Almost Ricci--Bourguignon Solitons on Contact Metric Three-Manifolds

Abstract

We investigate almost Ricci--Bourguignon solitons on three-dimensional contact metric manifolds. Under natural curvature assumptions, we show that the additional freedom introduced by allowing the soliton function to vary is rigidly constrained by the contact geometry. Using a local orthonormal \(\varphi\)-basis on the non-Sasakian region, we derive the full component form of the almost Ricci--Bourguignon soliton equation. As applications, we consider the cases where the potential vector field is pointwise collinear with, or orthogonal to, the Reeb vector field. For contact metric three-manifolds satisfying \(Q\xi=\sigma\xi\), we prove that a collinear potential field must vanish on the non-Sasakian region whenever \(\xi(\sigma)=0\). In the orthogonal case, when \(\sigma\) is constant and the manifold is non-Sasakian, the almost soliton function is forced to be constant; hence the soliton reduces to a Ricci--Bourguignon soliton. In fact, the metric is Einstein, and if the orthogonal potential field is not identically zero, then the metric is flat.