Conformal η -Ricci-Yamabe Solitons within the Framework of ϵ -LP-Sasakian 3-Manifolds

Abstract

In the present note, we study $ϵ$ -LP-Sasakian 3-manifolds $M^3\left(ϵ\right)$ whose metrics are conformal $\eta$ -Ricci-Yamabe solitons (in short, CERYS), and it is proven that if an $M^3\left(ϵ\right)$ with a constant scalar curvature admits a CERYS, then ${\pounds }_U\zeta$ is orthogonal to $\zeta$ if and only if $\Lambda -ϵ\sigma =-2ϵl+\left(mr/2\right)+\left(1/2\right)\left(p+\left(2/3\right)\right)$ . Further, we study gradient CERYS in $M^3\left(ϵ\right)$ and proved that an $M^3\left(ϵ\right)$ admitting gradient CERYS is a generalized conformal $\eta$ -Einstein manifold; moreover, the gradient of the potential function is pointwise collinear with the Reeb vector field $\zeta$ . Finally, the existence of CERYS in an $M^3\left(ϵ\right)$ has been drawn by a concrete example.