A note on trans-Sasakian manifolds

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 $\alpha\not=0$ 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.