Significance of Solitonic Fibers in Riemannian Submersions and Some Number Theoretic Applications

In this manifestation, we explain the geometrisation of η-Ricci–Yamabe soliton and gradient η-Ricci–Yamabe soliton on Riemannian submersions with the canonical variation. Also, we prove any fiber of the same submersion with the canonical variation (in short CV) is an η-Ricci–Yamabe soliton, which is called the solitonic fiber. Also, under the same setting, we inspect the η-Ricci–Yamabe soliton in Riemannian submersions with a φ(Q)-vector field. Moreover, we provide an example of Riemannian submersions, which illustrates our findings. Finally, we explore some applications of Riemannian submersion along with cohomology, Betti number, and Pontryagin classes in number theory.