Abstract
In this paper, we obtain a necessary and sufficient condition for a 3 dimensional compact connected positively curved trans-Sasakian manifold of type (alpha, beta), to be homothetic to a Sasakian manifold. The necessary and sufficient condition is that the smooth function alpha not equal 0 and satisfies the differential equation H-alpha (xi,xi) = 0. Similarly, using the smooth function beta, necessary and sufficient conditions are found for a trans-Sasakian manifold to be homothetic to a Sasakian manifold.