodesic completeness      351(Ex. 1)
Hyperbolic plane, geodesies      334Ч336
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      285Ч286
Induced inner product      308 313
Initial velocity      21(Ex. 6)
Inner product      304 316Ч317
Integral curve      186Ч187
Integral of function      286 292(Ex.
Integration of differential forms      167Ч176 283Ч297
Integration of differential forms, 1-forms over 1-segments      167Ч169 172Ч173
Integration of differential forms, 1-forms over oriented regions      285 292(Ex.
Integration of differential forms, 2-forms over 2-segments      169Ч172 174(Ex. 297(Ex.
Interior angle      374
Intrinsic distance      264Ч265 269(Ex. 351
Intrinsic geometry      271 304
Inverse function      2
Inverse function theorem      39 161Ч162
Isometric imbedding      367Ч369
Isometric immersion      367Ч369
Isometric invariant      271 304
Isometric surfaces      265
Isometry group      365
Isometry of Euclidean space      98Ч111
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      263Ч266 270Ч275 306
Isometry of surfaces, Euclidean isometries and      297Ч299
Isometry of surfaces, isometric immersions and      367
Isothermal coordinates      279(Ex. 2)
J (rotation operator)      309(Ex. 5)
Jacobi equation      355Ч357 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      265Ч270 362Ч365
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      352Ч358
Manifold      184Ч187
Mapping of Euclidean spaces      32Ч41
Mapping of surfaces      158Ч166
Mean curvature      203 205Ч208 212 217 252
Mercator projection      271(Ex. 13)
Metric tensor      305
Minimal surface      207 275(Ex.
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