The Geometry of Curvature Homogeneous Pseudo-Riemannian Manifolds.

By: Gilkey, Peter BMaterial type: TextTextSeries: Icp Advanced Texts in Mathematics SerPublisher: Singapore : Imperial College Press, 2007Copyright date: ©2007Description: 1 online resource (389 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9781860948589Subject(s): Curvature | Geometry, Differential | Riemannian manifoldsGenre/Form: Electronic books.Additional physical formats: Print version:: The Geometry of Curvature Homogeneous Pseudo-Riemannian ManifoldsDDC classification: 516.373 LOC classification: QA671.G55 2007Online resources: Click to View
Contents:
Intro -- Contents -- Preface -- 1. The Geometry of the Riemann Curvature Tensor -- 1.1 Introduction -- 1.2 Basic Geometrical Notions -- 1.2.1 Vector spaces with symmetric inner products -- 1.2.2 Vector bundles, connections, and curvature -- 1.2.3 Holonomy and parallel translation -- 1.2.4 A ne manifolds, geodesics, and completeness -- 1.2.5 Pseudo-Riemannian manifolds -- 1.2.6 Scalar Weyl invariants -- 1.3 Algebraic Curvature Tensors and Homogeneity -- 1.3.1 Algebraic curvature tensors -- 1.3.2 Canonical curvature tensors -- 1.3.3 The Weyl conformal curvature tensor -- 1.3.4 Models -- 1.3.5 Various notions of homogeneity -- 1.3.6 Killing vector fields -- 1.3.7 Nilpotent curvature -- 1.4 Curvature Homogeneity - a Brief Literature Survey -- 1.4.1 Scalar Weyl invariants in the Riemannian setting -- 1.4.2 Relating curvature homogeneity and homogeneity -- 1.4.3 Manifolds modeled on symmetric spaces -- 1.4.4 Historical survey -- 1.5 Results from Linear Algebra -- 1.5.1 Symmetric and anti-symmetric operators -- 1.5.2 The spectrum of an operator -- 1.5.3 Jordan normal form -- 1.5.4 Self-adjoint maps in the higher signature setting -- 1.5.5 Technical results concerning differential equations -- 1.6 Results from Differential Geometry -- 1.6.1 Principle bundles -- 1.6.2 Geometric realizability -- 1.6.3 The canonical algebraic curvature tensors -- 1.6.4 Complex geometry -- 1.6.5 Rank 1-symmetric spaces -- 1.6.6 Conformal complex space forms -- 1.6.7 K ahler geometry -- 1.7 The Geometry of the Jacobi Operator -- 1.7.1 The Jacobi operator -- 1.7.2 The higher order Jacobi operator -- 1.7.3 The conformal Jacobi operator -- 1.7.4 The complex Jacobi operator -- 1.8 The Geometry of the Curvature Operator -- 1.8.1 The skew-symmetric curvature operator -- 1.8.2 The conformal skew-symmetric curvature operator -- 1.8.3 The Stanilov operator.
1.8.4 The complex skew-symmetric curvature operator -- 1.8.5 The Szabo operator -- 1.9 Spectral Geometry of the Curvature Tensor -- 1.9.1 Analytic continuation -- 1.9.2 Duality -- 1.9.3 Bounded spectrum -- 1.9.4 The Jacobi operator -- 1.9.5 The higher order Jacobi operator -- 1.9.6 The conformal and complex Jacobi operators -- 1.9.7 The Stanilov and the Szabo operators -- 1.9.8 The skew-symmetric curvature operator -- 1.9.9 The conformal skew-symmetric curvature operator -- 2. Curvature Homogeneous Generalized Plane Wave Manifolds -- 2.1 Introduction -- 2.2 Generalized Plane Wave Manifolds -- 2.2.1 The geodesic structure -- 2.2.2 The curvature tensor -- 2.2.3 The geometry of the curvature tensor -- 2.2.4 Local scalar invariants -- 2.2.5 Parallel vector elds and holonomy -- 2.2.6 Jacobi vector fields -- 2.2.7 Isometries -- 2.2.8 Symmetric spaces -- 2.3 Manifolds of Signature (2 -- 2) -- 2.3.1 Immersions as hypersurfaces in at space -- 2.3.2 Spectral properties of the curvature tensor -- 2.3.3 A complete system of invariants -- 2.3.4 Isometries -- 2.3.5 Estimating kp,q if min(p, q) = 2 -- 2.4 Manifolds of Signature (2, 4) -- 2.5 Plane Wave Hypersurfaces of Neutral Signature (p -- p) -- 2.5.1 Spectral properties of the curvature tensor -- 2.5.2 Curvature homogeneity -- 2.6 Plane Wave Manifolds with Flat Factors -- 2.7 Nikcevic Manifolds -- 2.7.1 The curvature tensor -- 2.7.2 Curvature homogeneity -- 2.7.3 Local isometry invariants -- 2.7.4 The spectral geometry of the curvature tensor -- 2.8 Dunn Manifolds -- 2.8.1 Models and the structure groups -- 2.8.2 Invariants which are not of Weyl type -- 2.9 k-Curvature Homogeneous Manifolds I -- 2.9.1 Models -- 2.9.2 Affine invariants -- 2.9.3 Changing the signature -- 2.9.4 Indecomposability -- 2.10 k-Curvature Homogeneous Manifolds II -- 2.10.1 Models -- 2.10.2 Isometry groups.
