Einstein-Weyl Geometry

Abstract

A Weyl manifold is a conformal manifold equipped with a torsion free connection preserving the conformal structure, called a Weyl connection. It is said to be Einstein-Weyl if the symmetric tracefree part of the Ricci tensor of this connection vanishes. In particular, if the connection is the Levi-Civita connection of a compatible Riemannian metric, then this metric is Einstein. Such an approach has two immediate advantages: firstly, the homothety invariance of the Einstein condition is made explicit by focusing on the connection rather than the metric; and secondly, not every Weyl connection is a Levi-Civita connection, and so Einstein-Weyl manifolds provide a natural generalisation of Einstein geometry. The simplest examples of this generalisation are the locally conformally Einstein manifolds. A Weyl connection on a conformal manifold is said to be closed if it is locally the Levi-Civita connection of a compatible metric; but it need not be a global metric connection unless the manifold is simply connected. Closed EinsteinWeyl structures are then locally (but not necessarily globally) Einstein, and provide an interpretation of the Einstein condition which is perhaps more appropriate for multiply connected manifolds. For example, S1×Sn−1 admits flat Weyl structures, which are therefore closed Einstein-Weyl. These closed structures arise naturally in complex and quaternionic geometry. Einstein-Weyl geometry not only provides a different way of viewing Einstein manifolds, but also a broader setting in which to look for and study them. For instance, few compact Einstein manifolds with positive scalar curvature and continuous isometries are known to have Einstein deformations, yet we shall see that it is precisely under these two conditions that nontrivial Einstein-Weyl deformations can be shown to exist, at least infinitesimally. The Einstein-Weyl condition is particularly interesting in three dimensions, where the only Einstein manifolds are the spaces of constant curvature. In contrast, three dimensional Einstein-Weyl geometry is extremely rich [16, 68, 72], and has an equivalent formulation in twistor theory [34] which provides a tool for constructing selfdual four dimensional geometries. In section 10, we shall discuss a construction relating Einstein-Weyl 3-manifolds and hyperKahler 4-manifolds [40, 29, 50, 79]. Twistor methods also yield complete selfdual Einstein metrics of negative scalar curvature with prescribed conformal infinity [48, 35]. An important special case of this construction is the case of an Einstein-Weyl conformal infinity [34, 61]. Although Einstein-Weyl manifolds can be studied, along with Einstein manifolds, in a Riemannian framework, the natural context is Weyl geometry [23]. We