Almost Kenmotsu metric as Ricci-Yamabe soliton

Abstract

The object of the present paper is to characterize two classes of almost Kenmotsu manifolds admitting Ricci-Yamabe soliton. It is shown that a $(k,\mu {)}^{\prime }$-almost Kenmotsu manifold admitting a Ricci-Yamabe soliton or gradient Ricci-Yamabe soliton is locally isometric to the Riemannian product ${\mathbb{H}}^{n+1}(-4)\times {\mathbb{R}}^n$. For the later case, the potential vector field is pointwise collinear with the Reeb vector field. Also, a $(k,\mu )$-almost Kenmotsu manifold admitting certain Ricci-Yamabe soliton with the curvature property $Q\cdot P=0$ is locally isometric to the hyperbolic space ${\mathbb{H}}^{2n+1}(-1)$ and the non-existense of the curvature property $Q\cdot R=0$ is proved.

16 pages