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Preface 1 Basic Algebraic Notions 1.1 Introduction 1.2 A Historical Perspective in the Algebraic Context 1.3 Algebraic Preliminaries 1.3.1 Jordan Normal Form 1.3.2 Indefinite Geometry 1.3.3 Algebraic Curvature Tensors 1.3.4 Hermitian and Para-Hermitian Geometry 1.3.5 The Jacobi and Skew Symmetric Curvature Operators 1.3.6 Sectional, Ricci, Scalar, and Weyl Curvature 1.3.7 Curvature Decompositions 1.3.8 Self-Duality and Anti-Self-Duality Conditions 1.4 Spectral Geometry of the Curvature Operator 1.4.1 Osserman and Conformally Osserman Models 1.4.2 Osserman Curvature Models in Signature (2, 2) 1.4.3 Ivanov-Petrova Curvature Models 1.4.4 Osserman Ivanov-Petrova Curvature Models 1.4.5 Commuting Curvature Models 2 Basic Geometrical Notions 2.1 Introduction 2.2 History 2.3 Basic Manifold Theory 2.3.1 The Tangent Bundle, Lie Bracket, and Lie Groups 2.3.2 The Cotangent Bundle and Symplectic Geometry 2.3.3 Connections, Curvature, Geodesics, and Holonomy 2.4 Pseudo-Riemannian Geometry 2.4.1 The Levi-Civita Connection 2.4.2 Associated Natural Operators 2.4.3 Weyl Scalar Invariants 2.4.4 Null Distributions 2.4.5 Pseudo-Riemannian Holonomy 2.5 Other Geometric Structures 2.5.1 Pseudo-Hermitian and Para-Hermitian Structures 2.5.2 Hyper-Para-Hermitian Structures 2.5.3 Geometric Realizations 2.5.4 Homogeneous Spaces, and Curvature Homogeneity 2.5.5 Technical Results in Differential Equations 3 Walker Structures 3.1 Introduction 3.2 Historical Development 3.3 Walker Coordinates 3.4 Examples of Walker Manifolds 3.4.1 Hypersurfaces with Nilpotent Shape Operators 3.4.2 Locally Conformally Flat Metrics with Nilpotent Ricci Operator 3.4.3 Degenerate Pseudo-Riemannian Homogeneous Structures 3.4.4 Para-Kaehler Geometry 3.4.5 Two-step Nilpotent Lie Groups with Degenerate Center 3.4.6 Conformally Symmetric Pseudo-Riemannian Metrics 3.5 Riemannian Extensions 3.5.1 The Affine Category 3.5.2 Twisted Riemannian Extensions Defined by Flat Connections 3.5.3 Modified Riemannian Extensions Defined by Flat Connections 3.5.4 Nilpotent Walker Manifolds 3.5.5 Osserman Riemannian Extensions 3.5.6 Ivanov-Petrova Riemannian Extensions 4 Three-Dimensional Lorentzian Walker Manifolds 4.1 Introduction 4.2 History 4.3 Three Dimensional Walker Geometry 4.3.1 Adapted Coordinates 4.3.2 The Jordan Normal Form of the Ricci Operator 4.3.3 Christoffel Symbols, Curvature, and the Ricci Tensor 4.3.4 Locally Symmetric Walker Manifolds 4.3.5 Einstein-Like Manifolds 4.3.6 The Spectral Geometry of the Curvature Tensor 4.3.7 Curvature Commutativity Properties 4.4 Local geometry of Walker manifolds with ¦Ó¡Ù 0 4.4.1 Foliated Walker Manifolds 4.4.2 Contact Walker Manifolds 4.5 Strict Walker Manifolds 4.6 Three dimensional homogeneous Lorentzian manifolds 4.6.1 Three dimensional Lie groups and Lie algebras 4.7 Curvature Homogeneous Lorentzian Manifolds 4.7.1 Diagonalizable Ricci Operator 4.7.2 Type II Ricci Operator 5 Four-Dimensional Walker Manifolds 5.1 Introduction 5.2 History 5.3 Four-Dimensional Walker Manifolds 5.4 Almost Para-Hermitian Geometry 5.4.1 Isotropic Almost Para-Hermitian Structures 5.4.2 Characteristic Classes 5.4.3 Self-Dual Walker Manifolds 6 The Spectral Geometry of the Curvature Tensor 6.1 Introduction 6.2 History 6.3 Four-Dimensional Osserman Metrics 6.3.1 Osserman Metrics with DiagonalizableJacobi Operator 6.3.2 Osserman Walker Type II Metrics 6.4 Osserman and Ivanov-Petrova Metrics 6.5 Riemannian Extensions of Affine Surfaces 6.5.1 Affine Surfaces with Skew Symmetric Ricci Tensor 6.5.2 affine Surfaces with Symmetric and Degenerate Ricci Tensor 6.5.3 Riemannian Extensions with Commuting Curvature Operators 6.5.4 Other Examples with Commuting Curvature Operators 7 Hermitian Geometry 7.1 Introduction 7.2 History 7.3 Almost Hermitian Geometry of Walker Manifolds 7.3.1 The Proper Almost Hermitian Structure of a Walker Manifold 7.3.2 Proper Almost Hyper-Para-Hermitian Structures 7.4 Hermitian Walker Manifolds of Dimension Four 7.4.1 Proper Hermitian Walker Structures 7.4.2 Locally Conformally Kaehler Structures 7.5 Almost Kaehler Walker Four-Dimensional Manifolds 8 Special Walker Manifolds 8.1 Introduction 8.2 History 8.3 Curvature Commuting Conditions 8.4 Curvature Homogeneous Strict Walker Manifolds Bibliography Glossary Biography Index ±à¼­ÊÖ¼Ç