Selfdual Spaces With Complex Structures, Einstein-Weyl Geometry And Geodesics

. We study the Jones and Tod correspondence between selfdual conformal 4-manifolds with a conformal vector field and abelian monopoles on Einstein-Weyl 3-manifolds, and prove that invariant complex structures correspond to shear-free geodesic congruences. Such congruences exist in abundance and so provide a tool for constructing interesting selfdual geometries with symmetry, unifying the theories of scalar-flat Kahler metrics and hypercomplex structures with symmetry. We also show that in the presence of such a congruence, the Einstein-Weyl equation is equivalent to a pair of coupled monopole equations, and we solve these equations in a special case. The new EinsteinWeyl spaces, which we call Einstein-Weyl "with a geodesic symmetry", give rise to hypercomplex structures with two commuting triholomorphic vector fields. 1. Introduction Selfdual conformal 4-manifolds play a central role in low dimensional differential geometry. The selfduality equation is integrable, in the sense that th..