The Einstein-Weyl equations in complex and quaternionic geometry (Q1317027)

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.