Einstein-Weyl structures on almost cosymplectic manifolds

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.