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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
A central area of study in Differential Geometry is the examination
of the relationship between the purely algebraic properties of the
Riemann curvature tensor and the underlying geometric properties of
the manifold. In this book, the findings of numerous investigations
in this field of study are reviewed and presented in a clear,
coherent form, including the latest developments and proofs. Even
though many authors have worked in this area in recent years, many
fundamental questions still remain unanswered. Many studies begin
by first working purely algebraically and then later progressing
onto the geometric setting and it has been found that many
questions in differential geometry can be phrased as problems
involving the geometric realization of curvature. Curvature
decompositions are central to all investigations in this area. The
authors present numerous results including the Singer-Thorpe
decomposition, the Bokan decomposition, the Nikcevic decomposition,
the Tricerri-Vanhecke decomposition, the Gray-Hervella
decomposition and the De Smedt decomposition. They then proceed to
draw appropriate geometric conclusions from these
decompositions.The book organizes, in one coherent volume, the
results of research completed by many different investigators over
the past 30 years. Complete proofs are given of results that are
often only outlined in the original publications. Whereas the
original results are usually in the positive definite (Riemannian
setting), here the authors extend the results to the
pseudo-Riemannian setting and then further, in a complex framework,
to para-Hermitian geometry as well. In addition to that, new
results are obtained as well, making this an ideal text for anyone
wishing to further their knowledge of the science of curvature.
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