Riemann solitons on almost co-Kähler manifolds

Abstract

The aim of the present paper is to characterize almost co-K?hler 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?hler manifold is a Riemann soliton with the soliton vector field ?, then the manifold is flat. It is also shown that if the metric of a (?, ?)-almost co-K?hler manifold with ? < 0 is a Riemann soliton, then the soliton is expanding and ?, ?, ? satisfies a relation. We also prove that there does not exist gradient almost Riemann solitons on (?, ?)-almost co-K?hler manifolds with ? < 0. Finally, the existence of a Riemann soliton on a three dimensional almost co-K?hler manifold is ensured by a proper example.

Almost Riemann Solitons with Vertical Potential on Conformal Cosymplectic Contact Complex Riemannian Manifolds

“…In [4,5], Venkatesha, Devaraja and Kumara studied the cases of almost-Kenmotsu manifolds and K-contact manifolds. Biswas, Chen and U. C. De characterized almost-co-Kähler manifolds whose metrics are Riemann solitons in [6]. K. De and U. C. De proved in [7] some geometric properties of almost-Riemann solitons on non-cosymplectic normal almost-contact metric manifolds and in particular on quasi-Sasakian 3-dimensional manifolds.…”