Eisenhart L.P. — An introduction to differential geometry with use of the tensor calculus :: Электронная библиотека попечительского совета мехмата МГУ

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Eisenhart L.P. — An introduction to differential geometry with use of the tensor calculus

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Название: An introduction to differential geometry with use of the tensor calculus

Автор: Eisenhart L.P.

Аннотация:

Since 1909, when my Differential Geometry of Curves and Surfaces was published, the tensor calculus, which had previously been invented by Ricci, was adopted by Einstein in his General Theory of Relativity, and has been developed further in the study of Rieraannian Geometry and various generalizations of the latter. In the present book the tensor
calculus of cuclidean 3-space is developed and then generalized so as to apply to a Riemannian space of any number of dimensions. The tensor calculus as here developed is applied in Chapters 111 and IV to the etudy of differential geometry of surfaces in 3-space, the material treated being equivalent to what appears in general in the first eight chapters of my former book with such additions as follow from the introduction of the concept of parallelism of Levi-Civita and the content of the tensor calculus.
Of the many exercises in the book some involve merely direct appli- cation of the text, but most of them constitute an extension of it.

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Рубрика: Математика/Геометрия и топология/Дифференциальная геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1940

Количество страниц: 304

Добавлена в каталог: 10.11.2004

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Предметный указатель

      12.
$a_{ij}$ , $a^{ij}$ , a      70 72
$d_{\alpha\beta}$ , $d^{\alpha\beta}$       215 216 221 234 262
$e_{ijk}$ , $e^{ijk}$       6 93 94
$e_{\alpha\beta}$ , $e_{\alpha\beta}$       134 138
$g_{ij}$ , $g^{ij}$       101
$g_{\alpha\beta}$ , $g^{\alpha\beta}$       123 125 137 151 262
$h_{\alpha\beta}$ , $h^{\alpha\beta}$       253 256
      187 245
      225
$R_{ijk}^h$ , $R_{hijk}$       100—103 150 151 207
$W_{\beta\gamma\delta}^\alpha$       211
$\beta^i$       18
$\delta^{\alpha}_{\beta}$       125
$\Delta_{1\varphi}$ , $\Delta_{i(\varphi_1,\varphi_2)}$ , $\Theta_{(\varphi_1,\varphi_2)}$       155 156 159 160
$\Delta_{2\varphi}$       158 159
$\delta_{ij}$ , $\delta^{ij}$ , $\delta_{\beta}^{\alpha}$       8 25 32
$\epsilon_{ijk}$ , $\epsilon^{ijk}$       94 97
$\epsilon_{\alpha\beta}$ , $\epsilon_{\alpha\beta}$       135 137 138 151.
$\gamma^i$       20
$\rho$       16
$\tau$       22
$\tau_g$       247
Angle, of curves in a surface      130—132 137 145
Angle, of curves in a surface, of vectors      133
Applicable surfaces      215; see Isometric surfaces
Arc of a curve      9
Area, element of      136
Area, minimum      290
Associate, minimal surface      293
Associate, vector      194 250 252
Asymptotic directions      237 241
Asymptotic lines      237—239 243
Asymptotic lines geodesic      248
Asymptotic lines orthogonal      238
Asymptotic lines straight      237 267
Asymptotic lines, coordinate      237 238 242 260 266
Asymptotic lines, osculating plane of      237
Asymptotic lines, plane      249
Asymptotic lines, spherical representation of      260 266 267
Asymptotic lines, tangential coordinates      264 265
Beltrami      158 189 208 275
Bertrand curves      30
Bianchi      194
Bianchi identities      121
Binormal of a curve      19
Bonnet      190 273
Calculus of variations      290
Carenold      128 242 294
Center of curvature, of a curve      18
Center of curvature, of a curve in a surface      227
Center of curvature, principal, of a surface      226
Characteristics of a family of surfaces      54
Christoffel symbols, as components of a tensor      100 102 103
Christoffel symbols, for a surface      149 153 154
Christoffel symbols, for space      98 102 121
