Riemannian maps whose base manifolds admit a Ricci soliton

Abstract

In this paper, we study Riemannian maps whose base manifolds admit a Ricci soliton and give a non-trivial example of such a Riemannian map. First, we find Riemannian curvature tensor for the base manifolds of Riemannian map $F$. Further, we obtain the Ricci tensor and calculate the scalar curvature of the base manifold. Moreover, we obtain necessary conditions for the leaves of $rangeF_\ast$ to be Ricci soliton, almost Ricci soliton, and Einstein. We also obtain necessary conditions for the leaves of $(rangeF_\ast)^\bot$ to be Ricci soliton and Einstein. Also, we calculate the scalar curvatures of $rangeF_\ast$ and $(rangeF_\ast)^\bot$ by using Ricci soliton. Finally, we study the harmonicity and biharmonicity of such a Riemannian map. We obtain a necessary and sufficient condition for such a Riemannian map between Riemannian manifolds to be harmonic. We also obtain necessary and sufficient conditions for a Riemannian map from a Riemannian manifold to a space form that admits Ricci soliton to be harmonic and biharmonic.