The Geometry of Curvature Homogeneous Pseudo-riemannian Manifolds

by Gilkey, Peter B.

• ISBN13: 9781860947858
• ISBN10: 1860947859
• Format: Hardcover
• Copyright: 2007-06-07
• Publisher: World Scientific Pub Co Inc

Summary

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

Preface	p. V
The Geometry of the Riemann Curvature Tensor	p. 1
Introduction	p. 1
Basic Geometrical Notions	p. 4
Vector spaces with symmetric inner products	p. 4
Vector bundles, connections, and curvature	p. 6
Holonomy and parallel translation	p. 10
Affine manifolds, geodesies, and completeness	p. 11
Pseudo-Riemannian manifolds	p. 12
Scalar Weyl invariants	p. 15
Algebraic Curvature Tensors and Homogeneity	p. 16
Algebraic curvature tensors	p. 17
Canonical curvature tensors	p. 21
The Weyl conformal curvature tensor	p. 23
Models	p. 24
Various notions of homogeneity	p. 26
Killing vector fields	p. 27
Nilpotent curvature	p. 28
Curvature Homogeneity - a Brief Literature Survey	p. 28
Scalar Weyl invariants in the Riemannian setting	p. 28
Relating curvature homogeneity and homogeneity	p. 29
Manifolds modeled on symmetric spaces	p. 30
Historical survey	p. 31
Results from Linear Algebra	p. 32
Symmetric and anti-symmetric operators	p. 32
The spectrum of an operator	p. 32
Jordan normal form	p. 33
Self-adjoint maps in the higher signature setting	p. 34
Technical results concerning differential equations	p. 35
Results from Differential Geometry	p. 38
Principle bundles	p. 39
Geometric realizability	p. 39
The canonical algebraic curvature tensors	p. 41
Complex geometry	p. 47
Rank 1-symmetric spaces	p. 51
Conformal complex space forms	p. 53
Kahler geometry	p. 54
The Geometry of the Jacobi Operator	p. 54
The Jacobs operator	p. 55
The higher order Jacobi operator	p. 57
The conformal Jacobi operator	p. 59
The complex Jacobi operator	p. 60
The Geometry of the Curvature Operator	p. 62
The skew-symmetric curvature operator	p. 62
The conformal skew-symmetric curvature operator	p. 65
The Stanilov operator	p. 66
The complex skew-symmetric curvature operator	p. 66
The Szabo operator	p. 68
Spectral Geometry of the Curvature Tensor	p. 69
Analytic continuation	p. 70
Duality	p. 72
Bounded spectrum	p. 75
The Jacobi operator	p. 78
The higher order Jacobi operator	p. 81
The conformal and complex Jacobi operators	p. 82
The Stanilov and the Szabo operators	p. 83
The skew-symmetric curvature operator	p. 84
The conformal skew-symmetric curvature operator	p. 86
Curvature Homogeneous Generalized Plane Wave Manifolds	p. 87
Introduction	p. 87
Generalized Plane Wave Manifolds	p. 90
The geodesic structure	p. 92
The curvature tensor	p. 93
The geometry of the curvature tensor	p. 94
Local scalar invariants	p. 94
Parallel vector fields and holonomy	p. 96
Jacobi vector fields	p. 96
Isometries	p. 97
Symmetric spaces	p. 99
Manifolds of Signature (2, 2)	p. 101
Immersions as hypersurfaces in flat space	p. 103
Spectral properties of the curvature tensor	p. 105
A complete system of invariants	p. 107
Isometries	p. 109
Estimating k[subscript p,q] if min(p, q) = 2	p. 114
Manifolds of Signature (2, 4)	p. 115
Plane Wave Hypersurfaces of Neutral Signature (p, p)	p. 119
Spectral properties of the curvature tensor	p. 123
Curvature homogeneity	p. 128
Plane Wave Manifolds with Flat Factors	p. 130
Nikcevic Manifolds	p. 135
The curvature tensor	p. 137
Curvature homogeneity	p. 139
Local isometry invariants	p. 141
The spectral geometry of the curvature tensor	p. 145
Dunn Manifolds	p. 149
Models and the structure groups	p. 151
Invariants which are not of Weyl type	p. 155
k-Curvature Homogeneous Manifolds I	p. 156
Models	p. 159
Affine invariants	p. 162
Changing the signature	p. 164
Indecomposability	p. 165
k-Curvature Homogeneous Manifolds II	p. 166
Models	p. 168
Isometry groups	p. 171
Other Pseudo-Riemannian Manifolds	p. 181
Introduction	p. 181
Lorentz Manifolds	p. 182
Geodesies and curvature	p. 185
Ricci blowup	p. 187
Curvature homogeneity	p. 188
Signature (2, 2) Walker Manifolds	p. 193
Osserman curvature tensors of signature (2, 2)	p. 194
Indefinite Kahler Osserman manifolds	p. 196
Jordan Osserman manifolds which are not nilpotent	p. 197
Conformally Osserman manifolds	p. 198
Geodesic Completeness and Ricci Blowup	p. 201
The geodesic equation	p. 201
Conformaliy Osserman manifolds	p. 202
Jordan Osserman Walker manifolds	p. 206
Fiedler Manifolds	p. 206
Geometric properties of Fiedler manifolds	p. 207
