Abstract
In this article, we study Einstein–Weyl structures on almost cosymplectic manifolds. First we prove that an almost cosymplectic (kappa ,mu )-manifold is Einstein or cosymplectic if it admits a closed Einstein–Weyl structure or two Einstein–Weyl structures. Next for a three dimensional compact almost alpha -cosymplectic manifold admitting closed Einstein–Weyl structures, we prove that it is Ricc-flat. Further, we show that an almost alpha -cosymplectic admitting two Einstein–Weyl structures is either Einstein or alpha -cosymplectic, provided that its Ricci tensor is commuting. Finally, we prove that a compact K-cosymplectic manifold with a closed Einstein–Weyl structure or two special Einstein–Weyl structures is cosymplectic.
Highlights
A Weyl structure W = (D, [g]) on a smooth manifold M is a torsion free affine connection D preserving a conformal structure [g]
Later on Hitchin [16] in studying 3-dimensional minitwistor theory observed that the minitwistor theory can be generalized over a 3-manifold endowed with a Weyl structure satisfying a certain Ricci tensor condition, called an Einstein–Weyl structure
A Weyl structure W = (D, [g]) is Einstein–Weyl if the symmetrized Ricci tensor is proportional to a metric g representing [g]: RicD(Y, X ) + RicD(X, Y ) = g(Y, X ), ∈ C∞(M)
Summary
Boyer and Galicki [2] proposed an odd-dimensional Goldberg conjecture that a compact Einstein K -contact manifold is Sasakian and proved that it is true. Matzeu proved that every (2n + 1)-dimensional cosymplectic manifold of constant φ-sectional curvature c > 0 admits two Ricci-flat Weyl structures where the 1-forms associated to the metric g. She generalized this result by proving that if a compact cosymplectic manifold (M, φ, ξ, η, g) admits a closed Einstein–Weyl structure D, M is necessarily η-Einstein [21].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
