Einstein-Weyl structures on contact metric manifolds

In this paper we study Einstein-Weyl structures in the framework of contact metric manifolds. First, we prove that a complete K-contact manifold admitting both the Einstein-Weyl structures W+/- = (g, +/-omega) is Sasakian. Next, we show that a compact contact metric manifold admitting an Einstein-Weyl structure is either K-contact or the dual field of omega is orthogonal to the Reeb vector field, provided the Reeb vector field is an eigenvector of the Ricci operator. We also prove that a contact metric manifold admitting both the Einstein-Weyl structures and satisfying Q phi = phi Q is either K-contact or Einstein. Finally, a couple of results on contact metric manifold admitting an Einstein-Weyl structure W = (g, f eta) are presented.