The object of the present paper is to characterize two classes of almost Kenmotsu manifolds admitting Ricci-Yamabe soliton. It is shown that a (k, µ) ′ -almost Kenmotsu manifold admitting a Ricci-Yamabe soliton or gradient Ricci-Yamabe soliton is locally isometric to the Riemannian product H n+1 (−4) × R n . For the later case, the potential vector field is pointwise collinear with the Reeb vector field. Also, a (k, µ)-almost Kenmotsu manifold admitting certain Ricci-Yamabe soliton with the curvature property Q · P = 0 is locally isometric to the hyperbolic space H 2n+1 (−1) and the non-existense of the curvature property Q · R = 0 is proved.Mathematics Subject Classification 2010: 53D15, 35Q51.

It is well known that a unit sphere admits Sasakian 3-structure. Also, Sasakian manifolds are locally isometric to a unit sphere under several curvature and critical conditions. So, a natural question is: Does there exist any curvature or critical condition under which a Sasakian 3-manifold represents a geometrical object other than the unit sphere? In this regard, as an extension of the * -Ricci soliton, the notion of * -Ricci-Yamabe soliton is introduced and studied on two classes contact metric manifolds. A (2n + 1)-dimensional non-Sasakian N(k)-contact metric manifold admitting * -Ricci-Yamabe soliton is completely classified. Further, it is proved that if a Sasakian 3-manifold M admits * -Ricci-Yamabe soliton (g, V, λ, α, β) under certain conditions on the soliton vector field V , then M is * -Ricci flat, positive Sasakian and the transverse geometry of M is Fano. In addition, the Sasakian 3-metric g is homothetic to a Berger sphere and the soliton is steady. Also, the potential vector field V is an infinitesimal automorphism of the contact metric structure.

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The object of the present paper is to characterize quasi-conformally flat and $\xi$-quasi-conformally flat almost Kenmotsu manifolds with  $(k,\mu)$-nullity and $(k,\mu)'$-nullity distributions respectively. Also we characterize almost Kenmotsu manifolds with vanishing extended quasi-conformal curvature tensor and extended $\xi$-quasi-conformally flat almost Kenmotsu manifolds such that the characteristic vector field $\xi$ belongs to the $(k,\mu)$-nullity distribution.