Conformal Submersions Whose Total Manifolds Admit a Ricci Soliton

TL;DR

This paper investigates conformal submersions from Ricci solitons, analyzing their properties, conditions for being totally geodesic, and the Ricci curvature of total manifolds, with applications to fibers and base manifolds.

Contribution

It provides new conditions for conformal submersions from Ricci solitons to be totally geodesic, harmonic, and characterizes the Ricci curvature of total manifolds.

Findings

Conditions for conformal submersion to be totally geodesic

Necessary conditions for fibers and base to be Ricci solitons or Einstein

Characterization of scalar curvature and harmonicity of the submersion

Abstract

In this paper, we study conformal submersions from Ricci solitons to Riemannian manifolds with non-trivial examples. First, we study some properties of the O'Neill tensor $A$ in the case of conformal submersion. We also find a necessary and sufficient condition for conformal submersion to be totally geodesic and calculate the Ricci tensor for the total manifold of such a map with different assumptions. Further, we consider a conformal submersion $F:M \to N$ from a Ricci soliton to a Riemannian manifold and obtain necessary conditions for the fibers of $F$ and the base manifold $N$ to be Ricci soliton, almost Ricci soliton and Einstein. Moreover, we find necessary conditions for a vector field and its horizontal lift to be conformal on $N$ and $(KerF_\ast)^\bot,$ respectively. Also, we calculate the scalar curvature of Ricci soliton $M$. Finally, we obtain a necessary and sufficient condition for $F$ to be harmonic.