Pseudo-Riemannian geometry is, to a large extent,
the study of the Levi-Civita connection, which is the
unique torsion-free connection compatible with the
metric structure. There are, however, other affine
connections which arise in different contexts, such as
conformal geometry, contact structures, Weyl structures,
and almost Hermitian geometry. In this book, we reverse
this point of view and instead associate an auxiliary
pseudo-Riemannian structure of neutral signature to
certain affine connections and use this correspondence
to study both geometries. We examine Walker structures,
Riemannian extensions, and Kähler--Weyl geometry from
this viewpoint. This book is intended to be accessible
to mathematicians who are not expert in the subject and
to students with a basic grounding in differential
geometry. Consequently, the first chapter contains a
comprehensive introduction to the basic results and
definitions we shall need---proofs are included of many
of these results to make it as self-contained as
possible. Para-complex geometry plays an important role
throughout the book and consequently is treated
carefully in various chapters, as is the representation
theory underlying various results. It is a feature of
this book that, rather than as regarding para-complex
geometry as an adjunct to complex geometry, instead, we
shall often introduce the para-complex concepts first
and only later pass to the complex setting. The second
and third chapters are devoted to the study of various
kinds of Riemannian extensions that associate to an
affine structure on a manifold a corresponding metric of
neutral signature on its cotangent bundle. These play a
role in various questions involving the spectral
geometry of the curvature operator and homogeneous
connections on surfaces. The fourth chapter deals with
Kähler--Weyl geometry, which lies, in a certain sense,
midway between affine geometry and Kähler geometry.
Another feature of the book is that we have tried
wherever possible to find the original references in the
subject for possible historical interest. Thus, we have
cited the seminal papers of Levi-Civita, Ricci,
Schouten, and Weyl, to name but a few exemplars. We have
also given different proofs of various results than
those that are given in the literature, to take
advantage of the unified treatment of the area given
herein. Table of Contents: Basic Notions and Concepts /
The Geometry of Deformed Riemannian Extensions / The
Geometry of Modified Riemannian Extensions /
(para)-Kähler--Weyl Manifolds
|
|