Almost Ricci-Yamabe soliton on Almost Kenmotsu Manifolds

This manuscript examines almost Kenmotsu manifolds (briefly, AKMs) endowed with the almost Ricci-Yamabe solitons (ARYSs) and gradient ARYSs. The condition for an AKM with ARYS to be $\eta$-Einstein is established. We also show that an ARYS on Kenmotsu manifold becomes Ricci-Yamabe soliton under certain restrictions. In this series, it is proven that a $(2n+1)$-dimensional $(\kappa ,\mu {)}^{\mathrm{\prime }}$-AKM equipped with a gradient ARYS is either locally isometric to ${\mathbb{H}}^{n+1}(-4)\times {\mathbb{R}}^n$ or the Reeb vector field and the soliton vector field are codirectional. The properties of $3$-dimensional non-Kenmotsu AKMs endowed with a gradient ARYS are studied.