η-Ricci–Yamabe Solitons along Riemannian Submersions

In this paper, we investigate the geometrical axioms of Riemannian submersions in the context of the η-Ricci–Yamabe soliton (η-RY soliton) with a potential field. We give the categorization of each fiber of Riemannian submersion as an η-RY soliton, an η-Ricci soliton, and an η-Yamabe soliton. Additionally, we consider the many circumstances under which a target manifold of Riemannian submersion is an η-RY soliton, an η-Ricci soliton, an η-Yamabe soliton, or a quasi-Yamabe soliton. We deduce a Poisson equation on a Riemannian submersion in a specific scenario if the potential vector field ω of the soliton is of gradient type =:grad(γ) and provide some examples of an η-RY soliton, which illustrates our finding. Finally, we explore a number theoretic approach to Riemannian submersion with totally geodesic fibers.