Curvature homogeneity and ball-homogeneity on almost coK\"{a}hler 3-manifolds
Bull. Korean Math. Soc. 2019 Vol. 56, No. 1, 253-263
https://doi.org/10.4134/BKMS.b180222
Published online 2019 Jan 31
Yaning Wang
Henan Normal University
Abstract : 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
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