Eisenhart L.P. — Riemannian geometry :: Электронная библиотека попечительского совета мехмата МГУ

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Eisenhart L.P. — Riemannian geometry

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Название: Riemannian geometry

Автор: Eisenhart L.P.

Язык:

Рубрика: Математика/Геометрия и топология/Дифференциальная геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: second edition

Год издания: 1949

Количество страниц: 306

Добавлена в каталог: 10.11.2004

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

$b_{ij}$       197
$b_{\sigma|^{ij}}$       189
$C^{h}_{ijk}$       90 94
$e_{\alpha}$       37
$g^{ij}$       14
$g_{ij}$       14 34
$R^{h}_{ijk}$       19
$R_{ij}$       21
$S_{n}$       84
$S_{n}$ , applicable hypersurfaces      200 201
$S_{n}$ , motions      87 192
$S_{n}$ , n-tuply orthogonal systems in      121
$S_{n}$ , spaces conformal to an      91—93 121—124 182 214—218
$S_{n}$ , sub-spaces      187—220
$V_{n}$       1
$V_{n}$ in a $V_{m}$ , associate directions      74 75 167
$V_{n}$ in a $V_{m}$ , asymptotic directions      166
$V_{n}$ in a $V_{m}$ , asymptotic lines      167 175 176
$V_{n}$ in a $V_{m}$ , conjugate directious      166 167 175 176
$V_{n}$ in a $V_{m}$ , equations      44
$V_{n}$ in a $V_{m}$ , equations of Gauss and Codazzi      163 172
$V_{n}$ in a $V_{m}$ , geodesics      75
$V_{n}$ in a $V_{m}$ , in an $S_{n+p}      195—197
$V_{n}$ in a $V_{m}$ , lines of curvature      168
$V_{n}$ in a $V_{m}$ , mean curvature      168 178
$V_{n}$ in a $V_{m}$ , mean curvature normal      169 170 178
$V_{n}$ in a $V_{m}$ , metric properties      45 48
$V_{n}$ in a $V_{m}$ , normal and relative curvatures of a curve      150—154 158 164 165
$V_{n}$ in a $V_{m}$ , normals      41 47 140 143—146 158
$V_{n}$ in a $V_{m}$ , of constant curvature      210—214
$V_{n}$ in a $V_{m}$ , parallelism      74 75 167
$V_{n}$ in a $V_{m}$ , relative curvature      174
$V_{n}$ in a $V_{m}$ , second fundamental form      166
$V_{n}$ in a $V_{m}$ , totally geodesic      183—186 249
$V_{n}$ in an $S_{n+p}$ , equations      187
$V_{n}$ in an $S_{n+p}$ , equations of Gauss and Codazzi      190 197 198
$V_{n}$ in an $S_{n+p}$ , evolutes      193 198
$V_{n}$ in an $S_{n+p}$ , lines of curvature      193 198 199
$V_{n}$ in an $S_{n+p}$ , motion      192
$V_{n}$ in an $S_{n+p}$ , principal radii of normal curvature      193 198
$V_{n}$ , class      188
$V_{n}$ , conformal to an $S_{n}$       91—95 121—124 182 214—218
$V_{n}$ , curvature      80 81 83 113 157 159
$V_{n}$ , elemental      8 39
$V_{n}$ , evolutes      193 198
$V_{n}$ , of class one      197—200
$W^{h}_{ijk}$       135 141
$\beta_{ij}$       120
$\delta^{i_{1}...i_{p}}_{\alpha_{1}...\alpha_{p}}$       101
$\delta^{i}_{j}$       2
$\delta^{i}_{j}$ , components of a tensor      10 15
$\delta^{i}_{j}$ , contraction      13
$\Delta_{1}\theta$ , $\Delta_{1}(\theta,\varphi)$ , $\Delta_{2}\varphi$       41
$\dfrac{\partial g_{ij}}{\partial x^{k}} = 0$ , along a curve      92
$\dfrac{\partial g_{ij}}{\partial x^{k}} = 0$ , at a point      55 56
$\gamma_{hijk}$       98
$\gamma_{hij}$       97
$\lambda_{n|ij}$ symmetric      115
$\lambda_{\alpha|^{i}}$       8 40
$\mu_{\tau\sigma|j}$       161 163
$\nu_{\tau\sigma|j}$       189
$\Omega_{ij}$       147 158 168
$\Omega_{\sigma|ij}$       160
$\varepsilon_{i_{1}...i_{n}}$ , $\varepsilon^{i_{1}...i_{n}}$       101
Abelian group      243 249
Angle, between hypersurfaces      41
Angle, between parametric curves      38
Angle, measure of      37 47
Applicable hypersurfaces, of a space of constant curvature      212