3. Other Pseudo-Riemannian Manifolds -- 3.1 Introduction -- 3.2 Lorentz Manifolds -- 3.2.1 Geodesics and curvature -- 3.2.2 Ricci blowup -- 3.2.3 Curvature homogeneity -- 3.3 Signature (2, 2) Walker Manifolds -- 3.3.1 Osserman curvature tensors of signature (2, 2) -- 3.3.2 Inde nite Kahler Osserman manifolds -- 3.3.3 Jordan Osserman manifolds which are not nilpotent -- 3.3.4 Conformally Osserman manifolds -- 3.4 Geodesic Completeness and Ricci Blowup -- 3.4.1 The geodesic equation -- 3.4.2 Conformally Osserman manifolds -- 3.4.3 Jordan Osserman Walker manifolds -- 3.5 Fiedler Manifolds -- 3.5.1 Geometric properties of Fiedler manifolds -- 3.5.2 Fiedler manifolds of signature (2 -- 2) -- 3.5.3 Nilpotent Jacobi manifolds of order 2r -- 3.5.4 Nilpotent Jacobi manifolds of order 2r + 1 -- 3.5.5 Szab o nilpotent manifolds of arbitrarily high order -- 4. The Curvature Tensor -- 4.1 Introduction -- 4.2 Topological Results -- 4.2.1 Real vector bundles -- 4.2.2 Bundles over projective spaces -- 4.2.3 Clifford algebras in arbitrary signatures -- 4.2.4 Riemannian Clifford algebras -- 4.2.5 Vector fields on spheres -- 4.2.6 Metrics of higher signatures on spheres -- 4.2.7 Equivariant vector elds on spheres -- 4.2.8 Geometrically symmetric vector bundles -- 4.3 Generators for the Spaces Alg0 and Alg1 -- 4.3.1 A lower bound for (m) and for 1(m) -- 4.3.2 Geometric realizability -- 4.4 Jordan Osserman Algebraic Curvature Tensors -- 4.4.1 Neutral signature Jordan Osserman tensors -- 4.4.2 Rigidity results for Jordan Osserman tensors -- 4.5 The Szabo Operator -- 4.5.1 Szabo 1-models -- 4.5.2 Balanced Szab o pseudo-Riemannian manifolds -- 4.6 Conformal Geometry -- 4.6.1 The Weyl model -- 4.6.2 Conformally Jordan Osserman 0-models -- 4.6.3 Conformally Osserman 4-dimensional manifolds -- 4.6.4 Conformally Jordan Ivanov-Petrova 0-models -- 4.7 Stanilov Models.
