Conformal Submersions Whose Total Manifolds Admit a Ricci Soliton

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.