Journal of the
Korean Mathematical Society JKMS

The aim of this article is to study the $k$-almost Ricci soliton and $k$-almost gradient Ricci soliton on contact metric manifold. First, we prove that if a compact $K$-contact metric is a $k$-almost gradient Ricci soliton, then it is isometric to a unit sphere $S^{2n+1}$. Next, we extend this result on a compact $k$-almost Ricci soliton when the flow vector field $X$ is contact. Finally, we study some special types of $k$-almost Ricci solitons where the potential vector field $X$ is point wise collinear with the Reeb vector field $\xi$ of the contact metric structure.

Keywords: contact metric manifold, $k$-almost Ricci soliton, $k$-almost gradient Ricci soliton, $K$-contact manifold, Sasakian manifold, Einstein manifold

MSC numbers: 53C25, 53C20, 53C15