This book provides an introduction to and survey of recent developments in pseudo-Riemannian geometry, including applications in mathematical physics, by leading experts in the field. Topics covered are: Classification of pseudo-Riemannian symmetric spaces Holonomy groups of Lorentzian and pseudo-Riemannian manifolds Hypersymplectic manifolds Anti-self-dual conformal structures in neutral signature and integrable systems Neutral Kahler surfaces and geometric optics Geometry and dynamics of the Einstein universe Essential conformal structures and conformal transformations in pseudo-Riemannian geometry The causal hierarchy of spacetimes Geodesics in pseudo-Riemannian manifolds Lorentzian symmetric spaces in supergravity Generalized geometries in supergravity Einstein metrics with Killing leaves The book is addressed to advanced students as well as to researchers in differential geometry, global analysis, general relativity and string theory. It shows essential differences between the geometry on manifolds with positive definite metrics and on those with indefinite metrics, and highlights the interesting new geometric phenomena, which naturally arise in the indefinite metric case. The reader finds a description of the present state of the art in the field as well as open problems, which can stimulate further research.
Author(s): Dmitri V. Alekseevsky and Helga Baum
Publisher: European Mathematical Society
Year: 2008
Language: English
Commentary: no
Pages: 551
Cover......Page 1
ESI Lectures in Mathematics and Physics......Page 3
Title......Page 4
ISBN 978-3-03719-051-7......Page 5
Preface......Page 6
Contents......Page 10
Introduction......Page 12
Examples of metric Lie algebras......Page 16
Metric Lie algebras and quadratic extensions......Page 18
Quadratic cohomology......Page 22
A classification scheme......Page 26
Classification results for metric Lie algebras......Page 28
Symmetric triples and quadratic extensions......Page 30
The geometry of quadratic extensions......Page 32
Symmetric spaces of index one and two......Page 36
Examples of geometric structures......Page 41
Pseudo-Hermitian symmetric spaces......Page 43
Quaternionic Kähler and hyper-Kähler symmetric spaces......Page 46
Extrinsic symmetric spaces......Page 51
Manin triples......Page 56
Implications of (h,K)-equivariance......Page 57
Proof of Proposition 2.12......Page 58
Proof of Proposition 3.8......Page 60
References......Page 61
Introduction......Page 64
Holonomy groups of linear connections......Page 67
Holonomy groups of semi-Riemannian manifolds......Page 69
Lorentzian holonomy groups......Page 71
Basic algebraic properties......Page 72
Four types of indecomposable subalgebras of p......Page 73
Berger algebras and weak-Berger algebras......Page 74
Decomposition of the space of curvature endomorphisms......Page 75
Consequences for Lorentzian holonomy......Page 76
The classification of weak-Berger algebras......Page 78
Real and complex weak-Berger algebras......Page 79
Weak-Berger algebras of unitary type......Page 80
Weak-Berger algebras of real type......Page 81
Local description and coordinates......Page 86
Metrics that realise all types of Lorentzian holonomy......Page 88
Some examples......Page 91
pp-waves and generalisations......Page 95
Holonomy of space-times......Page 99
Parallel spinors on Lorentzian manifolds......Page 100
Holonomy of indecomposable, non-irreducible Einstein manifolds......Page 101
Open problems and outlook on higher signatures......Page 103
References......Page 104
Introduction......Page 108
The split quaternions......Page 109
Hypersymplectic manifolds......Page 110
Hypersymplectic quotients......Page 112
Toric constructions......Page 114
Cut constructions......Page 119
References......Page 120
Introduction......Page 124
Conformal compactification......Page 127
Spinors......Page 128
Anti-self-dual conformal structures in spinors......Page 129
Integrable systems and Lax pairs......Page 130
Curvature restrictions and their Lax pairs......Page 131
Non-null case......Page 138
Null case......Page 142
Twistor theory......Page 143
The analytic case......Page 144
LeBrun–Mason construction......Page 150
Topological restrictions......Page 152
Tod's scalar-flat Kähler metrics on S^2 x S^2......Page 153
Compact neutral hyper-Kähler metrics......Page 154
Ooguri–Vafa metrics......Page 155
References......Page 156
A neutral Kähler surface with applications
in geometric optics by
Brendan Guilfoyle andWilhelm Klingenberg......Page 160
Introduction......Page 161
Coordinates on L......Page 162
The correspondence space......Page 164
Jacobi fields along a line in E^3......Page 165
The symplectic structure......Page 166
The neutral metric......Page 167
Line congruences......Page 168
Parametric line congruences......Page 169
Normal line congruences......Page 170
Surfaces given by zero of a function......Page 171
Example: elliptic and hyperbolic paraboloids......Page 172
Reflection in a surface......Page 173
Example: plane wave scattered off a paraboloid......Page 174
Focal points of a line congruence......Page 175
Focal sets and the Kähler metric......Page 176
Further geometric properties......Page 177
The focal set of a plane wave reflected off the inside of a cylinder......Page 178
A point source reflected off a cylinder......Page 179
The focal set of a point source reflected off the inside of a cylinder......Page 180
Discussion of results......Page 181
Generalizations......Page 183
Higher dimensions......Page 184
Geodesics on 3-manifolds other than E^3......Page 185
Neutral Kähler structures on TN......Page 186
References......Page 188
A primer on the .2 C 1/ Einstein universe by
Thierry Barbot, Virginie Charette, Todd Drumm,William M. Goldman,
and Karin Melnick......Page 190
Introduction......Page 191
Synthetic geometry of Einn^n,1......Page 192
Lorentzian vector spaces......Page 193
Minkowski space......Page 194
Einstein space......Page 195
2-dimensional case......Page 197
