Ricci tensors on trans-Sasakian 3-manifolds

Abstract

In this paper, it is proved that a trans-Sasakian $3$-manifold is locally symmetric if and only if it is locally isometric to the sphere space $\mathbb{S}^3(c^2)$, the hyperbolic space $\mathbb{H}^3(-c^2)$, the Euclidean space $\mathbb{R}^3$, the product space $\mathbb{R}\times\mathbb{S}^2(c^2)$ or $\mathbb{R}\times\mathbb{H}^2(-c^2)$, where $c$ is a non-zero constant. Some examples are constructed to illustrate main results. We also give some new conditions for a compact trans-Sasakian $3$-manifold to be proper.