Eisenhart L.P. — Continuous groups of transformations :: Электронная библиотека попечительского совета мехмата МГУ

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Eisenhart L.P. — Continuous groups of transformations

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Название: Continuous groups of transformations

Автор: Eisenhart L.P.

Аннотация:

This book sets forth the general theory of Lie and his contemporaries and the results of recent investigations with the aid of the methods of the tensor calculus and concepts of the new differential geometry. The first three chapters contain in the main the results of the first period. Chapter Four is devoted to the theory of the adjoint group and the sequence of theorems basic to the characterization of semi-simple groups, as developed by Cartan and recently by Weyl and Schouten. Although geometrical ideas are used throughout the book, Chapter Five contains the geometrical applications of the theory both in the space of the transformations and in the group-space, and here are to be found particularly the concepts of the new differential geometry. The theory of contact transformations with applications to geometry and mechanics is set forth in the closing chapter.

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Рубрика: Математика/Симметрия и группы/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1961

Количество страниц: 159

Добавлена в каталог: 29.04.2005

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Предметный указатель

$A_{\alpha}^{a}$ , $A_{\alpha}^{a}$ , $\bar{A}_{\alpha}^{a}$ , $\bar{A}^{\alpha}_{a}$       20—23 27—30 32 161 198—201 206 206 229
$T_{\alpha}$       14
$U_{\alpha}^{a}$ , $U^{\alpha}_{a}$ , $\bar{U}_{\alpha}^{a}$ , $\bar{U}^{\alpha}_{a}$       51—55
Abelian group      48—50 86 88 113 137 166 160 162 183 184 192 218—220 227
Absolute, invariant of a group      61 62 67 68 93 96 96 107
Adjoint group      151—166 160—163 170 183 199
Affine group      43 67 66
Affine group, parameter      199
Amaldi      267 296
ARC      199 212
Asymmetric connection      197 198 230 234
Asystatic group      84 86 123
Automorphism, $B^{h}_{ijk}$       196—198 232 233
Automorphism, $B_{\beta\gamma}$       206
Automorphism, of a group      151
Automorphism, of a space      230 231 236
B $\hat{o}$ her      69 146 177 294
Base, vector      141
Base, vector, change of      141 142
Basis of a function group      282 285 288
Basis, change of      28
Basis, of a group      28
Berwald      277 295
Bianchi      26 44 66 66 71 82 86 108 121 123 133 137 138 212 216 218 220 228 229 236 293 294
Bianchi, identity of      232
BOM      280 296
Bortolotti      237 296
Burnside      183 294
Burnside, $C^{e}_{ab}$       22 26 27
Burnside, $C_{abe}$       162 206
Canonical form of a function- group      286 290
Canonical, form of transformation      145—160
Canonical, parameter      46 47 62 160 161 160
Cartesian coordinates      188
Carton      66 167 169 173 174 179 180 182 184 199 201—208 213 236 237 293—296
Characteristic, equation      140—142 165—160 164 169 172 177-179
Characteristic, function      262 260 262 274 278 280 292
Characteristic, matrix      166
Christoffel symbol      188 236
Classification of groups      166 180—182
Clifford, parallelism of      237
Coefficient of linear connection      47 192
Cohen      91 294
Comitant      161
Commutative,      113
Commutative, function-groups      283 287 288 290
Commutative, functions      283
Commutative, groups      112 114—117 121 122 256
Commutative, transformations      110 112
Commutator      6
Complete system of differential equations      7 8 12 86
Completely integrable differential equations      1 4
Complex, linear      186
Composition, constants of      22
Composition, indices of      128
Composition, series of      127—134
Conformal spaces      236
