Remark on quasi Sasakian structures

Abstract

In this work, we revisit quasi-Sasakian geometry in dimension three and examine how these structures interact with the foliation generated by the Reeb vector field and its basic cohomology. Through a deformation-based approach, we show that a closed, orientable $3$-manifold admits a quasi-Sasakian structure precisely when it is either Sasakian or arises as a K\"ahler mapping torus. In particular, every quasi-Sasakian structure in this setting can be deformed into a Sasakian or a co-K\"ahler one. This result leads to a complete classification of quasi-Sasakian manifolds in dimension three and highlights the geometric and topological features that distinguish the two cases.