Trans-Sasakian Structures with Certain Restrictions

We find restrictions on a trans-Sasakian structure F,u,γ,α,β on a 3-dimensional Riemannian manifold M3,g so that M3,g is homothetic to a Sasakian manifold. In that, first we show that if the vector u of the trans-Sasakian structure F,u,γ,α,β on a 3-dimensional Riemannian manifold M3,g is an affine conformal vector with affine potential α≠0 and the condition uα=−β2 holds, necessarily implies M3,g is homothetic to a Sasakian manifold. Similarly, it is shown that if the vector u of the trans-Sasakian structure F,u,γ,α,β on a 3-dimensional Riemannian manifold M3,g is a projective vector and the sectional curvatures of the plane sections containing u are positive constant, then M3,g is homothetic to a Sasakian manifold. Finally, we find certain generic conditions on a 3-dimensional Riemannian manifold M3,g possessing a trans-Sasakian structure F,u,γ,α,β so that M3,g is homothetic to a Sasakian manifold.