Einstein-Weyl structures on almost cosymplectic manifolds

TL;DR

This paper investigates Einstein-Weyl structures on almost cosymplectic manifolds, establishing conditions under which these manifolds are Einstein, cosymplectic, or Ricc-flat, with specific results for three-dimensional and compact cases.

Contribution

It provides new characterizations of almost cosymplectic manifolds admitting Einstein-Weyl structures, including conditions for being Einstein, cosymplectic, or Ricc-flat, and explores cases with multiple Einstein-Weyl structures.

Findings

Almost cosymplectic $(ppa,rac{mu)$-manifolds are Einstein or cosymplectic with Einstein-Weyl structures.

Three-dimensional compact almost $lpha$-cosymplectic manifolds with closed Einstein-Weyl structures are Ricc-flat.

Manifolds with two Einstein-Weyl structures are either Einstein or $lpha$-cosymplectic if Ricci tensor commutes.

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.