Remark on quasi Sasakian structures

TL;DR

This paper classifies three-dimensional quasi-Sasakian manifolds, showing they are either Sasakian or Kähler mapping tori, and demonstrates their deformation into these structures.

Contribution

It provides a complete classification of 3D quasi-Sasakian manifolds and establishes their deformation into Sasakian or co-K"ahler structures.

Findings

A 3D quasi-Sasakian manifold is either Sasakian or a K"ahler mapping torus.

Every quasi-Sasakian structure in this setting can be deformed into a Sasakian or co-K"ahler structure.

The classification highlights key geometric and topological distinctions.

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.