Connection,       199 230
Connection,       199 230
Connection, (0)      199 206 231
Connection, asymmetric      197 198 230 234
Connection, coefficients of      47 192
Connection, linear      45 47 192
Connection, symmetric      194 198 233 234 236
Constant, of composition      22
Constant, of composition of structure      22 26 28 43 48 56 112 126 151
Contact transformation, geometrical properties      242—246 267 267—277
Contact transformation, geometrical properties of maximum rank      263—273 276
Contact transformation, geometrical properties, homogeneous      239—266 263—278 291 292
Contact transformation, geometrical properties, infinitesimal      251—266 260 262 281 292
Contact transformation, geometrical properties, non- homogeneous      255—263 281
Contact transformation, geometrical properties, restricted non-homogeneous      260—263 280 281 286 287 291
Continuous group, finite      16 17
Continuous group, finite, infinite      15
Continuous group, finite, mixed      68
Contravariant, tensor      187
Contravariant, vector      26 36 187
Cosine of angle      209 212
Covariant, derivative      189
Covariant, differentiation      189 196 198 232 233
Covariant, normal      239 242—244 246
Covariant, tensor      187
Covariant, vector      27
Curvature,      195—198 206 207
Curvature, of a curve      98
Curvature, of a curve, $\delta_{b}^{a}$       8
Curvature, tensor      188
Derivative, covariant      189
Derived group      132—135 154 164 169 173 184
Dickson      15 91 294
Differential equations of a group      20
Differential equations of definition of a group      56 57
Differential equations of second order      100 107 108
Differential equations, admitting a group      82 86 88 89 95 96 100 101 107 135
Differential equations, admitting linear operators      85—91 106—108
Differential equations, associated      5 9
Differential equations, complete      7 8 12 86
Differential equations, completely integrable      1 4
Differential equations, linear partial      7 12 63 82 85—88
Differential equations, mixed      4
Differential equations, ordinary of first order      89 95—97
Differential equations, Pfaffian      93—95
Differential equations, system of      1 7
Differential equations, total      1
Differential invariant      96—98 100—107
Differentiation, covariant      189 195
Differentiation, covariant,      232 233
Dilatation      42
Direct product of groups      120—122 174 184
Displacement, parallel      199
Distance      191
Dummy index      7
Dynamics      277—280
Dynamics,       186
Dynamics, $\eta_{j}^{i}(u)$       60 152
Eiesland      236 295
Einstein      196 295
Einstein space      206
Eisenhart      80 199 236 239 294—296
Elementary, divisor      146
Elementary, hyperplane      238
Elementary, hypersurface      267
Engel      54 91 164 293 296
Envelope      244 245 268 271 272
Equations, of a group      13 16 20 56 57
Equations, of a motion      209
Equipollent vectors      201—203 207 208
Equivalent function-groups      287 291
Equivalent groups      34 76—80 84 115
Equivalent points      67
Essential parameters      9—12 56
Euclidean space      186 188 190 191
Euclidean space,      208 212 218
Exceptional $G_{1}$ of a $G_{r}$       113 118 152 162 183
Extended group      92—97 107 252
Extension of a group      97—101 105 125
Factor group      129—131 137
Fermat’s principle      280
Fine      13 15 243 246 258 264 294 295
Finsler      277 294
Flat space      188 193 203—205
Franklin      91 295
Frobenius      183 294
Fubini      217 218 221 229 294
Function group, basis of      282 285—291
Function group, commutative      283 287 288 290
Function group, equivalent      287 291
Function group, homogeneous      287—292
Function group, non-homogene- ous      281—287 291 292
Function group, reciprocal      283 285 288 290 291

