The Geometry of Walker Manifolds

Book description


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 Riemannian manifolds and the geometry of pseudo-Riemannian manifolds and thereby illustrate phenomena in pseudo-Riemannian geometry that are quite different from those which occur in Riemannian geometry, i.e. for indefinite as opposed to positive definite metrics. Indefinite metrics are important in many diverse physical contexts: classical cosmological models (general relativity) and string theory to name but two. Walker manifolds appear naturally in numerous physical settings and provide examples of extremal mathematical situations as will be discussed presently. To describe the geometry of a pseudo-Riemannian manifold, one must first understand the curvature of the manifold. We shall analyze a wide variety of curvature properties and we shall derive both geometrical and topological results. Special attention will be paid to manifolds of dimension 3 as these are quite tractable. We then pass to the 4 dimensional setting as a gateway to higher dimensions. Since the book is aimed at a very general audience (and in particular to an advanced undergraduate or to a beginning graduate student), no more than a basic course in differential geometry is required in the way of background. To keep our treatment as self-contained as possible, we shall begin with two elementary chapters that provide an introduction to basic aspects of pseudo-Riemannian geometry before beginning on our study of Walker geometry. An extensive bibliography is provided for further reading.

Math subject classifications : Primary: 53B20 -- (PACS: 02.40.Hw) Secondary: 32Q15, 51F25, 51P05, 53B30, 53C50, 53C80, 58A30, 83F05, 85A04

Table of Contents: Basic Algebraic Notions / Basic Geometrical Notions / Walker Structures / Three-Dimensional Lorentzian Walker Manifolds / Four-Dimensional Walker Manifolds / The Spectral Geometry of the Curvature Tensor / Hermitian Geometry / Special Walker Manifolds

