Pseudo-Riemannian homogeneous structures (Q2632429)

Homogeneous and locally homogeneous spaces are among the most important objects in differential geometry. They have been extensively investigated using many methods and techniques since many geometric properties can be translated into algebraic properties. There are several reasons to read this book. One is to have a modern point of view for the study of homogeneous structures with a special attention on holonomy. Both classical and recent results are presented and this aspect will lead the reader to a better perception of these notions. Another reason to have this book on the reading list is that it is self-contained and describes the notions to everybody having a basic knowledge of smooth manifolds, Lie groups and theory of representations. A large number of motivations as well as applications appear in the book. The outline of this book is the following. The first chapter is a collection of some fundamental results from the theory of \(G\)-structures, principal bundles and connections, holonomy, the last one having the main role in the rest of the book. A panoramic view of the needed results is given, comprising the classical constructions and new approaches. Chapter 2 describes homogeneous spaces and their Ambrose-Singer connections. The pseudo-Riemannian version of the Ambrose-Singer theorem is given, being a tool for the study of this geometry. The Kiričenko's theorem is presented since it extends the Ambrose-Singer results to the case when a homogeneous manifold is endowed with a geometric structure. Chapter 3 is devoted to the description of locally homogeneous pseudo-Riemannian manifolds in terms of homogeneous structures. It is shown how to reconstruct a strongly reductive locally homogeneous space, possible equipped with some additional structure \(P\), from the knowledge of its curvature tensor field, the tensor field \(P\), and their covariant derivatives up to some finite order. Chapter 4 illustrates a procedure to classify homogeneous structures associated to Ambrose-Singer-connections and Ambrose-Singer-Kiričenko-connections with integrable underlying geometric structure. Later on, homogeneous structures are used to obtain a complete characterization of three-dimensional homogeneous (Riemannian and Lorentzian) manifolds. Chapter 5 is devoted to some recent results obtained for a special class of homogeneous structures, called of linear type. The aim of Chapter 6 is the study of homogeneous structures under reduction by subgroups of the group of isometries. In particular, this gives rise to a new homogeneous structure tensor on the orbit space of the action. This procedure is a source of new homogeneous structure tensors and provides interesting examples. Moreover, reduction can furnish information about homogeneous structures in the unreduced space starting form the homogeneous structures in the quotient space and conversely. Applications are described for cosymplectic and Sasakian homogeneous structures of linear type. The last chapter is suggestively titled: ``Where all this fails: non-reductive homogeneous pseudo-Riemannian manifolds''. So, there exist homogeneous pseudo-Riemannian manifolds which do not admit any reductive decomposition. Therefore, homogeneous structures cannot be used in their study. These spaces appear starting with dimension four. Thus, the authors highlight the classification, the explicit description of the homogeneous metrics and some aspects of the geometry of the four-dimensional examples.