Fiedler manifolds of signature (2, 2)	p. 209
Nilpotent Jacobi manifolds of order 2r	p. 209
Nilpotent Jacobi manifolds of order 2r + 1	p. 213
Szabo nilpotent manifolds of arbitrarily high order	p. 216
The Curvature Tensor	p. 219
Introduction	p. 219
Topological Results	p. 221
Real vector bundles	p. 221
Bundles over projective spaces	p. 222
Clifford algebras in arbitrary signatures	p. 223
Riemannian Clifford algebras	p. 224
Vector fields on spheres	p. 226
Metrics of higher signatures on spheres	p. 226
Equivariant vector fields on spheres	p. 227
Geometrically symmetric vector bundles	p. 228
Generators for the Spaces Alg[subscript 0] and Alg[subscript 1]	p. 229
A lower bound for [nu](m) and for [nu subscript 1](m)	p. 231
Geometric realizability	p. 233
Jordan Osserman Algebraic Curvature Tensors	p. 234
Neutral signature Jordan Ossermem tensors	p. 235
Rigidity results for Jordan Osserman tensors	p. 238
The Szabo Operator	p. 241
Szabo 1-models	p. 242
Balanced Szabo pseudo-Riemannian manifolds	p. 243
Conformal Geometry	p. 245
The Weyl model	p. 245
Conformally Jordan Osserman 0-models	p. 246
Conformally Osserman 4-dimensional manifolds	p. 247
Conformally Jordan Ivanov-Petrova 0-models	p. 249
Stanilov Models	p. 251
Complex Geometry	p. 253
Complex Osserman Algebraic Curvature Tensors	p. 257
Introduction	p. 257
Clifford families	p. 257
Complex Osserman tensors	p. 258
Classification results in the algebraic setting	p. 259
Geometric examples	p. 260
Chapter outline	p. 261
Technical Preliminaries	p. 261
Criteria for complex Osserman models	p. 262
Controlling the eigenvalue structure	p. 263
Examples of complex Osserman 0-models	p. 264
Reparametrization of a Clifford family	p. 265
The dual Clifford family	p. 265
Compatible complex models given by Clifford families	p. 266
Linearly independent endomorphisms	p. 269
Technical results, concerning Clifford algebras	p. 272
Clifford Families of Rank 1	p. 276
Clifford Families of Rank 2	p. 278
The tensor c[subscript 1]A[subscript J[subscript 1]] + c[subscript 2]A[subscript J[subscript 2]]	p. 279
The tensor c[subscript 0]A[subscript [less than].,.[greater than]] + c[subscript 1]A[subscript J[subscript 1]] + c[subscript 2]A[subscript J[subscript 2]]	p. 286
Clifford Families of Rank 3	p. 288
Technical results	p. 288
The tensor A = c[subscript 1]A[subscript J[subscript 1]] + c[subscript 2]A[subscript J[subscript 2]] + c[subscript 3]A[subscript J[subscript 3]]	p. 291
The tensor A = c[subscript 0]A[subscript [less than].,.[greater than]] + c[subscript 1]A[subscript J[subscript 1]] + c[subscript 2]A[subscript J[subscript 2]] + c[subscript 3]A[subscript J[subscript 3]]	p. 292
Tensors A = c[subscript 1]A[subscript J[subscript 1]] +...+ c[subscript l]A[subscript J[subscript l]] for l [greater than or equal] 4	p. 295
Tensors A = c[subscript 0]A[subscript [less than].,.[greater than]] + c[subscript 1]A[subscript J[subscript 1]] +...+ c[subscript l]A[subscript J[subscript l]] for l [greater than or equal] 4	p. 301
Stanilov-Tsankov Theory	p. 309
Introduction	p. 309
Jacobi Tsankov manifolds	p. 310
Skew Tsankov manifolds	p. 311
Stanilov-Videv manifolds	p. 312
Jacobi Videv manifolds and 0-models	p. 313
Riemannian Jacobi Tsankov Manifolds and 0-Models	p. 313
Riemannian Jacobi Tsankov 0-models	p. 314
Riemannian orthogonally Jacobi Tsankov 0-models	p. 315
Riemannian Jacobi Tsankov manifolds	p. 322
Pseudo-Riemannian Jacobi Tsankov 0-Models	p. 323
Jacobi Tsankov 0-models	p. 324
Non Jacobi Tsankov 0-models with J[Characters not reproducible] = 0 [forall] x	p. 325
0-models with J[subscript x]J[subscript y] = 0 [forall] x, y [isin] V	p. 326
0-models with A[subscript xy]A[subscript zw] = 0 [forall] x, y, z, w [isin] V	p. 328
A Jacobi Tsankov 0-Model with J[subscript x]J[subscript y] [not equal] 0 for some x, y	p. 331
The model M[subscript 14]	p. 333
A geometric realization of M[subscript 14]	p. 338
Isometry invariants	p. 340
A symmetric space with model M[subscript 14]	p. 343
Riemannian Skew Tsankov Models and Manifolds	p. 345
Riemannian skew Tsankov models	p. 347
3-dimensional skew Tsankov manifolds	p. 349
Irreducible 4-dimensional skew Tsankov manifolds	p. 351
Flats in a Riemannian skew Tsankov manifold	p. 353
Jacobi Videv Models and Manifolds	p. 356
Equivalent properties characterizing Jacobi Videv models	p. 357
Decomposing Jacobi Videv models	p. 359
Bibliography	p. 361
Index	p. 373
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