Riemannian maps whose base manifolds admit a Ricci soliton

TL;DR

This paper investigates Riemannian maps with base manifolds admitting Ricci solitons, deriving curvature properties, conditions for Ricci soliton structures on leaves, and harmonicity criteria for such maps.

Contribution

It provides new conditions and examples for Riemannian maps related to Ricci solitons, expanding understanding of their geometric properties and harmonicity.

Findings

Derived Riemannian curvature tensor for base manifolds

Established conditions for leaves to be Ricci soliton or Einstein

Provided criteria for harmonic and biharmonic Riemannian maps

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.