A Study on Contact Metric Manifolds Admitting a Type of Solitons

Abstract

The principal aim of the present article is to characterize certain properties of η‐Ricci–Bourguignon solitons on three types of contact manifolds, that are K‐contact manifolds, (κ, μ)‐contact metric manifolds, and N(κ)‐contact metric manifolds. It is shown that if a K‐contact manifold admits an η‐Ricci–Bourguignon soliton whose potential vector field is the Reeb vector field, then the scalar curvature of the manifold is constant and the manifold becomes an η‐Einstein manifold. In this regard, it is shown that if a K‐contact manifold admits a gradient η‐Ricci–Bourguignon soliton, then the scalar curvature is a constant and either the soliton reduces to gradient Ricci–Bourguignon soliton, or the potential function is a constant. We also deduce that if a (κ, μ)‐contact metric manifold admits a gradient η‐Ricci–Bourguignon soliton, then either the manifold is locally En+1 × Sn(4), or the gradient of the potential function is pointwise collinear with the Reeb vector field, or the potential function is a constant. η‐Ricci–Bourguignon solitons on three‐dimensional N(κ)‐contact metric manifolds are also considered. Finally, an example using differential equations has been constructed to verify a result.