Quasi-Einstein structures and almost cosymplectic manifolds

TL;DR

This paper investigates the properties of almost cosymplectic manifolds with quasi-Einstein structures, establishing conditions under which these manifolds are locally isomorphic to Lie groups or are $ ext{eta}$-Einstein, and exploring related structures.

Contribution

It provides new results on the structure of almost cosymplectic manifolds admitting quasi-Einstein structures, including classification and non-existence results.

Findings

Almost cosymplectic $( abla, V, m, abla)$-manifolds are locally isomorphic to Lie groups under certain conditions.

No quasi-Einstein structures with collinear potential and Reeb vector on compact almost $( abla, abla)$-cosymplectic manifolds.

K-cosymplectic manifolds with non-steady quasi-Einstein structures are $ ext{eta}$-Einstein.

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.