On Compact Trans-Sasakian Manifolds

We study 3-dimensional compact and simply connected trans-Sasakian manifolds and find necessary and sufficient conditions under which these manifolds are homothetic to Sasakian manifolds. The first two results deal with finding necessary and sufficient conditions on a compact and simply connected trans-Sasakian manifold to be homothetic to an Einstein Sasakian manifold and in the third result deals with finding necessary and sufficient condition on a compact and simply connected trans-Sasakian manifold to be homothetic to a Sasakian manifold.