Gradient ρ-Einstein solitons on almost Co-Kähler manifolds

Abstract

The aim of this paper is to characterize almost co-Kähler manifolds and co-Kähler three-manifolds whose metrices are the gradient [Formula: see text]-Einstein solitons. At first we prove that a proper [Formula: see text]-almost co-Kähler manifold with [Formula: see text] does not admit gradient [Formula: see text]-Einstein soliton. It is also shown that if a proper [Formula: see text]-Einstein almost co-Kähler manifold with constant coefficients admits a gradient [Formula: see text]-Einstein soliton, then either the manifold is a [Formula: see text]-almost co-Kähler manifold or the soliton is trivial. Next, we prove that in case of co-Kähler three-manifold the manifold is of constant scalar curvature. Moreover, either the manifold is flat or the gradient of the potential function is collinear with the Reeb vector field [Formula: see text]. Finally, we construct two examples to illustrate our results.