Almost Ricci-Yamabe Soliton on Contact Metric Manifolds

TL;DR

This paper investigates almost Ricci-Yamabe solitons on contact metric manifolds, establishing conditions under which these manifolds are Einstein, Sasakian, or flat, and providing examples to illustrate these results.

Contribution

It characterizes the geometric structure of contact metric manifolds admitting almost Ricci-Yamabe solitons, including conditions for compactness, Einstein property, and specific manifold types.

Findings

Complete contact metric manifolds with certain solitons are compact Einstein Sasakian.

Complete K-contact manifolds with gradient Ricci-Yamabe solitons are isometric to spheres.

Non-Sasakian (k,μ)-contact manifolds with gradient solitons are flat or locally isometric to Euclidean and spherical products.

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.