Function group, subgroup of      282—286 290
Functions, commutative      283
Functions, in involution      250 283
Functions, singular      283 284 287 291 292
Fundamental theorem,      216 277
Fundamental theorem, $g^{ij}$       188 189 194 195
Fundamental theorem, $G_{2}$       60 61 90 91 135 137 155 184 228
Fundamental theorem, $G_{3}$       66 67 123 133 137 138 154 184 227 236 255
Fundamental theorem, $G_{4}$       66 123 136 137 155 184
Fundamental theorem, $g_{ij}$       156 174 186—189 194 195 205
Fundamental theorem, $G_{r}$       17
Fundamental theorem, $\Gamma_{\beta\gamma}^{\alpha}$       44 194 199 206 231—233
Fundamental Theorem, First      24
Fundamental Theorem, Second      54
Fundamental theorem, third      55
Fundamental,      208 226 236
Fundamental, quadratic form      186
Fundamental, tensor      187 195
Fundamental, variety      244
Fundamental, vector      148 166 170
Generic, rank      49 64
Generic, root      156 160
Geodesic      191 193 206 207 209—212 216 217 235 274—277 279
Geodesic, correspondence      236
Geodesic, hypersphere      275 279
Geodesic, sub-space      203
Geodesically parallel hypersunaces      211 235
Graham      182
Group      14
Group, admits a transformation      110 112 121
Group, basis of      28
Group, finite continuous      16 17
Group, generator of      40
Group, infinite      15 249
Group, infinitesimal transformation of a      36 40 41
Group, invariant under a transformation      110—112 121
Group, mixed      58
Group, one- parameter      32—36
Group, rank of a      159 160
Group, semi      15 17 18 21 28 29 31
Group, structure of a      28
Group, symbols of a      28
Group, trajectories of a      35 40 44 45 253 274 affine asystatic derived extended extension factor imprimitive induced integrable intransitive linear motion primitive projective semi-simple simple systatic transitive)
Group-space      27 31 44—47 198—206
Group-space,      213 229—231 236
Groups with the same structure      28 29 32 76 78 80 93 98 125 equivalent isomorphism reciprocal)
Groups, classification of      165 180—183
Groups, classification of, product of      120—122 174 184
Groups, similar      76
Hamiltonian equations      263 279
Hamiltonian equations, function      263 279
Holonomic system      263 277
Homothetic transformation      42
Huntington      15 294
Huygen’s principle      272
Hyperplane      238
Hyperplane, tangential      239 267
Hypersphere      275 279
Hypersurface      210 237
Hypersurface of origin      271 272
Hypersurface, elementary      267
Hypersurface, geodesically parallel      210 211 235
Identity      15
Identity of Bianchi      232
Identity of Jacobi      6 250
Identity of Ricci      189 232
Identity, transformation      14
Imprimitive group      80—83 117 123 137
Imprimitivity, system of      80—85 117 137
Independent (cpnstant coefficients)      21
Indices of composition      128
Induced group      70—72 84 85 119 123 124 137—139
Infinitesimal, motion      208 209 231
Infinitesimal, motion, transformations of a group      36 40 41
Integrable group      134—137 144 145 162—165 172 173 183 184
Intransitive group      71—73 78—80 98
Intransitive group,      210 216—226 235 236
Invariant of points      108
Invariant sub-group      118—124 129—134 137—139 144 153 154 158 161—163 173 174 183 184 203 235
Invariant sub-group, maximum      122 127—131
Invariant variety      67—72 80 81 85
Invariant variety,      123 138 153 210 211
Invariant variety, isolated      119
Invariant variety, minimum      67—71 211 221 235
Invariant vector-spaces      142—145 163
Invariant, absolute, of a group      61 62 67—69 93 95 96 98 107 159 161
Invariant, differential      96—98 100—107
Invariant, direction      140 142 144
Invariant, numerical      291
Invariant, relative, of a group      63 66 67 69 93 95 107
Inverse transformation      13 249
Inverse transformation of a semi-group      17
Involution, functions in      250 283
Isolated variety      119
Isomorphism of groups, simple      124—127 131 137 151 208
Isomorphism of groups, simple, multiple      124—127 129 136 151 184
jacobi      263 293
Jacobi, form of a complete system      8
Jacobi, identity      6 250
Jacobi, relations      26
Killing      158 164 180 183 184 208 293
Killing, equations of      208 215—221 237
Knebelman      236 295 296
Kronecker delta      8
Kronecker delta, $L_{\beta\gamma\delta}^{\alpha}$       198
Kronecker delta, $L_{\beta\gamma}^{\alpha}$       29 198 199 229
Kronecker delta, $\Lambda_{\beta\gamma\delta}^{\alpha}$       75 193 195
Kronecker delta, $\Lambda_{\beta\gamma}^{\alpha}$       74 192—197 217 218 230
Lagrangian function      263 278
Length, element of      186
Levi — Civita      190 267 294 295
Lie      36 39 54 55—58 66 80 84 91 106—108 119 122 123 131 137 145 154 155 163 164 180 183—185 242 248—252 255 258—261 278 281 282 285 287-292 293
Lie group      40
Linear, complex      185
Linear, complex, group      43 138 143—150 153 163 184 234 292
Matrix, characteristic      155
Matrix, equation      142—144 148—150
Matrix, trace of a      157
Mattoli      194 295
Maurer      23 293
Maximum invariant sub-group      122 127—131
Metric, generalized      277
Metric, Riemannian      187 206
Michal      43 295
Minimum invariant variety      67—71
Minimum invariant variety,      221 223 235
Mixed, group      58
Mixed, system of differential equations      4
Moore      15 294
Motion, in 2—space      42 80 98 226—228
Motion, in 3—space      43 80 228
Motion, infinitesimal      209
Motions, group of      57 208—221 230—237
Motions, in a linearly connected manifold      229—235 237
Motions, intransitive      210 216—226 235 236
Motions, multiply transitive      221
Motions, of maximum order      215 224 236
Motions, simply transitive      218 235 236
Multiply isomorphic groups      see Isomorphism
Normal, covariant      239 242 243
Null vector      174 187 212
Null vector, $\Omega_{jk}^{i}$       195—198 205—207 231—235 237
Operator, linear      6
Parallelism,       199—205
Parallelism,       199—205
Parallelism, (0)      199 204 205
Parallelism, absolute      190 193 194
Parallelism, absolute,      217 234 235
Parallelism, relative      190 191 193 199
Parameter affine      199

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