On a type of almost Kenmotsu manifolds with harmonic curvature tensors

Abstract

Let $M^{2n+1}$ be an almost Kenmotsu manifold with the characteristic vector field belonging to the $(k,\mu)'$-nullity distribution. We prove that the curvature tensor of $M^{2n+1}$ is harmonic if and only if $M^{2n+1}$ is locally isometric to either a product space $\mathbb{H}^{n+1}(-4)\times\mathbb{R}^n$, or an Einstein warped product $C\times_{f}N^{2n}$ of an open interval and a Ricci-flat almost Kähler manifold.