Trans-Sasakian 3-manifolds with Einstein-like Ricci operators

Abstract

Let M be a trans-Sasakian 3-manifold such that the Ricci curvature of the structure vector field vanishes. In this paper, it is proved that if M is compact, then it is homothetic to a cosymplectic manifold. Without the compactness assumption, we prove that M is locally isometric to the product of R and a Kӓhler surface of constant curvature if the Ricci operator of M is of Codazzi type or cyclic parallel. We classify trans-Sasakian 3-manifolds with Killing type Ricci tensors. We construct some concrete examples to illustrate the above theorems.