Kenmotsu $3-$manifold admitting gradient Ricci-Yamabe solitons and $ \ast-\eta-$Ricci-Yamabe solitons

Abstract

In this paper, we classify Kenmotsu manifolds admitting gradient Ricci-Yamabe solitons and $\ast-\eta-$Ricci-Yamabe solitons. We find some conditions of Kenmotsu manifold about when it shrink, expand and steady. It is shown that Kenmotsu 3-manifold endowed with gradient Ricci-Yamabe soliton and with constant scalar curavature becomes an Einstein manifold. We, also study Kenmotsu manifold admitting $\ast-\eta-$ Ricci-Yamabe solitons becomes generalized $\eta-$Einstein manifold and solve the curvature condition $ R.S=0 $. Finally, we provide two examples which proves existence of gradient Ricci-Yamabe soliton and $\ast-\eta-$ Ricci-Yamabe soliton in Kenmotsu manifolds.