Clairaut Riemannian maps whose total manifolds admit a Ricci soliton

In this paper, we study Clairaut Riemannian maps whose total manifolds admit a Ricci soliton and give a nontrivial example of such Clairaut Riemannian maps. First, we calculate Ricci tensors and scalar curvature of total manifolds of Clairaut Riemannian maps. Then we obtain necessary conditions for the fibers of such Clairaut Riemannian maps to be Einstein and almost Ricci solitons. We also obtain a necessary condition for vector field a to be conformal, where a is a geodesic curve on total manifold of Clairaut Riemannian map. Further, we show that if total manifolds of Clairaut Riemannian maps admit a Ricci soliton with the potential mean curvature vector field of ker F. then the total manifolds of Clairaut Riemannian maps also admit a gradient Ricci soliton and obtain a necessary and sufficient condition for such maps to be harmonic by solving Poisson equation.