In this paper, it is shown that for a 3-dimensional compact simply connected trans-Sasakian manifold of type (α, β), the smooth functions α, β satisfy the Poisson equations Δα = β, Δα = α2β and Δβ = α2β, 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 (α, β) in terms of a differential equation satisfied by the smooth function α to be homothetic to a Sasakian manifold

Recalling the recent results obtained by Sharma, Deshmukh and
Al-Solamy [SD] for a canonically induced almost contact metric structure by
a global unit tangent vector field on a Lagrangian submanifold of the nearly