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The purpose of this study is to characterize Kenmotsu manifolds that satisfy specific curvature conditions. We give the Kenmotsu manifold curvature tensors satisfying the conditions RW5 = 0, RW7 = 0, RW9 = 0 and φ-RW0* = 0. Also we consider a W0*-flat and a φ-W0*-flat Kenmotsu metric manifolds. As a result, M is an η-Einstein Kenmotsu metric manifold.

In the present paper, we study the curvature tensors of (k,mu)-paracontact manifold satisfying the conditions P(ξ ,Y)Z=0, P(ξ ,Y)P=0, P(ξ ,Y)S =0 and (ξ,Y)R=0. According these cases, (k,μ)-paracontact manifolds have been characterized. We obtain the necessary and sufficient conditions of a (k,μ)-paracontact manifold to be η-Einsteins.

The aim of this paper is to study (k,μ)-Paracontact metric manifold. We introduce the curvature tensors of a (k,μ)-paracontact metric manifold satisfying the conditions R⋅P_*=0, R⋅L=0, R⋅W_1=0, R⋅W_0=0 and R⋅M=0. According to these cases, (k,μ)-paracontact manifolds have been characterized such as η-Einstein and Einstein. We get the necessary and sufficient conditions of a (k,μ)-paracontact metric manifold to be η-Einstein. Also, we consider new conclusions of a (k,μ)-paracontact metric manifold contribute to geometry. We think that some interesting results on a (k,μ)-paracontact metric manifold are obtained.

In this paper, we consider $\left(LCS\right)_{n}$ manifold admitting almost $\eta-$Ricci solitons by means of curvature tensors. Ricci pseudosymmetry concepts of $\left(LCS\right)_{n}$ manifold admitting $\eta-$Ricci soliton have introduced according to the choice of some special curvature tensors such as pseudo-projective, $W_{1}$, $W_{1}^{\ast}$ and $W_{2}.$ Then, again according to the choice of the curvature tensor, necessary conditions are searched for $\left(LCS\right)_{n}$ manifold admitting $\eta-$Ricci soliton to be Ricci semisymmetric. Then some characterizations are obtained and some classifications have made.