O'Neill B. — Semi-Riemannian Geometry: With Applications to Relativity :: Электронная библиотека попечительского совета мехмата МГУ

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O'Neill B. — Semi-Riemannian Geometry: With Applications to Relativity

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Название: Semi-Riemannian Geometry: With Applications to Relativity

Автор: O'Neill B.

Аннотация:

This book is an exposition of semi-Riemannian geometry (also called pseudo-Riemannian geometry) — the study of a smooth manifold furnished with a metric tensor of arbitrary signature. The principal special cases are Riemannian geometry, where the metric is positive definite, and Lorentz geometry. For many years these two geometries have developed almost independently: Riemannian geometry reformulated in coordinate-free fashion and directed toward global problems, Lorentz geometry in classical tensor notation devoted to general relativity. More recently, this divergence has been reversed as physicists, turning increasingly toward invariant methods, have produced results of compelling mathematical interest.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 1st edition

Год издания: 1983

Количество страниц: 468

Добавлена в каталог: 04.07.2008

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Предметный указатель

Hyperbolic space      111 113 156 228 278
Hyperbolic space, upper and lower imbeddings of      111
Hyperquadric      108—110
Hyperquadric, curvature of      113
Hyperquadric, frame-homogeneity of      113
Hyperquadric, geodesics in      112—113 149—150
Hyperquadric, isometrics of      113 239
Hyperquadric, product manifold structure of      110
Hyperquadric, sign of      108
Hypersurface      20 106—108 124
Hypersurface, totally umbilic      116—118
Identity component $G_{0}$       447
Imbedding, isometric      122
Imbedding, smooth      19
Immersed submanifold      19
Immersion, isometric      121—122
Immersion, smooth      19
Impact parameter      381
incomplete      see “Complete entries”
Indefinite (or semi-) unitary group U(p,q)      324
Indefinite (or semi-) unitary group U(p,q), fundamental group      329
Indefinite (or semi-) unitary group U(p,q), Lie algebra      324
Index of scalar product      47
Index of scalar product space      51
Index of semi-Riemannian manifold      55
Index, form      269
Induced connection on submanifold      98—99
Induced topology      15—16
Inextendible      see “Extendible entries”
Inner product      47
Instantaneous observer      180 333
Integral curve      27
Integral curve, maximal      28
Interpretations of tensors      36
Inverse function theorem      10
Involutive map      231
Isometric invariant      58
Isometry      58
Isometry group      233
Isometry group as Lie group      255
Isometry, local      90—91
Isotropy subgroup      255
Isotropy, local spatial      342
Isotropy, observed      341
Jacobi equation      216
Jacobi field      216 232
Jacobi identity      13 447
Jacobian function      196
Jacobian matrix      10
Kaehler manifold      325
Kepler’s laws      453—455
Killing form      302
Killing vector field      250—256
kinetic energy      159
Kobayashi’s proposition      321
Koszul formula      61
