- Turkish Journal of Mathematics
- Vol: 40 Issue: 4
- Almost co-K"{a}hler manifolds satisfying some symmetry conditions
Almost co-K"{a}hler manifolds satisfying some symmetry conditions
Authors : Yaning Wang
Pages : 740-752
View : 12 | Download : 7
Publication Date : 9999-12-31
Article Type : Makaleler
Abstract :Let $M^{2n+1}$ be an almost co-K\"{a}hler manifold of dimension $>3$ with K\"{a}hlerian leaves. In this paper, we first prove that if $M^{2n+1}$ is locally symmetric, then either it is a co-K\"{a}hler manifold with locally symmetric K\"{a}hlerian leaves, or the Reeb vector field $\xi$ is harmonic and in this case $M^{2n+1}$ is non-co-K\"{a}hler. We also prove that any almost co-K\"{a}hler manifold of dimension $3$ is $\phi$-symmetric if and only if it is locally isometric to either a flat Euclidean space $\mathbb{R}^3$ or a Riemannian product $\mathbb{R}\times N^2(c)$, where $N^2(c)$ denotes a K\"{a}hler surface of constant curvature $c\neq0$.Keywords : Locally symmetric, $phi$-symmetric, almost co-K"{a}hler manifold, K"{a}hlerian leaves