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.
