Quasi-Einstein structures and almost cosymplectic manifolds

Abstract

In this article, we study almost cosymplectic manifolds admitting quasi-Einstein structures $(g, V, m, \lambda)$. First we prove that an almost cosymplectic $(\kappa,\mu)$-manifold is locally isomorphic to a Lie group if $(g, V, m, \lambda)$ is closed and on a compact almost $(\kappa,\mu)$-cosymplectic manifold there do not exist quasi-Einstein structures $(g, V, m, \lambda)$, in which the potential vector field $V$ is collinear with the Reeb vector filed $\xi$. Next we consider an almost $\alpha$-cosymplectic manifold admitting a quasi-Einstein structure and obtain some results. Finally, for a $K$-cosymplectic manifold with a closed, non-steady quasi-Einstein structure, we prove that it is $\eta$-Einstein. If $(g, V, m, \lambda)$ is non-steady and $V$ is a conformal vector field, we obtain the same conclusion.