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This paper aims to study almost Ricci–Yamabe soliton in the context of certain contact metric manifolds.

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.

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 S²ⁿ⁺¹ and in the case of (k, μ)-contact metric manifold, it is flat in three-dimension and locally isometric to Eⁿ⁺¹ × Sⁿ(4) in higher dimension.

All results are novel and generalizations of previously obtained results.

Besse first conjectured that the solution of the critical point equation (CPE) must be Einstein. The CPE conjecture on some other types of Riemannian manifolds, for instance, odd-dimensional Riemannian manifolds has considered by many geometers. Hence, it deserves special attention to consider the CPE on a certain class of almost contact metric manifolds. In this direction, the authors considered CPE on almost f-cosymplectic manifolds.

The paper opted the tensor calculus on manifolds to find the solution of the CPE.

In this paper, in particular, the authors obtained that a connected f-cosymplectic manifold satisfying CPE with \lambda=\tilde{f} is Einstein. Next, the authors find that a three dimensional almost f-cosymplectic manifold satisfying the CPE is either Einstein or its scalar curvature vanishes identically if its Ricci tensor is pseudo anti‐commuting.

The paper proved that the CPE conjecture is true for almost f-cosymplectic manifolds.