Riemannian manifolds in three dimensions and $*$-$\eta$-Ricci-Yamabe solitons

In the domain of Riemannian Geometry, we explore $*$-$\eta$-Ricci-Yamabe soliton on a Riemannian manifold $(G^3, g)$. Initially, we establish that if the metric $g$ of $G^3$ constitutes a $*$-$\eta$-Ricci-Yamabe soliton, then $G^3$ is necessarily Einstein, when the soliton vector field $V$ is contact. Additionally, we investigated that the Riemannian manifold $(G^3, g),$ accommodates a gradient almost $*$-$\eta$-Ricci-Yamabe soliton, concluding that it must be Einstein with a consistent scalar curvature $r=-6.$ The associated functions of the $*$-$\eta$-Ricci soliton are characterized by $\alpha=1$ and $\beta=0.$