Riemann Solitons on Almost Co-Kahler Manifolds

The aim of the present paper is to characterize almost co-Kahler 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-Kahler 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-Kahler 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-Kahler manifolds with kappa < 0. Finally, the existence of a Riemann soliton on a three dimensional almost co-Kahler manifold is ensured by a proper example.