Almost Ricci-Yamabe Soliton on Contact Metric Manifolds

Abstract

We consider almost Ricci-Yamabe soliton in the context of certain contact metric manifolds. Firstly, we prove that if the metric $g$ admits an almost $(\alpha,\beta)$-Ricci-Yamabe soliton with $\alpha\neq 0$ and potential vector field collinear with the Reeb vector field $\xi$ on a complete contact metric manifold with the Reeb vector field $\xi$ as an eigenvector of the Ricci operator, then the manifold is compact Einstein Sasakian and the potential vector field is a constant multiple of the Reeb vector field $\xi$. Next, if complete $K$-contact manifold admits gradient Ricci-Yamabe soliton with $\alpha\neq 0$, then it is compact Sasakian and isometric to unit sphere $S^{2n+1}$. Finally, gradient almost Ricci-Yamabe soliton with $\alpha\neq 0$ in non-Sasakian $(k,\mu)$-contact metric manifold is assumed and found that $M^3$ is flat and for $n>1$, $M$ is locally isometric to $E^{n+1}\times S^n(4)$ and the soliton vector field is tangential to the Euclidean factor $E^{n+1}$. An illustrative example is given to support the obtained result.