Almost Ricci–Yamabe soliton on contact metric manifolds

Purpose This paper aims to study almost Ricci–Yamabe soliton in the context of certain contact metric manifolds. Design/methodology/approach The paper is designed as follows: In Section 3, a complete contact metric manifold with the Reeb vector field ξ as an eigenvector of the Ricci operator admitting almost Ricci–Yamabe soliton is considered. In Section 4, a complete K -contact manifold admits gradient Ricci–Yamabe soliton is studied. Then in Section 5, gradient almost Ricci–Yamabe soliton in non-Sasakian ( k , μ )-contact metric manifold is assumed. Moreover, the obtained result is verified by constructing an example. Findings We prove that if the metric g admits an almost ( α , β )-Ricci–Yamabe soliton with α ≠ 0 and potential vector field collinear with the Reeb vector field ξ on a complete contact metric manifold with the Reeb vector field ξ as an eigenvector of the Ricci operator, then the manifold is compact Einstein Sasakian and the potential vector field is a constant multiple of the Reeb vector field ξ . For the case of complete K -contact, we found that it is isometric to unit sphere S 2 n +1 and in the case of ( k , μ )-contact metric manifold, it is flat in three-dimension and locally isometric to E n +1 × S n (4) in higher dimension. Originality/value All results are novel and generalizations of previously obtained results.