Christoffel symbols, relations between for a surface and its spherical representation      257 260
Circle, of curvature      18
Circle, osculating      19
Circle, superosculating      24
Circles, orthogonal system, on the sphere      267
Codazzi equations      219 230 236 238
Combeseure transformation of curves      29
Complementary surfaces      275
Cone      53 57
Conformal correspondence, of a plane with itself      204
Conformal correspondence, of a sphere with itself      204
Conformal correspondence, of a sphere with the plane      204
Conformal correspondence, of a surface and its spherical representation      255
Conformal correspondence, of a surface of constant curvature with the plane      288
Conformal correspondence, of a surface with itself      203
Conformal correspondence, of two surfaces      201—205
Congruence of curves      78
Conjugate directions      231 241
Conjugate net      232 243
Conjugate net, coordinate      232 235 236
Conjugate net, isometric-      235 236
Conjugate net, mean-      240 243 260.
Conjugate net, of plane curves      266
Conjugate net, orthogonal      232
Conjugate net, spherical representation      260
Conjugate net, tangential coordinates      263
Conjugate systems in correspondence      243
Conoid, right      50 53 129 149 221 266
Contraction of indices      95
Contravariant, components      85 88 127 134
Contravariant, index      89
Contravariant, tensor      89 90 91
Contravariant, vector      77 82 126
Coordinate curves, in a surface      46 127
Coordinate curves, in space      69
Coordinate net      49
Coordinate surface      68 81
Coordinates cartesian      63 70 88
Coordinates cylindrical      83; see Polar coordinates Tangential
Coordinates, , as      137 221
Coordinates, in a surface      48
Coordinates, in space      63—68
Covariant components      85 86 88 127 134
Covariant differentiation      107—112
Covariant differentiation, of $a_{ij}$ , $a^{ij}$ , $\delta_j^i$       110
Covariant differentiation, of $g_{\alpha\beta}$ , $g^{\alpha\beta}$ , $\epsilon_{\alpha\beta}$ , $\epsilon^{\alpha\beta}$       151
Covariant differentiation, of sum, difference, outer and inner product of tensors      111
Covariant index      89
Covariant tensor      89 90 91
Covariant vector      84 127
Cubic, twisted      5 7 8 16 44
Curvature normal, center of      224
Curvature normal, of a surface      224
Curvature normal, principal centers of      226
Curvature normal, principal radii of      225
Curvature normal, radius of      224 227
Curvature of a curve      16
Curvature of a curve, center of      18
Curvature of a curve, circle of      18
Curvature of a curve, constant      19 23 24 28
Curvature of a curve, first      22
Curvature of a curve, geodesic      see Geodesic curvature
Curvature of a curve, of a triangle      183
Curvature of a curve, radius of      16
Curvature of a curve, second      22
Curvature of a surface, total      151; see Gaussian curvature Mean
Curvature tensor      see Riemann tensor
Curvature, of a quadratic form      151
Curve twisted or skew      4
Curve, arc of      9
Curve, definition      3
Curve, form of      26
Curve, length of      9
Curve, minimal, or of length zero      10
Curve, of constant curvature      19 24 28
Curve, of constant torsion      24 28 29 30
Curve, plane      4
Curvilinear coordinates      47 69
Cylinder      3 53 61
D, D’, D’’      216
Darboux      115 173 233
Developable surface      54 57—61 152
Developable surface, edge of regression      61
Developable surface, Gaussian curvature      150
Developable surface, geodesic in a      178
Developable surface, isometric with a plane      147
Developable surface, isotropic      61
Developable surface, lines of curvature      229
Developable surface, of normals to a surface      229
Developable surface, polar      61 149
Developable surface, rectifying      61
Developable surface, tangent planes      147 148
Developable surface, tangent surface of a curve      57
Differential parameters, of the first order      155 156 159 160