Applicable hypersurfaces, of an $S_{n}$       200 201
Associate curvature of a vector      73 164
Associate curvature of a vector of orders 1,...,n-1      106
Associate directions of a vector      73-78
Associate directions of a vector, in a sub-space      74 75 167
Associate directions of a vector, of normals to a hypersurface      157
Associate directions of a vector, of orders 1,...,n-1      105
Associate tensor      16
Asymptotic, directions in a hypersurface      156
Asymptotic, in a $V_{n}$ in a $V_{m}$       166 167
Asymptotic, lines of a $V_{n}$ in a $V_{m}$       167 175 176
Asymptotic, lines of a hypersurface      156—158
Beltrami      134 219 252
Bianchi      21 59 60 73 75 77 79 82 88 94 121 150 154 158 159 179 200 206 210 213 218 219—222 224 225 227 236 239 242—244 246 247 249 253—255
Bianchi, identity of      82
Birkhoff      55 255
Blaschke      107 254
BLISS      49 256
Bocher      23 109 144 145 201 253
Bolza      49 253
Bompiani      167 170 175 179 184—186 254—256
Brinkmann      94 95 215 255 256
Bromwich      108 111 253
Cartesian coordinates      34
Cartesian coordinates, generalized      84 87 120 187
Christoffel      5 17 21 26 28 252
Christoffel symbols      17
Christoffel symbols, as components of a tensor      22
Christoffel symbols, in two coordinate systems      19
Class of a $V_{n}$       188
Codazzi, equations of      149 150 162 172 190 198 211 212
Coefficients of rotation      97—100
Components, number      11
Components, of a tensor      9 10
Components, of a vector      4 7
Conformal, correspondence      89—91 182
Conformal, curvature tensor      91
Conformal, spaces      89—95 121—124 182 214—219
Conformal, transformations      230—233
Congruence of curves      6
Congruence of curves, canonical      125—128 139 140 154 241
Congruence of curves, geodesic      100 115 139 241
Congruence of curves, normal      114—125 128 140 142 154 185
Congruence of curves, normal to sub-spaces      41 47 140 144—146 158
Congruence of curves, principal      110 118
Conjugate directions, in a $V_{n}$ in a $V_{m}$       166 167 175 176
Conjugate directions, in a hypersurface      156
Conjugate tensor      15
Continuous group, Abelian      243 249
Continuous group, complete      244
Continuous group, conformal      230—233
Continuous group, constants of composition      224
Continuous group, generator      222
Continuous group, infinitesimal transformations      222
Continuous group, intransitive      225 233 235 236 244 249
Continuous group, invariant variety      225—227 235
Continuous group, minimum invariant variety      225 227 233 236 244 249
Continuous group, necessary and sufficient conditions      224
Continuous group, of one parameter      221
Continuous group, of r parameters      223
Continuous group, order of transformations of a      224
Continuous group, paths      222 231 235 239 240 241 249
Continuous group, simply transitive      247 249
Continuous group, sub-group of stability      225
Continuous group, transitive      225 236
Contraction      13
Contravariant, differentiation      31
Contravariant, tensor      9 10
Contravariant, vector      4 5 10 36 39
coordinates      see “Cartesian”
Coordinates, geodesic      56
Coordinates, normal      55
Coordinates, of Weierstrass      204—210
Coordinates, polar      34
Coordinates, Riemannian      53—55
Coordinates, transformation of      1 8 23 25 55 141
Correspondence, conformal      89 182
Correspondence, geodesic      131—139 141 227—230
Correspondence, isometric      84—88 184
Covariant derivative      27—31
Covariant derivative, zero      28 29 107 124 141 142

