The Einstein–Weyl equations in complex and quaternionic geometry

Abstract

Einstein–Weyl manifolds with compatible complex structures are shown to be given as torus bundles on Kähler–Einstein manifolds, extending known results on locally conformal Kähler manifolds. The Weyl structure is derived from a Ricci-flat metric constructed by Calabi on the canonical bundle of the Kähler–Einstein manifold. Similar questions are addressed when the Weyl geometry admits compatible hypercomplex or quaternionic structures.