Table of contents

  1. Preface
  2. Basic Algebraic Notions
    1. Introduction
    2. A Historical Perspective in the Algebraic Context
    3. Algebraic Preliminaries
      1. Jordan Normal Form
      2. Indefinite Geometry
      3. Algebraic Curvature Tensors
      4. Hermitian and Para-Hermitian Geometry
      5. The Jacobi and Skew Symmetric Curvature Operators
      6. Sectional, Ricci, Scalar, and Weyl Curvature
      7. Curvature Decompositions
      8. Self-Duality and Anti-Self-Duality Conditions
    4. Spectral Geometry of the Curvature Operator
      1. Spectral Geometry of the Curvature Operator
        1. Osserman and Conformally Osserman Models
        2. Osserman Curvature Models in Signature (2,2)
        3. Ivanov--Petrova Curvature Models
        4. Osserman Ivanov--Petrova Curvature Models
        5. Commuting Curvature Models
      2. Basic Geometrical Notions (1/4)
      3. Basic Geometrical Notions (2/4)
      4. Basic Geometrical Notions (3/4)
      5. Basic Geometrical Notions (4/4)
        1. Introduction
        2. History
        3. Basic Manifold Theory
          1. The Tangent Bundle, Lie Bracket, and Lie Groups
          2. The Cotangent Bundle and Symplectic Geometry
          3. Connections, Curvature, Geodesics, and Holonomy
        4. Pseudo-Riemannian Geometry
          1. The Levi-Civita Connection
          2. Associated Natural Operators
          3. Weyl Scalar Invariants
          4. Null Distributions
          5. Pseudo-Riemannian Holonomy
        5. Other Geometric Structures
          1. Pseudo-Hermitian and Para-Hermitian Structures
          2. Hyper-Para-Hermitian Structures
          3. Geometric Realizations
          4. Homogeneous Spaces, and Curvature Homogeneity
          5. Technical Results in Differential Equations
      6. Walker Structures (1/4)
      7. Walker Structures (2/4)
      8. Walker Structures (3/4)
      9. Walker Structures (4/4)
        1. Introduction
        2. Historical Development
        3. Walker Coordinates
        4. Examples of Walker Manifolds
          1. Hypersurfaces with Nilpotent Shape Operators
          2. Locally Conformally Flat Metrics with Nilpotent Ricci Operator
          3. Locally Conformally Flat Metrics with Nilpotent Ricci Operator
          4. Degenerate Pseudo-Riemannian Homogeneous Structures
          5. Degenerate Pseudo-Riemannian Homogeneous Structures
          6. Para-Kaehler Geometry
          7. Two-step Nilpotent Lie Groups with Degenerate Center
          8. Conformally Symmetric Pseudo-Riemannian Metrics
        5. Riemannian Extensions
          1. The Affine Category
          2. Twisted Riemannian Extensions Defined by Flat Connections
          3. Twisted Riemannian Extensions Defined by Flat Connections
          4. Modified Riemannian Extensions Defined by Flat Connections
          5. Modified Riemannian Extensions Defined by Flat Connections
          6. Nilpotent Walker Manifolds
          7. Osserman Riemannian Extensions
          8. Ivanov--Petrova Riemannian Extensions
      10. Three-Dimensional Lorentzian Walker Manifolds (1/21)
      11. Three-Dimensional Lorentzian Walker Manifolds (2/21)
      12. Three-Dimensional Lorentzian Walker Manifolds (3/21)
      13. Three-Dimensional Lorentzian Walker Manifolds (4/21)
      14. Three-Dimensional Lorentzian Walker Manifolds (5/21)
      15. Three-Dimensional Lorentzian Walker Manifolds (6/21)
      16. Three-Dimensional Lorentzian Walker Manifolds (7/21)
      17. Three-Dimensional Lorentzian Walker Manifolds (8/21)
      18. Three-Dimensional Lorentzian Walker Manifolds (9/21)
      19. Three-Dimensional Lorentzian Walker Manifolds (10/21)
      20. Three-Dimensional Lorentzian Walker Manifolds (11/21)
      21. Three-Dimensional Lorentzian Walker Manifolds (12/21)
      22. Three-Dimensional Lorentzian Walker Manifolds (13/21)
      23. Three-Dimensional Lorentzian Walker Manifolds (14/21)
      24. Three-Dimensional Lorentzian Walker Manifolds (15/21)
      25. Three-Dimensional Lorentzian Walker Manifolds (16/21)
      26. Three-Dimensional Lorentzian Walker Manifolds (17/21)
      27. Three-Dimensional Lorentzian Walker Manifolds (18/21)
      28. Three-Dimensional Lorentzian Walker Manifolds (19/21)
      29. Three-Dimensional Lorentzian Walker Manifolds (20/21)
      30. Three-Dimensional Lorentzian Walker Manifolds (21/21)
        1. Three-Dimensional Lorentzian Walker Manifolds
          1. Introduction
          2. History
          3. Three Dimensional Walker Geometry
            1. Adapted Coordinates
            2. The Jordan Normal Form of the Ricci Operator
            3. Christoffel Symbols, Curvature, and the Ricci Tensor
            4. Locally Symmetric Walker Manifolds
            5. Einstein-Like Manifolds
            6. The Spectral Geometry of the Curvature Tensor
            7. Curvature Commutativity Properties
          4. Local geometry of Walker manifolds with =0
            1. Local geometry of Walker manifolds with =0
              1. Foliated Walker Manifolds
              2. Contact Walker Manifolds
            2. Strict Walker Manifolds
          5. Three dimensional homogeneous Lorentzian manifolds
            1. Three dimensional Lie groups and Lie algebras
          6. Curvature Homogeneous Lorentzian Manifolds
            1. Curvature Homogeneous Lorentzian Manifolds
              1. Diagonalizable Ricci Operator
              2. Type II Ricci Operator
            2. Four-Dimensional Walker Manifolds
              1. Four-Dimensional Walker Manifolds
                1. Introduction
                2. History
                3. Four-Dimensional Walker Manifolds
                4. Almost Para-Hermitian Geometry
                  1. Isotropic Almost Para-Hermitian Structures
                  2. Characteristic Classes
                  3. Self-Dual Walker Manifolds
              2. The Spectral Geometry of the Curvature Tensor
                1. Introduction
                2. History
                3. Four-Dimensional Osserman Metrics
                  1. Osserman Metrics with Diagonalizable Jacobi Operator
                  2. Osserman Walker Type II Metrics
                4. Osserman and Ivanov--Petrova Metrics
                5. Riemannian Extensions of Affine Surfaces
                  1. Affine Surfaces with Skew Symmetric Ricci Tensor
                  2. Affine Surfaces with Symmetric and Degenerate Ricci Tensor
                  3. Affine Surfaces with Symmetric and Degenerate Ricci Tensor
                  4. Riemannian Extensions with Commuting Curvature Operators
                  5. Riemannian Extensions with Commuting Curvature Operators
                  6. Other Examples with Commuting Curvature Operators
              3. Hermitian Geometry
                1. Introduction
                2. History
                3. Almost Hermitian Geometry of Walker Manifolds
                  1. Almost Hermitian Geometry of Walker Manifolds
                    1. The Proper Almost Hermitian Structure of a Walker Manifold
                    2. The Proper Almost Hermitian Structure of a Walker Manifold
                    3. Proper Almost Hyper-Para-Hermitian Structures
                  2. Hermitian Walker Manifolds of Dimension Four
                    1. Hermitian Walker Manifolds of Dimension Four
                    2. Almost Kaehler Walker Four-Dimensional Manifolds
                  3. Special Walker Manifolds
                    1. Introduction
                    2. History
                    3. Curvature Commuting Conditions
                    4. Curvature Homogeneous Strict Walker Manifolds
                    5. Curvature Homogeneous Strict Walker Manifolds
                4. Bibliography
                5. Glossary
                6. Biography
                7. Index

Product information

  • Title: The Geometry of Walker Manifolds
  • Author(s): Peter Gilkey, Miguel Brozos-Vazquez, Eduardo García-Río, Stana Nikčević
  • Release date: July 2009
  • Publisher(s): Morgan & Claypool Publishers
  • ISBN: 9781598298208