The Geometry of Curvature Homogeneous Pseudo-Riemannian Manifolds

Peter B. Gilkey9781860947858, 1-86094-785-9

Pseudo-Riemannian geometry is an active research field not only in differential geometry but also in mathematical physics where the higher signature geometries play a role in brane theory. An essential reference tool for research mathematicians and physicists, this book also serves as a useful introduction to students entering this active and rapidly growing field. The author presents a comprehensive treatment of several aspects of pseudo-Riemannian geometry, including the spectral geometry of the curvature tensor, curvature homogeneity, and Stanilov-Tsankov-Videv theory.

---

Table of contents :
Contents……Page 8
Preface……Page 6
1.1 Introduction……Page 14
1.2.1 Vector spaces with symmetric inner products……Page 17
1.2.2 Vector bundles, connections, and curvature……Page 19
1.2.3 Holonomy and parallel translation……Page 23
1.2.4 A ne manifolds, geodesics, and completeness……Page 24
1.2.5 Pseudo-Riemannian manifolds……Page 25
1.2.6 Scalar Weyl invariants……Page 28
1.3 Algebraic Curvature Tensors and Homogeneity……Page 29
1.3.1 Algebraic curvature tensors……Page 30
1.3.2 Canonical curvature tensors……Page 34
1.3.3 The Weyl conformal curvature tensor……Page 36
1.3.4 Models……Page 37
1.3.5 Various notions of homogeneity……Page 39
1.3.6 Killing vector fields……Page 40
1.4.1 Scalar Weyl invariants in the Riemannian setting……Page 41
1.4.2 Relating curvature homogeneity and homogeneity……Page 42
1.4.3 Manifolds modeled on symmetric spaces……Page 43
1.4.4 Historical survey……Page 44
1.5.2 The spectrum of an operator……Page 45
1.5.3 Jordan normal form……Page 46
1.5.4 Self-adjoint maps in the higher signature setting……Page 47
1.5.5 Technical results concerning differential equations……Page 48
1.6 Results from Differential Geometry……Page 51
1.6.2 Geometric realizability……Page 52
1.6.3 The canonical algebraic curvature tensors……Page 54
1.6.4 Complex geometry……Page 60
1.6.5 Rank 1-symmetric spaces……Page 64
1.6.6 Conformal complex space forms……Page 66
1.7 The Geometry of the Jacobi Operator……Page 67
1.7.1 The Jacobi operator……Page 68
1.7.2 The higher order Jacobi operator……Page 70
1.7.3 The conformal Jacobi operator……Page 72
1.7.4 The complex Jacobi operator……Page 73
1.8.1 The skew-symmetric curvature operator……Page 75
1.8.2 The conformal skew-symmetric curvature operator……Page 78
1.8.4 The complex skew-symmetric curvature operator……Page 79
1.8.5 The Szabo operator……Page 81
1.9 Spectral Geometry of the Curvature Tensor……Page 82
1.9.1 Analytic continuation……Page 83
1.9.2 Duality……Page 85
1.9.3 Bounded spectrum……Page 88
1.9.4 The Jacobi operator……Page 91
1.9.5 The higher order Jacobi operator……Page 94
1.9.6 The conformal and complex Jacobi operators……Page 95
1.9.7 The Stanilov and the Szabo operators……Page 96
1.9.8 The skew-symmetric curvature operator……Page 97
1.9.9 The conformal skew-symmetric curvature operator……Page 99
2.1 Introduction……Page 100
2.2 Generalized Plane Wave Manifolds……Page 103
2.2.1 The geodesic structure……Page 105
2.2.2 The curvature tensor……Page 106
2.2.4 Local scalar invariants……Page 107
2.2.6 Jacobi vector fields……Page 109
2.2.7 Isometries……Page 110
2.2.8 Symmetric spaces……Page 112
2.3 Manifolds of Signature (2; 2)……Page 114
2.3.1 Immersions as hypersurfaces in at space……Page 116
2.3.2 Spectral properties of the curvature tensor……Page 118
2.3.3 A complete system of invariants……Page 120
2.3.4 Isometries……Page 122
2.3.5 Estimating kp,q if min(p, q) = 2……Page 127
2.4 Manifolds of Signature (2, 4)……Page 128
2.5 Plane Wave Hypersurfaces of Neutral Signature (p; p)……Page 132
2.5.1 Spectral properties of the curvature tensor……Page 136
2.5.2 Curvature homogeneity……Page 141
2.6 Plane Wave Manifolds with Flat Factors……Page 143
2.7 Nikcevic Manifolds……Page 148
2.7.1 The curvature tensor……Page 150
2.7.2 Curvature homogeneity……Page 152
2.7.3 Local isometry invariants……Page 154
2.7.4 The spectral geometry of the curvature tensor……Page 158
2.8 Dunn Manifolds……Page 162
2.8.1 Models and the structure groups……Page 164
2.8.2 Invariants which are not of Weyl type……Page 168
2.9 k-Curvature Homogeneous Manifolds I……Page 169
2.9.1 Models……Page 172
2.9.2 Affine invariants……Page 175
2.9.3 Changing the signature……Page 177
2.9.4 Indecomposability……Page 178