4.8 Complex Geometry -- 5. Complex Osserman Algebraic Curvature Tensors -- 5.1 Introduction -- 5.1.1 Clifford families -- 5.1.2 Complex Osserman tensors -- 5.1.3 Classification results in the algebraic setting -- 5.1.4 Geometric examples -- 5.1.5 Chapter outline -- 5.2 Technical Preliminaries -- 5.2.1 Criteria for complex Osserman models -- 5.2.2 Controlling the eigenvalue structure -- 5.2.3 Examples of complex Osserman 0-models -- 5.2.4 Reparametrization of a Clifford family -- 5.2.5 The dual Clifford family -- 5.2.6 Compatible complex models given by Clifford families -- 5.2.7 Linearly independent endomorphisms -- 5.2.8 Technical results concerning Cli ord algebras -- 5.3 Clifford Families of Rank 1 -- 5.4 Clifford Families of Rank 2 -- 5.4.1 The tensor c1AJ1 + c2AJ -- 5.4.2 The tensor c0A(.,.) + c1AJ1 + c2AJ2 -- 5.5 Clifford Families of Rank 3 -- 5.5.1 Technical results -- 5.5.2 The tensor A = c1AJ1 + c2AJ2 + c3AJ3 -- 5.5.3 The tensor A = c0A(.,.) + c1AJ1 + c2AJ2 + c3AJ3 -- 5.6 Tensors A = c1AJ1 + ... + c`AJ` for ` 4 -- 5.7 Tensors A = c0A(.,.) + c1AJ1 + ... + c`AJ` for ` 4 -- 6. Stanilov-Tsankov Theory -- 6.1 Introduction -- 6.1.1 Jacobi Tsankov manifolds -- 6.1.2 Skew Tsankov manifolds -- 6.1.3 Stanilov-Videv manifolds -- 6.1.4 Jacobi Videv manifolds and 0-models -- 6.2 Riemannian Jacobi Tsankov Manifolds and 0-Models -- 6.2.1 Riemannian Jacobi Tsankov 0-models -- 6.2.2 Riemannian orthogonally Jacobi Tsankov 0-models -- 6.2.3 Riemannian Jacobi Tsankov manifolds -- 6.3 Pseudo-Riemannian Jacobi Tsankov 0-Models -- 6.3.1 Jacobi Tsankov 0-models -- 6.3.2 Non Jacobi Tsankov 0-models with Jx2 = 0 x -- 6.3.3 0-models with JxJy = 0 x, y V -- 6.3.4 0-models with AxyAzw = 0 x, y, z, w V -- 6.4 A Jacobi Tsankov 0-Model with JxJy 6= 0 for some x, y -- 6.4.1 The model M14 -- 6.4.2 A geometric realization of M14 -- 6.4.3 Isometry invariants.
6.4.4 A symmetric space with model M14 -- 6.5 Riemannian Skew Tsankov Models and Manifolds -- 6.5.1 Riemannian skew Tsankov models -- 6.5.2 3-dimensional skew Tsankov manifolds -- 6.5.3 Irreducible 4-dimensional skew Tsankov manifolds -- 6.5.4 Flats in a Riemannian skew Tsankov manifold -- 6.6 Jacobi Videv Models and Manifolds -- 6.6.1 Equivalent properties characterizing Jacobi Videv models -- 6.6.2 Decomposing Jacobi Videv models -- Bibliography -- Index.
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.
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Intro -- Contents -- Preface -- 1. The Geometry of the Riemann Curvature Tensor -- 1.1 Introduction -- 1.2 Basic Geometrical Notions -- 1.2.1 Vector spaces with symmetric inner products -- 1.2.2 Vector bundles, connections, and curvature -- 1.2.3 Holonomy and parallel translation -- 1.2.4 A ne manifolds, geodesics, and completeness -- 1.2.5 Pseudo-Riemannian manifolds -- 1.2.6 Scalar Weyl invariants -- 1.3 Algebraic Curvature Tensors and Homogeneity -- 1.3.1 Algebraic curvature tensors -- 1.3.2 Canonical curvature tensors -- 1.3.3 The Weyl conformal curvature tensor -- 1.3.4 Models -- 1.3.5 Various notions of homogeneity -- 1.3.6 Killing vector fields -- 1.3.7 Nilpotent curvature -- 1.4 Curvature Homogeneity - a Brief Literature Survey -- 1.4.1 Scalar Weyl invariants in the Riemannian setting -- 1.4.2 Relating curvature homogeneity and homogeneity -- 1.4.3 Manifolds modeled on symmetric spaces -- 1.4.4 Historical survey -- 1.5 Results from Linear Algebra -- 1.5.1 Symmetric and anti-symmetric operators -- 1.5.2 The spectrum of an operator -- 1.5.3 Jordan normal form -- 1.5.4 Self-adjoint maps in the higher signature setting -- 1.5.5 Technical results concerning differential equations -- 1.6 Results from Differential Geometry -- 1.6.1 Principle bundles -- 1.6.2 Geometric realizability -- 1.6.3 The canonical algebraic curvature tensors -- 1.6.4 Complex geometry -- 1.6.5 Rank 1-symmetric spaces -- 1.6.6 Conformal complex space forms -- 1.6.7 K ahler geometry -- 1.7 The Geometry of the Jacobi Operator -- 1.7.1 The Jacobi operator -- 1.7.2 The higher order Jacobi operator -- 1.7.3 The conformal Jacobi operator -- 1.7.4 The complex Jacobi operator -- 1.8 The Geometry of the Curvature Operator -- 1.8.1 The skew-symmetric curvature operator -- 1.8.2 The conformal skew-symmetric curvature operator -- 1.8.3 The Stanilov operator.

