Journal of the
Korean Mathematical Society JKMS

In this paper, we prove that the Ricci tensor of a three-dimensional almost Kenmotsu manifold satisfying $\nabla_\xi h=0$, $h\neq0$, is $\eta$-parallel if and only if the manifold is locally isometric to either the Riemannian product $\mathbb{H}^{2}(-4)\times\mathbb{R}$ or a non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure.

Keywords: $3$-dimensional almost Kenmotsu manifold, $\eta$-parallel parallel Ricci tensor, non-unimodular Lie group

MSC numbers: Primary 53D15; Secondary 53C25