Some Applications of $\eta-$Ricci Solitons to Contact Riemannian Submersions

Abstract

The aim of this paper is to study a contact Riemannian submersion $\pi$ from Sasakian manifold $M$ admits an $\eta-$Ricci soliton to an almost contact metric manifold $B$. Here, we provide the Ricci tensor of the total space $M$ and classify about any fiber of $\pi$ and the manifold $B$ are Ricci soliton, $\eta-$Ricci soliton, Einstein, $\eta-$Einstein, quasi-Einstein or generalized quasi-Einstein. Finally, we investigate the total space $M$ is endowed with a torqued vector field and obtain some characterizations about contact Riemannian submersions depending on whether such a vector field is horizontal or vertical.