Ricci solitons and gradient Ricci solitons in three-dimensional trans-Sasakian manifolds

The object of the present paper is to study 3-dimensional trans-Sasakian manifolds admitting Ricci solitons and gradient Ricci solitons. We prove that if (1, V, lambda) is a Ricci soliton where V is collinear with the characteristic vector field xi, then V is a constant multiple of xi and the manifold is of constant scalar curvature provided alpha, beta = constant. Next we prove that in a 3-dimensional trans-Sasakian manifold with constant scalar curvature if 1 is a gradient Ricci soliton, then the manifold is either a beta-Kenmotsu manifold or an Einstein manifold. As a consequence of this result we obtain several corollaries.