Covariant differentiation,      27—31
Covariant differentiation, of the inner product      29
Covariant differentiation, of the outer product      28
Covariant differentiation, of the sum or difference of tensors      28
Covariant tensor      9 10
Covariant vector      7 10
Covariant vector, geometrical interpretation      39
Covariant vector, null      36
Covariant vector, unit      36
Curl of a vector      27 31
Curvature of a $V_{n}$ , invariant      83 157
Curvature of a $V_{n}$ , mean      113
Curvature of a $V_{n}$ , Riemannian      79—81 113 159
Curvature of a $V_{n}$ , scalar      83 157
Curvature of a curve      61
Curvature of a curve, (n-1)      107
Curvature of a curve, geodesic      152
Curvature of a curve, in a hypersurface      150—152
Curvature of a curve, normal      150—154 158 165 193
Curvature of a curve, relative      151 165
Curvature tensor      see “Riemann tensor”
Curvature tensor, conformal      91
Curvature tensor, projective      135
Curvature, associate      see “Associate curvature”
Curvature, lines of      see “Lines of curvature”
Curvature, mean, of a $V_{n}$ in a $V_{m}$       168 178
Curvature, mean, of a hypersurface      168
Curvature, relative, of a $V_{n}$ in a $V_{m}$       174
Curve      6
Curve, curvature of      61 107
Curve, length      36 37
Curve, minimal      37
Curve, mutually orthogonal normals      107
Curve, of length zero      37
Curve, osculating geodesic surface      62
Curve, principal normal      61
Curve, tangent to      6
Curves, congruence of      see “Congruence”
Curves, orthogonal to a hypersurface      43
Darboux      120 253
Deltas, Kronecker      2
Determinants      101—103
Di Pirro      131 252
Dienes      65 255
Differential forms      see “Quadratic differential forms”
Differential parameters      41 47 124
Directions, associate      see “Associate”
Directions, asymptotic      see “Asymptotic”
Directions, conjugate      see “Conjugate”
Directions, principal      see “Principal”
Divergence, of a tensor      32
Divergence, of a vector      32 47
Dummy index      3
e      35 36
Eddington      69 101 255
Einstein      32 113 188 254
Einstein space      92 93 189 200
Einstein space, conformal to an $S_{n}$       93
Einstein space, spaces conformal to an      94 95
Eisenhart      31 46 69 72 118 124 133 140 142 184 251 253—256
Element, of length      34 35
Element, of volume      177 179
Ennuple      see “Orthogonal ennuple”
Equations, of Codazzi      see “Codazzi”
Equations, of Euler      see “Euler”
Equations, of Gauss      see “Gauss”
Equations, of Killing      see “Killing”
Euclidean space      34
Euclidean space, conformal representation of      94
Euclidean space, hypersurfaces      218
Euclidean space, motion      87 192
Euler, equations of      49 177
Euler, formula of      154
Evolutes of a $V_{n}$       193 198
Fermi      92 255
First integrals of the equations of geodesics      128-131 138 141 238
Flat space      84
Formulas of Frenet      107
Fubini      231 233 236 237 244 246 253
Fundamental form      34 57
Fundamental form, second      150 155 156 166
Fundamental form, third      219
G      14
Gauss      45 57
Gauss, equations of      149 150 163 172 190 197 200
Gaussian curvature of a geodesic surface      81
Geodesic, congruences      100 115 139 241
Geodesic, coordinates      56
Geodesic, correspondence      131—139 141
Geodesic, curvature      152
Geodesic, form of the linear element      57
Geodesic, surface      79 81
Geodesic, surface osculating a curve      62
Geodesic, torsion      174
Geodesic, triangle      79
Geodesic, variety      166 167 176 183—186 249
Geodesically parallel hypersurfaces      57 158 249
Geodesics      49—52; see also “First integrals etc.”
Geodesics, as paths of a motion      239
Geodesics, characteristic property      64
Geodesics, equations in finite form      59
Geodesics, equations of      50—52 141
Geodesics, in a space of constant curvature      139 207—210 213 219 230
Geodesics, in Riemannian coordinates      53
Geodesics, infinitesimal transformations preserving      227—230
Geodesics, minimal      51 210
Geodesics, of a sub-space      75 156
Geodesics, spaces with corresponding      131—139 141
Goursat      1 5 42 58 70 252 253
Gradient      7 27
Group      see “Continuous group”
Group of motions      233—251
Group of motions, intransitive      235 236 244 249
Group of motions, invariant varieties      235 245
Group of motions, minimum invariant varieties      236 244 249
Group of motions, of a regular space      250
Group of motions, of a space of constant curvature      207 238
Group of motions, order of transformations      238
Group of motions, paths      235 239 240 241 249
Group of motions, simply transitive      249
Group of motions, spaces $V_{2}$ admitting a      241—244
Group of motions, spaces $V_{3}$ admitting a      245
Group of motions, transitive      236 250 251
Group of motions, translations      239 241 249
hadamard      184 253
Herglotz      93 254
Homogeneous space      114
Hypercone, fundamental      215
Hyperquadric, fundamental      202 203
Hypersphere      202
Hypersurface, conjugate and asymptotic directions      156
Hypersurface, curvature of a curve in a      150—152
Hypersurface, curves orthogonal to a      43
Hypersurface, definition      8 41
Hypersurface, equations of Gauss and Codazzi      149 150 197
Hypersurface, geodesics      75 156
Hypersurface, isothermic      116 241
Hypersurface, lines of curvature      153—157 159 213 219
Hypersurface, mean curvature      168
Hypersurface, minimal      178
Hypersurface, normals parallel along a curve      158
Hypersurface, normals to a      41
Hypersurface, null normals to a      41
Hypersurface, of a flat space      197—201 218 219
Hypersurface, of a space of constant curvature      157 159 212 213 218—220
Hypersurface, principal radii of normal curvature      153 158
Hypersurface, relative and normal curvatures      151
Hypersurface, second fundamental form      150 155 156
Hypersurface, third fundamental form      219
Hypersurface, totally geodesic      183 184 249

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