Clairaut conformal submersions from Ricci solitons

Abstract

In the present article, we characterize Clairaut conformal submersions whose total manifolds admit a Ricci soliton and provide a non-trivial example of such Clairaut conformal submersions. We firstly calculate scalar curvature and Ricci tensors of total manifolds of Clairaut conformal submersions and provide necessary conditions for the fibres of such Clairaut conformal submersions to be almost Ricci solitons and Einstein. Further, we provide necessary conditions for the base manifold to be Ricci soliton and Einstein. Then, we find a necessary condition for vector field $\dot{\zeta}$ to be conformal vector field and killing vector field. Besides, we indicate that if total manifolds of Clairaut conformal submersions admit a Ricci soliton with the potential mean curvature vector field $H$ of $Ker\vartheta _{\ast }, $ then the total manifolds of Clairaut conformal submersions admit a gradient Ricci soliton. Finally, by solving Poisson equation, we acquire a necessary and sufficient condition for Clairaut conformal submersions to be harmonic.