1.8.4 The complex skew-symmetric curvature operator -- 1.8.5 The Szabo operator -- 1.9 Spectral Geometry of the Curvature Tensor -- 1.9.1 Analytic continuation -- 1.9.2 Duality -- 1.9.3 Bounded spectrum -- 1.9.4 The Jacobi operator -- 1.9.5 The higher order Jacobi operator -- 1.9.6 The conformal and complex Jacobi operators -- 1.9.7 The Stanilov and the Szabo operators -- 1.9.8 The skew-symmetric curvature operator -- 1.9.9 The conformal skew-symmetric curvature operator -- 2. Curvature Homogeneous Generalized Plane Wave Manifolds -- 2.1 Introduction -- 2.2 Generalized Plane Wave Manifolds -- 2.2.1 The geodesic structure -- 2.2.2 The curvature tensor -- 2.2.3 The geometry of the curvature tensor -- 2.2.4 Local scalar invariants -- 2.2.5 Parallel vector elds and holonomy -- 2.2.6 Jacobi vector fields -- 2.2.7 Isometries -- 2.2.8 Symmetric spaces -- 2.3 Manifolds of Signature (2 -- 2) -- 2.3.1 Immersions as hypersurfaces in at space -- 2.3.2 Spectral properties of the curvature tensor -- 2.3.3 A complete system of invariants -- 2.3.4 Isometries -- 2.3.5 Estimating kp,q if min(p, q) = 2 -- 2.4 Manifolds of Signature (2, 4) -- 2.5 Plane Wave Hypersurfaces of Neutral Signature (p -- p) -- 2.5.1 Spectral properties of the curvature tensor -- 2.5.2 Curvature homogeneity -- 2.6 Plane Wave Manifolds with Flat Factors -- 2.7 Nikcevic Manifolds -- 2.7.1 The curvature tensor -- 2.7.2 Curvature homogeneity -- 2.7.3 Local isometry invariants -- 2.7.4 The spectral geometry of the curvature tensor -- 2.8 Dunn Manifolds -- 2.8.1 Models and the structure groups -- 2.8.2 Invariants which are not of Weyl type -- 2.9 k-Curvature Homogeneous Manifolds I -- 2.9.1 Models -- 2.9.2 Affine invariants -- 2.9.3 Changing the signature -- 2.9.4 Indecomposability -- 2.10 k-Curvature Homogeneous Manifolds II -- 2.10.1 Models -- 2.10.2 Isometry groups.

3. Other Pseudo-Riemannian Manifolds -- 3.1 Introduction -- 3.2 Lorentz Manifolds -- 3.2.1 Geodesics and curvature -- 3.2.2 Ricci blowup -- 3.2.3 Curvature homogeneity -- 3.3 Signature (2, 2) Walker Manifolds -- 3.3.1 Osserman curvature tensors of signature (2, 2) -- 3.3.2 Inde nite Kahler Osserman manifolds -- 3.3.3 Jordan Osserman manifolds which are not nilpotent -- 3.3.4 Conformally Osserman manifolds -- 3.4 Geodesic Completeness and Ricci Blowup -- 3.4.1 The geodesic equation -- 3.4.2 Conformally Osserman manifolds -- 3.4.3 Jordan Osserman Walker manifolds -- 3.5 Fiedler Manifolds -- 3.5.1 Geometric properties of Fiedler manifolds -- 3.5.2 Fiedler manifolds of signature (2 -- 2) -- 3.5.3 Nilpotent Jacobi manifolds of order 2r -- 3.5.4 Nilpotent Jacobi manifolds of order 2r + 1 -- 3.5.5 Szab o nilpotent manifolds of arbitrarily high order -- 4. The Curvature Tensor -- 4.1 Introduction -- 4.2 Topological Results -- 4.2.1 Real vector bundles -- 4.2.2 Bundles over projective spaces -- 4.2.3 Clifford algebras in arbitrary signatures -- 4.2.4 Riemannian Clifford algebras -- 4.2.5 Vector fields on spheres -- 4.2.6 Metrics of higher signatures on spheres -- 4.2.7 Equivariant vector elds on spheres -- 4.2.8 Geometrically symmetric vector bundles -- 4.3 Generators for the Spaces Alg0 and Alg1 -- 4.3.1 A lower bound for (m) and for 1(m) -- 4.3.2 Geometric realizability -- 4.4 Jordan Osserman Algebraic Curvature Tensors -- 4.4.1 Neutral signature Jordan Osserman tensors -- 4.4.2 Rigidity results for Jordan Osserman tensors -- 4.5 The Szabo Operator -- 4.5.1 Szabo 1-models -- 4.5.2 Balanced Szab o pseudo-Riemannian manifolds -- 4.6 Conformal Geometry -- 4.6.1 The Weyl model -- 4.6.2 Conformally Jordan Osserman 0-models -- 4.6.3 Conformally Osserman 4-dimensional manifolds -- 4.6.4 Conformally Jordan Ivanov-Petrova 0-models -- 4.7 Stanilov Models.

