The $k$-almost Yamabe solitons and contact metric manifolds

Abstract

We introduce the concept of a $k$-almost Yamabe soliton which extends naturally from Yamabe solitons. Our aim is to study the $k$-almost Yamabe soliton $\left(g,V,k,\lambda \right)$ on a contact metric manifold $M^{2n+1}$. Firstly, for a general contact metric manifold, it is proved that $V$ is Killing if the potential vector field $V$ is a contact vector field and that $M$ is $K$-contact if $V$ is collinear with Reeb vector field. Secondly, we prove that a compact $K$-contact manifold, admitting a $k$-almost Yamabe gradient soliton, is isometric to a standard unit sphere. Moreover, for a complete Sasakian manifold admitting a $k$-almost Yamabe soliton, we show that it is isometric to a standard unit sphere ${\mathbb{𝕊}}^{2n+1}\left(1\right)$ when $n>1$ and for $n=1$, $M$ is also isometric to a standard unit sphere if it admits a closed $k$-almost Yamabe soliton. Finally, we consider a contact metric $\left(\kappa ,\mu \right)$-manifold with a nontrivial $k$-almost Yamabe gradient soliton and show that it is flat in dimension $3$ and in higher dimension $M$ is locally isometric to $E^{n+1}\times {\mathbb{𝕊}}^n\left(4\right)$. In the end, we construct two examples of contact metric manifolds with a $k$-almost Yamabe soliton.