Remarks on Almost Cosymplectic 3-Manifolds with RICCI Operators

Abstract

Let M be an almost cosymplectic 3-h-a-manifold. In this paper, we prove that the Ricci operator of M is transversely Killing if and only if M is locally isometric to a product space of an open interval and a surface of constant Gauss curvature, or a unimodular Lie group equipped with a left invariant almost cosymplectic structure. Some corollaries of this result and some examples illustrating main results are given.where X and ∇ denote an arbitrary vector field and the Levi-Civita connection on a Riemannian manifold (M, g), respectively. Relation (1) is much weaker than parallelism of Blair. In [20], Blair applied (1) for the structure tensor field to give a characterization for an almost contact metric manifold to be cosymplectic. In this paper, applying relation (1) for the Ricci operator, we obtain some local classification results of