4.8 Complex Geometry -- 5. Complex Osserman Algebraic Curvature Tensors -- 5.1 Introduction -- 5.1.1 Clifford families -- 5.1.2 Complex Osserman tensors -- 5.1.3 Classification results in the algebraic setting -- 5.1.4 Geometric examples -- 5.1.5 Chapter outline -- 5.2 Technical Preliminaries -- 5.2.1 Criteria for complex Osserman models -- 5.2.2 Controlling the eigenvalue structure -- 5.2.3 Examples of complex Osserman 0-models -- 5.2.4 Reparametrization of a Clifford family -- 5.2.5 The dual Clifford family -- 5.2.6 Compatible complex models given by Clifford families -- 5.2.7 Linearly independent endomorphisms -- 5.2.8 Technical results concerning Cli ord algebras -- 5.3 Clifford Families of Rank 1 -- 5.4 Clifford Families of Rank 2 -- 5.4.1 The tensor c1AJ1 + c2AJ -- 5.4.2 The tensor c0A(.,.) + c1AJ1 + c2AJ2 -- 5.5 Clifford Families of Rank 3 -- 5.5.1 Technical results -- 5.5.2 The tensor A = c1AJ1 + c2AJ2 + c3AJ3 -- 5.5.3 The tensor A = c0A(.,.) + c1AJ1 + c2AJ2 + c3AJ3 -- 5.6 Tensors A = c1AJ1 + ... + c`AJ` for ` 4 -- 5.7 Tensors A = c0A(.,.) + c1AJ1 + ... + c`AJ` for ` 4 -- 6. Stanilov-Tsankov Theory -- 6.1 Introduction -- 6.1.1 Jacobi Tsankov manifolds -- 6.1.2 Skew Tsankov manifolds -- 6.1.3 Stanilov-Videv manifolds -- 6.1.4 Jacobi Videv manifolds and 0-models -- 6.2 Riemannian Jacobi Tsankov Manifolds and 0-Models -- 6.2.1 Riemannian Jacobi Tsankov 0-models -- 6.2.2 Riemannian orthogonally Jacobi Tsankov 0-models -- 6.2.3 Riemannian Jacobi Tsankov manifolds -- 6.3 Pseudo-Riemannian Jacobi Tsankov 0-Models -- 6.3.1 Jacobi Tsankov 0-models -- 6.3.2 Non Jacobi Tsankov 0-models with Jx2 = 0 x -- 6.3.3 0-models with JxJy = 0 x, y V -- 6.3.4 0-models with AxyAzw = 0 x, y, z, w V -- 6.4 A Jacobi Tsankov 0-Model with JxJy 6= 0 for some x, y -- 6.4.1 The model M14 -- 6.4.2 A geometric realization of M14 -- 6.4.3 Isometry invariants.

6.4.4 A symmetric space with model M14 -- 6.5 Riemannian Skew Tsankov Models and Manifolds -- 6.5.1 Riemannian skew Tsankov models -- 6.5.2 3-dimensional skew Tsankov manifolds -- 6.5.3 Irreducible 4-dimensional skew Tsankov manifolds -- 6.5.4 Flats in a Riemannian skew Tsankov manifold -- 6.6 Jacobi Videv Models and Manifolds -- 6.6.1 Equivalent properties characterizing Jacobi Videv models -- 6.6.2 Decomposing Jacobi Videv models -- Bibliography -- Index.

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.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2018. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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