3-dimensional case......Page 199
The conformal Riemannian sphere......Page 200
The conformal Lorentzian quadric......Page 202
Involutions......Page 203
Time orientation......Page 205
Future and past......Page 207
Geometry of the universal covering......Page 208
Improper points of Minkowski patches......Page 209
The inner product on the second exterior power......Page 210
Lagrangian subspaces and the Einstein universe......Page 211
Positive complex structures and the Siegel space......Page 213
The contact projective structure on photons......Page 215
Summary......Page 216
Structure theory......Page 217
Symplectic splittings......Page 219
Parabolic subalgebras......Page 220
Weyl groups......Page 221
Projective singular limits......Page 222
Cartan's decomposition G = K A K......Page 223
Maximal domains of properness......Page 225
Crooked surfaces......Page 227
Crooked planes in Minkowski space......Page 228
An example......Page 229
Topology of a crooked surface......Page 230
Automorphisms of a crooked surface......Page 234
Spine reflections......Page 235
Actions on photon space......Page 237
Some questions......Page 238
References......Page 239
Introduction......Page 242
Conjecture in the pseudo-Riemannian framework......Page 245
Lichnerowicz's conjecture for parabolic geometries......Page 247
Some words about the proof of Theorem 1.2......Page 250
Conformal dynamics on compact manifolds......Page 253
Geometry of Einstein's universe......Page 254
Examples of essential dynamics on Einstein's universe......Page 257
Schottky groups on Einstein's universe......Page 259
More complicated essential dynamics......Page 260
Essential versus isometric dynamics......Page 263
Stable conformal dynamics and its consequences on the geometry......Page 264
Examples where stability imposes conformal flatness......Page 266
Essential actions of simple groups on compact manifolds......Page 268
References......Page 269
Introduction......Page 272
Basic concepts......Page 274
Flat and conformally flat spaces......Page 277
The Riemannian case......Page 280
Conformal transformations of Einstein spaces......Page 281
Spaces which are conformally Einstein......Page 283
Conformal gradient fields......Page 288
4-dimensional Lorentzian manifolds......Page 293
The transition to the Penrose limit......Page 297
Conformal vector fields and twistor spinors......Page 299
References......Page 304
Introduction......Page 310
First definitions and conventions......Page 312
Conformal/classical causal structure......Page 314
Causal relations. Local properties......Page 316
Further properties of causal relations......Page 319
Time-separation and maximizing geodesics......Page 321
Lightlike geodesics and conjugate events......Page 323
The causal hierarchy......Page 327
Non-totally vicious spacetimes......Page 328
Chronological spacetimes......Page 329
Causal spacetimes......Page 330
Distinguishing spacetimes......Page 331
Continuous causal curves......Page 333
Strongly causal spacetimes......Page 335
A break: volume functions, continuous I, reflectivity......Page 339
Stably causal spacetimes......Page 344
Causally continuous spacetimes......Page 348
Causally simple spacetimes......Page 349
Globally hyperbolic spacetimes......Page 351
The ladder of isocausality......Page 362
Some examples......Page 364
References......Page 366
Introduction......Page 370
Preliminaries......Page 373
Special properties of geodesics in spacetimes depending on their causal character......Page 375
Conformal changes......Page 378
First results......Page 379
Completeness under conformal symmetries......Page 382
Heuristic construction of incomplete examples......Page 384
Surfaces......Page 387
Influence of curvature......Page 389
Warped products......Page 393
GRW spacetimes......Page 396
Stationary spacetimes......Page 399
The Riemannian framework for geodesics connecting two points......Page 402
Variational principles for static and stationary spacetimes: extrinsic and intrinsic approaches......Page 407
Time dependent metrics and saddle critical points......Page 415
References......Page 424
Introduction......Page 430
Lorentzian symmetric spaces......Page 431
Lorentzian parallelisable manifolds......Page 434
Flat metric connections with closed torsion......Page 435
Bi-invariant Lorentzian metrics on Lie groups......Page 438
Supergravity......Page 440
Eleven-dimensional supergravity......Page 442
Ten-dimensional IIB supergravity......Page 444
Six-dimensional (2,0) and (1,0) supergravities......Page 446
Maximally supersymmetric backgrounds......Page 447
Eleven-dimensional supergravity......Page 448
Ten-dimensional IIB supergravity......Page 451
Six-dimensional (2,0) and (1,0) supergravities......Page 453
Five-dimensional N=2 supergravity......Page 455
Parallelisable type II backgrounds......Page 457
Ten-dimensional parallelisable geometries......Page 458
Type II backgrounds......Page 459
References......Page 462
Introduction......Page 466
G-structures......Page 470
Clifford algebras and spin structures......Page 474
Compactification in supergravity......Page 480
The group Spin(n,n)......Page 484
Special orbits......Page 486
Twisting with an H-flux......Page 491
Spinors......Page 493
The field equations......Page 496
Integrable generalised G-structures......Page 497
Geometric properties......Page 501
References......Page 504
Introduction......Page 506
Metrics of (G_2,2)-type......Page 508
Geometric aspects......Page 509
Global solutions......Page 511
-complex structures......Page 513
Global properties of solutions......Page 515
Algebraic solutions......Page 517
Kruskal–Szekeres type solutions......Page 518
Metrics of G_2,1 )-type......Page 519
Metrics of (G_2,0 )-type......Page 520
The standard linearized theory......Page 521
A simple example......Page 530
Appendix: The Petrov classification......Page 531
References......Page 535
List of Contributors......Page 538
Index......Page 540
Back Cover......Page 551
