A Study on Contact Metric Manifolds Admitting a Type of Solitons

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 Nκ-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×Sn4, 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 Nκ-contact metric manifolds are also considered. Finally, an example using differential equations has been constructed to verify a result.