Geometry of almost conformal Ricci solitons on K-contact manifolds

Abstract

UDC 515.1

We study geometric aspects of an almost conformal Ricci soliton and an almost conformal gradient Ricci soliton on K-contact manifolds. Among others, we first obtain the nature of almost conformal Ricci soliton under the conditions: (i) the potential vector field is a contact vector field, and (ii) the potential vector field is pointwise collinear with the Reeb vector field $\xi$. Moreover, we present an example of almost conformal Ricci soliton on a K-contact manifold with potential vector field as a contact vector field. We also find a necessary and sufficient condition for the existence of cyclic Ricci tensor on a K-contact manifold. Further, we give a necessary and sufficient condition for the potential vector field $V$ of a conformal Ricci soliton to be Jacobi along $\xi$ on the K-contact $\eta$-Einstein manifold, and study the nature of almost conformal Ricci soliton on the K-contact $\eta$-Einstein manifold when the  potential vector field is a conformal vector field. Finally, we prove that if a complete K-contact metric is an almost conformal gradient Ricci soliton, then the manifold is isometric to a hyperbolic space $H^{2n+1}(\,-1)\,$.