Kruskal plane      386—388 399 400
Kruskal spacetime      389—391
Kruskal spacetime, geodesics in      395—398
Kruskal spacetime, isometries of      399
Kruskal spacetime, truncated      392
Kulkarni’s theorem      229
Laplacian      86 213
Leaf of warped product      205
Left- and right-multiplication      447—448
Left-invariant vector field      448
Levi-Civita connection      61
Lie algebra      447
Lie algebra of Lie group      448
Lie algebra, abelian      301
Lie algebra, semisimple      306
Lie bracket      see “Bracket operation”
Lie derivative      46 53 195 250
Lie exponential map      449 450
Lie groups, basic theory and examples      446—452
Lie groups, further properties      300—304 (see also individual Lie groups e.g. “Orthogonal
Lie subspace      310
Lift of functions, vectors, and vector fields      25 205
Lift of mapping      443
Lift of tensors      210
Light-like particle      163 380—384 392—393
Lightcone      see “Nullcone”
Lightlike      see “Null entries”
Lightlike submanifold      142
Lightlike subspace      141
Limit sequence      405
Line element      56
Linear isometry      51
Linear isomorphism      xiii
Linear isotropy group      311
Linear operator      242—243
Local diffeomorphism      10
Local isometry      90 91
Local section      32
Locally symmetric semi-Riemannian manifold      215 219—224
Loop (or closed curve)      186
Lorentz coordinate system      164 167
Lorentz group      235 240
Lorentz manifold      55 126 143—149
Lorentz surface      150—153
Lorentz vector space      140
Lorentz — FitzGerald contraction      175
Lorentz, H.A.      158
Manifold, Lorentz      55
Manifold, Riemannian      55
Manifold, semi-Riemannian      54
Manifold, smooth      3
Manifold, smooth, construction of      23
Manifold, topological      413
Map (mapping), smooth      5
Marsden’s proposition      258
Mass of particle      159 163 177
Mass of Schwarzschild and Kruskal spacetimes      367 389
Matched covering      203
Material particle      163
Matter      335
Maximal geodesic      68
Maximal integral curve      30
Mean curvature      101 123 124
Metric equivalence      60
Metric tensor      54
Microwave background radiation      357
Minimizing geodesic      136
Minkowski space(time)      55 163
Minkowski, H.      158
Misner-completeness      156
Momentum, Newtonian      159
Momentum, relativistic      177—178 180
Myers’ theorem      279
Natural coordinate functions on $R^{n}$       1
Newtonian gravitation      453—455
Newtonian motion      159—161
Newtonian space      159
Newtonian time      159
Nondegenerate bilinear form      46
Nondegenerate subspace      49
nor (normal projection)      98 205 344 369
Normal bundle      198
Normal bundle, exponential map of      199
Normal connection      114 118
Normal connection, covariant derivative      114 119
Normal connection, curvature tensor      125
Normal connection, parallel translation      119
Normal coordinates      72—73
Normal curvature vector      105 106 108
Normal neighborhood of point      71
Normal neighborhood of submanifold      199
Null geodesic and causality      404 430—431 435—437
Null geodesic in hyperquadrics      149—150
Null geodesic in surfaces      152 153

