Abstract
A systematic study of eta-Einstein trans-Sasakian manifold is performed. We find eight necessary and sufficient conditions for the structure vector field xi of a trans-Sasakian manifold to be an eigenvector field of the Ricci operator. We show that for a 3-dimensional almost contact metric manifold (M, phi, xi, eta, g), the conditions of being normal, trans-K-contact, trans-Sasakian are all equivalent to del xi o phi = phi 0 del xi. In particular, the conditions of being quasi-Sasakian, normal with 0 = 2 beta = div xi, trans-K-contact of type (alpha, 0), trans-Sasakian of type (alpha, 0), and C-6-class are all equivalent to del xi = -alpha phi, where 2 alpha = Trace(phi del xi). In last, we give fifteen necessary and sufficient conditions for a 3-dimensional trans-Sasakian manifold to be eta-Einstein.