Almost Kenmotsu 3-Manifolds Satisfying Some Generalized Nullity Conditions

Abstract

In this paper, a three-dimensional almost Kenmotsu manifold $M^3$ satisfying the generalized $(\kappa,\mu)'$-nullity condition is investigated. We mainly prove that on $M^3$ the following statements are equivalent: $M^3$ is $\phi$-symmetric; (2) the Ricci tensor of $M^3$ is cyclic-parallel; (3) the Ricci tensor of $M^3$ is of Codazzi type; (4) $M^3$ is conformally flat with scalar curvature invariant along the Reeb vector field; (5) $M^3$ is locally isometric to either the hyperbolic space $\mathbb{H}^3(-1)$ or the Riemannian product $\mathbb{H}^{2}(-4)\times\mathbb{R}$.