
By Eduardo GarcíaRío, Peter Gilkey, Stana Nik?evi?
, 168 pages
© 2013
ISBN 1608457591
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Price: US $45.00
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PseudoRiemannian geometry is, to a large extent, the study of the LeviCivita connection, which is the unique torsionfree 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 pseudoRiemannian structure of neutral signature to certain affine connections and use this correspondence to study both geometries. We examine Walker structures, Riemannian extensions, and KählerWeyl 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 needproofs are included of many of these results to make it as selfcontained as possible. Paracomplex 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 paracomplex geometry as an adjunct to complex geometry, instead, we shall often introduce the paracomplex 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 KählerWeyl 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 LeviCivita, 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)KählerWeyl Manifolds