Null geodesic, closed nonperiodic      193
Null geodesic, focal points on      290—298
Null geodesic, Kruskal      400
Null geodesic, Robertson — Walker      353—357
Null vector      48 56
Nullcone      53 56 109 128
NullSpace      53
Nullspace of index form      272 285
Observer      167
Observer field      358
Observer field, geodesic      358
Observer field, irrotational      358
Observer field, proper time synchronizable      359
Observer field, synchronizable      359
Observer, instantaneous      180 333
Observer, Schwarzschild      371
One-form      15
One-parameter subgroup      449
Open submanifold      3—4
Orbit of point      187
Orbit, free fall, in Schwarzschild spacetime      374—384
Orbit, manifold      187 188 191
Order of conjugate point      271
Order of focal point      283 291
Orientation (orientability, oriented) of hypersurface      189 197
Orientation (orientability, oriented) of manifold      23 189 195
Orientation (orientability, oriented) of vector bundle      198
Orientation (orientability, oriented) of vector space      189
Orientation (orientability, oriented), covering manifold      190
Orientation (orientability, oriented), natural, of $R^{n}$       189
Orientation (orientability, oriented), space-orientation      237 240—242
Orientation (orientability, oriented), time-orientation      see “Time-orientation”
Orientation-preserving map      190
Orientation-reversing map      190
Orthogonal coordinate system      64
Orthogonal group O(n)      451
Orthogonal group O(n), components of      238
Orthogonal group O(n), Lie algebra of      451
Orthogonal projection      50
Orthogonal vectors      48
Orthonormal basis      50
Orthonormal expansion      50
Pair isometry      102
Paracompactness      22
Parallel tensor field      65
Parallel translation      66
Parallel vector field on curve      66
Parameter curve      122
Partial derivative on manifold      7
Particle, Newtonian      159
Particle, relativistic (material and lightlike)      163
Partition of unity      22
Past (dual of future)      163 402
Past (dual of future) for Past entries      see “Future entries”
Penrose, R.      401
Penrose’s singularity theorem      434—437
Perfect fluid      337—339 361 362
Perfect fluid, energy density of      339
Perfect fluid, pressure of      339
Perfect fluid, Robertson — Walker model of      345—347 362
Perihelion advance      378—380
Perp operation      49
Photon      163
Photon, frequency and wavelength of      179
Physical equivalence      336
Physical singularity      348
Piecewise smooth curve      11
Piecewise smooth curve segment      11
Piecewise smooth variation      264
Poincare half-plane      94 151 260
Poincare, H.      158 442
Polar map      221
Position vector field      26 128
Pregeodesic      69
Product manifold, semi-Riemannian      57 89
Product manifold, smooth      4 24
Product rule      44
Projective space, complex      327
Projective space, real      188 192 247 259
Proper time      163
Properly discontinuous group      188
Pseudohyperbolic space      110 (see also “Hyperquadric”)
Pseudosphere      110 (see also “Hyperquadric”)
Pullback      42
Q(v,w)      77
Quadratic form      47
Quasi-limit      404
Radiation cosmological model      353 362—363
Redshift, cosmological      354
Reductive homogeneous space      310
Reductive homogeneous space, naturally      312
Relative speed      172
Reparametrization of a curve      11 132
Rest photon in Kruskal spacetime      393 395
Restspace in general relativity      358
Restspace in special relativity      171
Ricci curvature      87
Ricci equation      125
Ricci flat      87
Riemann, G.F.B.      54
Riemannian manifold (metric)      55
Riemannian manifold (metric), completeness of      138
Riemannian manifold (metric), existence of      140
Riemannian symmetric space      319—321
Riemannian symmetric space of compact type      319
Riemannian symmetric space of noncompact type      319
Robertson — Walker spacetime, construction of      341—343
Robertson — Walker spacetime, cosmology of      347—350
Robertson — Walker spacetime, geodesics in      353
Robertson — Walker spacetime, perfect fluid in      345—346
Robertson — Walker spacetime, space of      343
Scalar curvature      88
Scalar product      47
Scalar product space      48—53
Scalar product space as semi-Riemannian manifold      58—59
Scalar product, Hermitian      324
Scalar product, natural Hermitian      324
Schwarz inequality      141
Schwarzschild black hole      367
Schwarzschild exterior      367
Schwarzschild free fall orbits      374—384
Schwarzschild half-plane $P_{I}$       251
Schwarzschild observers      371
Schwarzschild radius function      365
Schwarzschild spacetime $N\bigcup B$ , construction of      364—367
Schwarzschild spacetime $N\bigcup B$ , curvature of      369
Schwarzschild spacetime $N\bigcup B$ , extension of      386—390
Schwarzschild spacetime $N\bigcup B$ , geodesics in      372—384
Schwarzschild strip $P_{II}$       367
Schwarzschild time function      364
Second countability      21
Second fundamental form      100 107
Sectional curvature      77 124
Self-adjoint linear operator (matrix)      243 260—262
Semi-Euclidean space      55
Semi-Euclidean space, geodesics in      69
Semi-Euclidean space, isometries of      239—240
Semi-Riemannian covering      191 201
Semi-Riemannian hypersurface      see “Hypersurface”
Semi-Riemannian manifold      54
Semi-Riemannian manifold, homogeneous      see “Homogeneous space”
Semi-Riemannian manifold, isotropic      260
Semi-Riemannian manifold, locally symmetric      215 219—224
Semi-Riemannian manifold, symmetric      see “Symmetric space”
Semi-Riemannian submanifold      57 97—125
Semi-Riemannian submersion      212—213
Semi-Riemannian surface      80—81 94 124 150—153 156 262
Semi-Riemannian warped product      see “Warped product”
Semiorthogonal group $O_{\nu}(n)=O(p,q)$       234
Semiorthogonal group $O_{\nu}(n)=O(p,q)$ , components      236—238

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