What is included with this book?
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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