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| Dubrovin B.A., Fomenko A.T., Novikov S.P. — Modern Geometry - Methods and Applications. Part 1. The Geometry of Surfaces, Transformation Groups and Fields |
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| Предметный указатель |
Linear vector field 223
Liouville's equations 118
Liouville's theorem on conformal transformations 138
Lobachevskian plane 93
Lobachevskian, isometries of 94 117
Lobachevskian, Klein's model of 94
Lobachevskian, Poincare model of 93
Local co-ordinates 63 65
Local group of diffeomorphisms 208
Local principle 443
Lorentz contraction 58
Lorentz transformation 53 57
Lorentz, matrix of 54
Lorentz, physical consequences of 57
Lowering of indices of a tensor 171
Loxodrome 76
Magnetic field 197 389
Magnetic strength 389
Mass hyperboloid 328 415 418 441
Maupertuis' Principle 345
Maxwell's as Euler — Lagrange equations 390—392
Maxwell's equations 246—247
Maxwell's, geometric content of 264—266
Metric induced on the sphere 87 115
Metric induced, conformal form of 114
Metric Lobachevsky, conformal form of 114
Metric, Euclidean 20
Metric, invariant (on a transformation group) 227
Metric, Kaehlerian 273
Metric, Lobachevsky 93
Metric, Minkowski 21
Metric, motion of see “Isometry”
Metric, pseudo-Euclidean 21
Metric, pseudo-Riemannian 20
Metric, Riemannian 18
Moebius trefoil 406
Moment four-tensor 336
Momentum 320
Momentum four-vector 329 385
Momentum, angular 333 336 387
Natural parameter 13 52 322
Neutral surface of a thin plate under bending 407—408
Nodal line 125
Non-singular point 6 62 63 65 67
Non-singular second-rank tensor 161
Normal section 81
Normal spherical map see “Gauss map”
Normal to a surface 70
One-parameter group of diffeomorphisms 208
One-parameter of matrices 130
One-parameter, velocity field of 209
Operation * on tensors 174
Operation of contraction of indices 158
Operation of permutation of indices 158
Orbit of a group 177
Ordinary point see “Non-singular point”
Orientation of a cube 267
Parallel equation of 289
Parallel tensor along a curve 288—289
Parallel transport 289—290
Pauli matrices 131 433
Periodic crystal 180
Perturbation of the arguments of a functional 381
Peterson — Codazzi equations 313
Pfaffian 349
Phase space 338
Phase, abstract 350
Phase, canonical co-ordinates for 350
Phase, canonical transformation of 359
Phase, extended 351
Phase, skew-symmetric metric on 350
Photon number 394
Plane transformations 25
Platonic polyhedra 191
Poincare group 53
Poisson bracket 352
Poisson coefficient 408
Poisson commutator see “Poisson bracket”
Poisson's equation 424
Poly vector 167
Poynting's vector 205
Principal curvatures 77 81
Principal directions of a pair of quadratic forms 80
Product of tensors 159
Proper time 52 327 418
Pseudosphere 54 90
Pseudosphere, metric in conformal form 114
Pseudosphere, metric induced on 92
Pullback see “Restriction of a tensor”
Quadratic form 17 37
Quadratic Hamiltonian as generalized oscillator 361
Quadratic, eigenvalues of 172
Quadratic, index of inertia of 21
Quaternions 131
Raising of indices of a tensor 171
Real analytic function 110
Realization map of a complex vector space 96
Realization, analogue for quaternions 134
Region 4
Region, boundary of 4
Region, bounded 5
Relativistic analogue of centre of mass 337
Relativistic correction to straight-line trajectory of light ray 427
Residue theorem 261
Restriction of a tensor 206
Riemann curvature tensor 301
Riemann surface 107
Right-invariant field 237
Rigidity of an elastic plate 412
Rotation group see “Special orthogonal group”
Rotation of a covector field see “Curl”
Scalar curvature 306
Scalar product in a Riemannian metric 18
Scalar product of vector and covector 160
Scalar product, invariant (on a Lie algebra) 227
| Schwarz derivative 119
Schwarzschild metric 426
Schwarzschild radius 426
Second variation of an arc 371—372
Semi-direct product 26 27
Semi-spinor representations of 50(1 3)
Serret — Frenet formulae 40 45 60
Serret — Frenet in matrix form 48
Sine — Gordon equation 314
Skew-symmetric tensor 164 165
Skew-symmetric, canonical form of 201
Skew-symmetric, divergence of 245
Slow particle 421
Slow time-like geodesic 330
Smooth curve 13
Smooth function 6
Space, Cartesian 2
Space, configuration 362
Space, Euclidean 9
Space, Minkowski 21 50
Space, pseudo-Euclidean 21 50
Space, pseudo-Hermitian 99
Space, pseudo-Riemannian 21
Space, Riemannian 17
Space-like curve 417
Space-like vector 50 417
Space-time continuum 2 21 50
Space-time interval 22
Special unitary group 98
Spinor representation of O(1,3) 436
Spinor representation of SO(3) 435
Spinor, Dirac conjugate of 439
Spinor, four-component 436
Spinor, two-component 433—435
Stationary function for a functional 381
Stationary group of a crystal lattice 186
Stereographic projection 87 91
Stokes' formula 264
Strain for small deformations 155 215
Strain tensor 155
Stress tensor 154
Structural constants 303
Summation convention 158
surface 62 64
Surface area 73 74
Surface area formula 75
Surface integral 249 256
Surface, closed 256
Surface, convex space-like 95
Surface, curvature form of 257
Surface, first fundamental form of 69
Surface, induced metric on 69
Surface, k-dimensional 66
Surface, local parameters on 62
Surface, minimal 398
Surface, non-singular point of 62 63 65 67
Surface, second fundamental form of 78
Surface, strictly convex 85
Surface, unit normal to 70
Symmetric tensor 163
Symmetry group of a crystal see “Crystallographic point group”
Symplectic group 135
Symplectic transformation 361
Tangent plane 66 67
Tangent vector 10
Tensor 152
Tensor, covariantly constant see “Parallel tensor”
Tensor, Ricci 306
Tensor, torsion 285
Tensor, trace of 158
Tensor, transformation rule for 152
Tetrad 303 452
Theorema egregium 77 306
Three-dimensional formalism 328 417
Time-like curve 417
Time-like vector 50 417
Torsion of a curve 45
Trajectory bundle 362—363
Trajectory, action of 365
Transformation 23
Transformation rule for an arbitrary tensor 152
Transformation rule for gradient of a function 15 147
Transformation rule for linear operators 150
Transformation rule for quadratic forms 17 148
Transformation rule for vectors 14 15 146
Transformation, aflfine 27 33
Transformation, complex aflfine 30
Transformation, conformal 30
Transformation, Galilean 37 52
Transformation, Lorentz 53 57
Transformation, orthochronous 55
Transformation, orthogonal 28
Transitive group 178
Translation 25
Translation group 182
Translation subgroup 181
Translation-invariant crystal lattice 177
Unit-speed curve 78
Unitary group 98
Unitary matrix 98
Unitary transformation 98
Universal gravitational constant 399 425
Variational class 340
Variational derivative 320 381
Vector field 15
Vector gradient 348
Vector product see “Sross product”
Velocity field 209
Velocity vector see “Tangent vector”
Wave front 369
Weyl's equations 441
World time 423
World-line of a point particle 3 50
Yang — Mills field 455
Young's elastic modulus 408
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