Differential parameters, of the second order      158 159
Dini      210
Dini, surface of      287
Direction cosines of principal normal      18
Direction cosines of tangent      12
Direction cosines, of binormal      20 23
Divergence of a vector      113 155
Dupin indicatrix of a surface      241
E, F, G      126
Edge of regression      56 61
Element linear      see Linear Element
Element, of area      see Area
Ellipsoid      50; see Quadries central
Elliptic point of a surface      242
Enneper      248 292
Enneper, minimal surface of      294
Envelope, edge of regression      56 62
Envelope, of a family of spheres      62
Envelope, of a one-parameter family of surfaces      54—56
Envelope, of a oneparameter family of planes      57 see
Envelope, of characteristics      56
Equations parametric, of a line      1
Equations, parametric, of a curve      3
Equations, parametric, of a surface      46
Equivalent representation of surfaces      205
Euler, equation of      240
Euler, equations of      177 289
Evolute of a curve      36 38
Family characteristic      54
Family of curves in a surface      138—141
Family of geodesics      174
Family of planes      57
Family of spheres      62
Family, one-parameter, of surfaces      54
Formula of Green      190 192
Formula of Liouville      193
Frenet formulas      25 27
Frenet formulas in a surface      199
Frenet formulas in general coordinates      106
Fundamental quadratic form, first, of a surface      124
Fundamental quadratic form, of space      70
Fundamental quadratic form, second, of a starface      215
Fundamental tensor, first, of a surface      125 262
Fundamental tensor, of space      91
Fundamental tensor, second, of a surface      215 262
G      101 125
Gauss      45 151 174 184 216 219 225 252 254
Gauss — Bonnet theorem      191
Gauss, equation of      219
Gauss, equations of      216
Gaussian curvature of a surface      151 154 193 225 255
Generator of a developable surface      59
Generator of a surface of translation      236
Geodesic correspondence of two surfaces      205—211
Geodesic correspondence of two surfaces, of constant curvature      209
Geodesic curvature      186 193 199 201 243
Geodesic curvature, center of      246 275
Geodesic curvature, curves of constant      192 267
Geodesic curvature, radius of      276
Geodesic parallels      174 200
Geodesic polar coordinates      180—182
Geodesic torsion      247—249
Geodesic triangle      183 184 200
Geodesic, circles      180 186
Geodesic, ellipses and hyperbolas      185 186
Geodesics      170—179 187 246
Geodesics coordinate      173
Geodesics plane      248
Geodesics, equations of      171 179 197
Geodesics, in a surface of Liouville      176
Gradient      84
Group property      68
H      253
Helicoid, skew      129 145 165 242 294
Helieoid      53 129 218
Helieoid minimal      294
Helieoid pseudospherical      287
Helieoid spherical      287
Helieoid, isometric with a surface of revolution      160
Helieoid, parameter of a      129
Helieoid, surfaces of center      276
Helix circular      14 16 19 23 30 34 37
Helix conical      15
Helix cylindrical      15 19 24 28 37 38 149 178
Hyperbolic point of a surface      242
Hyperboloid      50 242
Hyperboloid, rulings      242; see Quadries central
Index , dummy      2
Index contravariant      89
Index covariant      89
Index free      2
Index lowering      95
Index raising      95
Indicatrix of binormal      22
Indicatrix of Dupin      see Dupin
Indicatrix spherical, of tangent      17
Inner product of tensors      95
Intrinsic derivative      195
Intrinsic equations of a curve      31 147
Intrinsic geometry of a surface      146
inversion      233 236
Inversion, preserves lines of curvature      233
Involute of a curve      35 37 39
Isometric surfaces      147 150 166—169 171 197
Isometric, orthogonal net      161—165
Isometric, orthogonal net, -conjugate net      235 236
Isometric, orthogonal net, coordinates      161
K      151 154 193 225
Kronecker deltas      25 63
Lagrange      290
lame      158
Lelieuvre, formulas of      266
Levi-Civita      196 199 251
Line of curvature      228 229
Line of curvature, conjugate      232 233
Line of curvature, coordinate      230
Line of curvature, equation      228 277
Line of curvature, geodesic      248
Line of curvature, geodesic torsion of      248
Line of curvature, normal curvature of      229
Line of curvature, osculating plane of      259
Line of curvature, plane      230 249 260 266 268 276
Line of curvature, spherical      249
Line of curvature, spherical representation of      255
Line of curvature, two surfaces intersecting in a      248
Line of curvature, under an inversion      233
Linear element of a curve      9
Linear element of a surface      124
Linear element of space      70
Linear element of the spherical representation      252
Linearly independent (constant coefficients)      262
Lines of length zero      see Minimal lines
Lines of shortest length      175
Liouville, formula of      193
Liouville, surface of      176
Loxodromic curve      145
Mainardi      219
McConnell      194
Mean curvature of a surface      225
Mean-conjugate net      240 243 260
Meridian of a surface of involution      49
Metric tensor, of a surface      124 125
Metric tensor, of space      91
Meusnier, theorem of      224
Minimal curve      10 16
Minimal curve on a sphere      154
Minimal curve on a surface      124
Minimal surface      238 249 288—295
Minimal surface adjoint      293
Minimal surface algebraic      293

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