Almost co-K¨ahler manifolds satisfying some symmetry conditions

Abstract

Let M2n+1 be an almost co-K¨ahler manifold of dimension > 3 with K¨ahlerian leaves. In this paper, we first prove that if M2n+1 is locally symmetric, then either it is a co-K¨ahler manifold with locally symmetric K¨ahlerian leaves, or the Reeb vector field ξ is harmonic and in this case M2n+1 is non-co-K¨ahler. We also prove that any almost co-K¨ahler manifold of dimension 3 is ϕ-symmetric if and only if it is locally isometric to either a flat Euclidean space R 3 or a Riemannian product R × N 2 (c), where N 2 (c) denotes a K¨ahler surface of constant curvature c ̸=0.