Riemann solitons on Almost Co-K\"ahler Manifolds

Abstract

The aim of the present paper is to characterize almost co-K\"ahler manifolds whose metrics are the Riemann solitons. At first we provide a necessary and sufficient condition for the metric of a 3-dimensional manifold to be Riemann soliton. Next it is proved that if the metric of an almost co-K\"ahler manifold is a Riemann soliton with the soliton vector field $\xi$, then the manifold is flat. It is also shown that if the metric of a $(\kappa,\mu)$-almost co-K\"ahler manifold with $\kappa<0$ is a Riemann soliton, then the soliton is expanding and $\kappa, \mu, \lambda$ satisfies a relation. We also prove that there does not exist gradient almost Riemann solitons on $(\kappa,\mu)$-almost co-K\"ahler manifolds with $\kappa<0$. Finally, the existence of a Riemann soliton on a three dimensional almost co-K\"ahler manifold is ensured by a proper example.