Einstein-Weyl Geometry

instein 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 innitesimally. 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 innity [48, 35]. An important special case of t