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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 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. Originality/value: All results are novel and generalizations of previously obtained results. [ABSTRACT FROM AUTHOR]

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 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. Originality/value – All results are novel and generalizations of previously obtained results.

The aim of the paper is to study conformal η-Ricci soliton on (2n + 1)- dimensional (k, μ)-contact metric manifold. At first, we establish various results related to (2n + 1)-dimensional (k, μ)-contact metric manifold that exhibit conformal η-Ricci soliton. Next we study some curvature conditions admitting conformal η-Ricci soliton on (2n + 1)-dimensional (k, μ)-contact metric manifold. Furthermore, we consider specific conditions associated with conformal η-Ricci soliton on (2n+1)-dimensional (k, μ)- contact metric manifold. Besides these geometrical point of view we consider this soliton in a perfect fluid spacetime and obtain some interesting physical properties. Finally, we show the existence of a conformal η-Ricci soliton on (k, μ)-contact metric manifold. [ABSTRACT FROM AUTHOR]

The object of the present paper is to introduce a new type of quarter symmetric non-metric connection on a (k, μ)-contact metric manifold and study some properties of quarter symmetric non-metric connection on a (k, μ)-contact metric manifold. Further, we obtain some properties of nearly Ricci recurrent on a (k, μ)-contact metric manifold with respect to quarter symmetric non-metric connection. Finally, we present an example to verify our result. [ABSTRACT FROM AUTHOR]

In this paper we study (k, μ)-contact metric manifold en- dowed with a C-Bochner curvature tensor. Here we describe C-Bochner pseudosymmetric (k, μ)-contact metric manifold, then the manifold is η- Einstein manifold. Moreover, we also investigates (k, μ)-contact metric manifold satisfying the conditions B(ξ, X). B = 0 and B(ξ, X) .R = 0. [ABSTRACT FROM AUTHOR]