Study of Riemannian maps with total manifolds admit a Ricci-Bourguignon soliton

Abstract

In this article, we look at Riemannian maps with total manifolds that accept a Ricci-Bourguignon soliton and provide an illustration. We have obtained the requirements for any fiber of such a Riemannian map to be Ricci-Bourguignon soliton, almost Ricci-Bourguignon soliton, and Einstein. We also find that the range space of such a Riemannian map must be Ricci-Bourguignon soliton and Einstein. Moreover, we have studied $\eta$-Ricci-Bourguignon soliton on a totally geodesic Riemannian map. Furthermore, we investigate the harmonicity and biharmonicity of a Riemannian map derived from the Ricci-Bourguignon soliton and determine the necessary and sufficient conditions for such a Riemannian map to be harmonic and biharmonic.