Bulletin of the
Korean Mathematical Society BKMS

Let $M$ be a curvature homogeneous or ball-homogeneous non-coK\"{a}hler almost coK\"{a}hler $3$-manifold. In this paper, we prove that $M$ is locally isometric to a unimodular Lie group if and only if the Reeb vector field $\xi$ is an eigenvector field of the Ricci operator. To extend this result, we prove that $M$ is homogeneous if and only if it satisfies $\nabla_\xi h=2f\phi h$, $f\in\mathbb{R}$.

Keywords: almost coK\"{a}hler 3-manifold, ball-homogeneity, curvature homogeneity, Locally homogeneity, Lie group

MSC numbers: Primary 53D15; Secondary 53C30, 53C25