Curvature Properties of Almost Kenmotsu Manifolds with Generalized Nullity Conditions

Abstract

In this paper, it is proved that on a generalized $(k,\mu)'$-almost Kenmotsu manifold of dimension $2n+1$, $n>1$, the conditions of local symmetry, semi-symmetry, pseudo-symmetry and quasi weak-symmetry are equivalent and this is also equivalent to that $M^{2n+1}$ is locally isometric to either the hyperbolic space $\mathbb{H}^{2n+1}(-1)$ or the Riemannian product $\mathbb{H}^{n+1}(-4)\times\mathbb{R}^n$. Moreover, we also prove that, a generalized $(k,\mu)$-almost Kenmotsu manifold of dimension $2n+1$, $n>1$, is pseudo-symmetric if and only if it is locally isometric to the hyperbolic space $\mathbb{H}^{2n+1}(-1)$.