Trans-Sasakian Manifolds Homothetic to Sasakian Manifolds

Abstract

In this paper, it is shown that for a 3-dimensional compact simply connected trans-Sasakian manifold of type $${(\alpha,\beta)}$$ , the smooth functions $${\alpha,\beta}$$ satisfy the Poisson equations $${\Delta \alpha = \beta}$$ , $${\Delta \alpha = \alpha ^{2}\beta}$$ and $${\Delta \beta = \alpha ^{2}\beta}$$ , respectively, if and only if it is homothetic to a Sasakian manifold. We also find a necessary and sufficient condition for a connected 3-dimensional trans-Sasakian manifold of type $${(\alpha,\beta)}$$ in terms of a differential equation satisfied by the smooth function $${\alpha}$$ to be homothetic to a Sasakian manifold.