Einstein–Weyl structures on contact metric manifolds

Abstract

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, ±ω) 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 ω 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 EinsteinWeyl structures and satisfying Qϕ = ϕ Q is either K -contact or Einstein. Finally, a couple of results on contact metric manifold admitting an Einstein-Weyl structure W = (g, f η) are presented.

On the Classification of Almost Kenmotsu Manifolds of Dimension 3

“…Recall that a Weyl structure [4,15] on a Riemannian manifold ( , ) of dimension ≥ 3 is defined by the pair + = ( , ) satisfying…”

“…Notice that Narita [15] proved that an -Einstein almost contact metric manifold satisfying ∇ = − admits an Einstein-Weyl structure + = ( , ). However, (4) implies that an almost Kenmotsu manifold never satisfies Narita's condition even if ℎ = 0. Since then, we present the following sufficient conditions to characterize Einstein-Weyl structure on an almost Kenmotsu manifold of dimension 3.…”

“…The Weyl structure − = ( , − ) is said to be an Einstein-Weyl structure if (45) holds for a smooth function Λ on . For an Einstein-Weyl structure − = ( , − ) it follows from [4] …”

“…However, in higher dimensions the classification of contact metric manifold with = is still open. It is worthy to point out that Ghosh [4] recently proved that a contact metric manifold admitting the Einstein-Weyl structures ± = ( , ± ) (see Section 4) and = is either a K-contact or an Einstein manifold.…”

On the Closed Einstein-Weyl Structure and Compact K-Contact Manifold

“…Indeed, by applying a D-homothetic deformation (Tanno [17]) to the metric of the unit sphere S 2n+1 (1) one can prove that the resulting metric becomes η-Einstein (i.e., the Ricci tensor S satisfies S = αg + βη ⊗ η for some constants α, β and η is the contact form), and it admits an Einstein-Weyl structure (g, f η) with β < 0 for some constant f . For details we refer to [4] and [8].…”