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 Kahler--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 Kahler--Weyl geometry, which lies, in a certain
sense, midway between affine geometry and Kahler 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.
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