The geometry of Walker manifolds

• Other Authors: Brozos-Vázquez, Miguel.
• Format: Electronic
• Language: English
• Published: San Rafael, Calif. (1537 Fourth Street, San Rafael, CA 94901 USA) : Morgan & Claypool Publishers, c2009.
• Series: Synthesis lectures on mathematics and statistics (Online), # 5.
• Subjects: Manifolds (Mathematics); Riemannian manifolds.; Curvature.
• Online Access: Abstract with links to full text

This book, which focuses on the study of curvature, is an introduction to various aspects of pseudo- Riemannian geometry. We shall use Walker manifolds (pseudo-Riemannian manifolds which admit a non-trivial parallel null plane field) to exemplify some of the main differences between the geometry of...

Table of Contents:

• Basic algebraic notions
• Introduction
• A historical perspective in the algebraic context
• Algebraic preliminaries
• Jordan normal form
• Indefinite geometry
• Algebraic curvature tensors
• Hermitian and para-Hermitian geometry
• The Jacobi and skew symmetric curvature operators
• Sectional, Ricci, scalar, and Weyl curvature
• Curvature decompositions
• Self-duality and anti-self-duality conditions
• Spectral geometry of the curvature operator
• Osserman and conformally Osserman models
• Osserman curvature models in signature (2, 2)
• Ivanov-Petrova curvature models
• Osserman Ivanov-Petrova curvature models
• Commuting curvature models
• Basic geometrical notions
• Introduction
• History
• Basic manifold theory
• The tangent bundle, lie bracket, and lie groups
• The cotangent bundle and symplectic geometry
• Connections, curvature, geodesics, and holonomy
• Pseudo-Riemannian geometry
• The Levi-Civita connection
• Associated natural operators
• Weyl scalar invariants
• Null distributions
• Pseudo-Riemannian holonomy
• Other geometric structures
• Pseudo-Hermitian and para-Hermitian structures
• Hyper-para-Hermitian structures
• Geometric realizations
• Homogeneous spaces, and curvature homogeneity
• Technical results in differential equations
• Walker structures
• Introduction
• Historical development
• Walker coordinates
• Examples of Walker manifolds
• Hypersurfaces with nilpotent shape operators
• Locally conformally flat metrics with nilpotent Ricci operator
• Degenerate pseudo-Riemannian homogeneous structures
• Para-Kaehler geometry
• Two-step nilpotent lie groups with degenerate center
• Conformally symmetric pseudo-Riemannian metrics
• Riemannian extensions
• The affine category
• Twisted Riemannian extensions defined by flat connections
• Modified Riemannian extensions defined by flat connections
• Nilpotent Walker manifolds
• Osserman Riemannian extensions
• Ivanov-Petrova Riemannian extensions
• Three-dimensional Lorentzian Walker manifolds
• Introduction
• History
• Three dimensional Walker geometry
• Adapted coordinates
• The Jordan normal form of the Ricci operator
• Christoffel symbols, curvature, and the Ricci tensor
• Locally symmetric Walker manifolds
• Einstein-like manifolds
• The spectral geometry of the curvature tensor
• Curvature commutativity properties
• Local geometry of Walker manifolds with
• Foliated Walker manifolds
• Contact Walker manifolds
• Strict Walker manifolds
• Three dimensional homogeneous Lorentzian manifolds
• Three dimensional lie groups and lie algebras
• Curvature homogeneous Lorentzian manifolds
• Diagonalizable Ricci operator
• Type II Ricci operator
• Four-dimensional Walker manifolds
• Introduction
• History
• Four-dimensional Walker manifolds
• Almost para-Hermitian geometry
• Isotropic almost para-Hermitian structures
• Characteristic classes
• Self-dual Walker manifolds
• The spectral geometry of the curvature tensor
• Introduction
• History
• Four-dimensional Osserman metrics
• Osserman metrics with diagonalizable Jacobi operator
• Osserman Walker type II metrics
• Osserman and Ivanov-Petrova metrics
• Riemannian extensions of affine surfaces
• Affine surfaces with skew symmetric Ricci tensor
• Affine surfaces with symmetric and degenerate Ricci tensor
• Riemannian extensions with commuting curvature operators
• Other examples with commuting curvature operators
• Hermitian geometry
• Introduction
• History
• Almost Hermitian geometry of Walker manifolds
• The proper almost Hermitian structure of a Walker manifold
• Proper almost hyper-para-Hermitian structures
• Hermitian Walker manifolds of dimension four
• Proper Hermitian Walker structures
• Locally conformally Kaehler structures
• Almost Kaehler Walker four-dimensional manifolds
• Special Walker manifolds
• Introduction
• History
• Curvature commuting conditions
• Curvature homogeneous strict Walker manifolds
• Bibliography.