2.10 k-Curvature Homogeneous Manifolds II……Page 179
2.10.1 Models……Page 181
2.10.2 Isometry groups……Page 184
3.1 Introduction……Page 194
3.2 Lorentz Manifolds……Page 195
3.2.1 Geodesics and curvature……Page 198
3.2.2 Ricci blowup……Page 200
3.2.3 Curvature homogeneity……Page 201
3.3 Signature (2, 2) Walker Manifolds……Page 206
3.3.1 Osserman curvature tensors of signature (2, 2)……Page 207
3.3.2 Inde nite Kahler Osserman manifolds……Page 209
3.3.3 Jordan Osserman manifolds which are not nilpotent……Page 210
3.3.4 Conformally Osserman manifolds……Page 211
3.4.1 The geodesic equation……Page 214
3.4.2 Conformally Osserman manifolds……Page 215
3.5 Fiedler Manifolds……Page 219
3.5.1 Geometric properties of Fiedler manifolds……Page 220
3.5.3 Nilpotent Jacobi manifolds of order 2r……Page 222
3.5.4 Nilpotent Jacobi manifolds of order 2r + 1……Page 226
3.5.5 Szab o nilpotent manifolds of arbitrarily high order……Page 229
4.1 Introduction……Page 232
4.2.1 Real vector bundles……Page 234
4.2.2 Bundles over projective spaces……Page 235
4.2.3 Clifford algebras in arbitrary signatures……Page 236
4.2.4 Riemannian Clifford algebras……Page 237
4.2.6 Metrics of higher signatures on spheres……Page 239
4.2.7 Equivariant vector elds on spheres……Page 240
4.2.8 Geometrically symmetric vector bundles……Page 241
4.3 Generators for the Spaces Alg0 and Alg1……Page 242
4.3.1 A lower bound for (m) and for 1(m)……Page 244
4.3.2 Geometric realizability……Page 246
4.4 Jordan Osserman Algebraic Curvature Tensors……Page 247
4.4.1 Neutral signature Jordan Osserman tensors……Page 248
4.4.2 Rigidity results for Jordan Osserman tensors……Page 251
4.5.1 Szabo 1-models……Page 255
4.5.2 Balanced Szab o pseudo-Riemannian manifolds……Page 256
4.6.1 The Weyl model……Page 258
4.6.2 Conformally Jordan Osserman 0-models……Page 259
4.6.3 Conformally Osserman 4-dimensional manifolds……Page 260
4.6.4 Conformally Jordan Ivanov–Petrova 0-models……Page 262
4.7 Stanilov Models……Page 264
4.8 Complex Geometry……Page 266
5.1.1 Clifford families……Page 270
5.1.2 Complex Osserman tensors……Page 271
5.1.3 Classification results in the algebraic setting……Page 272
5.1.4 Geometric examples……Page 273
5.2 Technical Preliminaries……Page 274
5.2.1 Criteria for complex Osserman models……Page 275
5.2.2 Controlling the eigenvalue structure……Page 276
5.2.3 Examples of complex Osserman 0-models……Page 277
5.2.5 The dual Clifford family……Page 278
5.2.6 Compatible complex models given by Clifford families……Page 279
5.2.7 Linearly independent endomorphisms……Page 282
5.2.8 Technical results concerning Cli ord algebras……Page 285
5.3 Clifford Families of Rank 1……Page 289
5.4 Clifford Families of Rank 2……Page 291
5.4.1 The tensor c1AJ1 + c2AJ……Page 292
5.4.2 The tensor c0A(.,.) + c1AJ1 + c2AJ2……Page 299
5.5.1 Technical results……Page 301
5.5.2 The tensor A = c1AJ1 + c2AJ2 + c3AJ3……Page 304
5.5.3 The tensor A = c0A(.,.) + c1AJ1 + c2AJ2 + c3AJ3……Page 305
5.6 Tensors A = c1AJ1 + … + c`AJ` for ` 4……Page 308
5.7 Tensors A = c0A(.,.) + c1AJ1 + … + c`AJ` for ` 4……Page 314
6.1 Introduction……Page 322
6.1.1 Jacobi Tsankov manifolds……Page 323
6.1.2 Skew Tsankov manifolds……Page 324
6.1.3 Stanilov–Videv manifolds……Page 325
6.2 Riemannian Jacobi Tsankov Manifolds and 0-Models……Page 326
6.2.1 Riemannian Jacobi Tsankov 0-models……Page 327
6.2.2 Riemannian orthogonally Jacobi Tsankov 0-models……Page 328
6.2.3 Riemannian Jacobi Tsankov manifolds……Page 335
6.3 Pseudo-Riemannian Jacobi Tsankov 0-Models……Page 336
6.3.1 Jacobi Tsankov 0-models……Page 337
6.3.2 Non Jacobi Tsankov 0-models with Jx2 = 0 x……Page 338
6.3.3 0-models with JxJy = 0 x, y V……Page 339
6.3.4 0-models with AxyAzw = 0 x, y, z, w V……Page 341
6.4 A Jacobi Tsankov 0-Model with JxJy 6= 0 for some x, y……Page 344
6.4.1 The model M14……Page 346
6.4.2 A geometric realization of M14……Page 351
6.4.3 Isometry invariants……Page 353
6.4.4 A symmetric space with model M14……Page 356
6.5 Riemannian Skew Tsankov Models and Manifolds……Page 358
6.5.1 Riemannian skew Tsankov models……Page 360
6.5.2 3-dimensional skew Tsankov manifolds……Page 362
6.5.3 Irreducible 4-dimensional skew Tsankov manifolds……Page 364
6.5.4 Flats in a Riemannian skew Tsankov manifold……Page 366
6.6 Jacobi Videv Models and Manifolds……Page 369
6.6.1 Equivalent properties characterizing Jacobi Videv models……Page 370
6.6.2 Decomposing Jacobi Videv models……Page 372
Bibliography……Page 374
Index……Page 386