Some type of semisymmetry on two classes of almost Kenmotsu manifolds

summary:The object of the present paper is to study some types of semisymmetry conditions on two classes of almost Kenmotsu manifolds. It is shown that a $(k,\mu )$-almost Kenmotsu manifold satisfying the curvature condition $Q\cdot R = 0$ is locally isometric to the hyperbolic space $\mathbb {H}^{2n+1}(-1)$. Also in $(k,\mu )$-almost Kenmotsu manifolds the following conditions: (1) local symmetry $(\nabla R = 0)$, (2) semisymmetry $(R\cdot R = 0)$, (3) $Q(S,R) = 0$, (4) $R\cdot R = Q(S,R)$, (5) locally isometric to the hyperbolic space $\mathbb {H}^{2n+1}(-1)$ are equivalent. Further, it is proved that a $(k,\mu )'$-almost Kenmotsu manifold satisfying $Q\cdot R = 0$ is locally isometric to $\mathbb {H}^{n+1}(-4) \times \mathbb {R}^n$ and a $(k,\mu )'$\HH almost Kenmotsu manifold satisfying any one of the curvature conditions $Q(S,R) = 0$ or $R\cdot R = Q(S,R)$ is either an Einstein manifold or locally isometric to $\mathbb {H}^{n+1}(-4) \times \mathbb {R}^n$. Finally, an illustrative example is presented