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 Decoitions
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
. Basic Manifold Theory
..1 The Tangent Bundle, Lie rct, and Lie Groups
..2 The Cotangent Bundle and Symplectic Geometry
.. 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
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