It suffices to say that all circle bundles on compact Riemann surfaces admit the structure of a closed Sasakian 3-manifold. The question is, the converse of this statement and/or what are the sufficient conditions on a Sasakian 3-manifold that ensure it will be such a circle bundle?



Is every closed Sasakian 3-manifold a circle bundle on a Riemann surface?
A complete topological classification is due to Geiges, and can be found in [this 2001 paper by Guilfoyle](http://arxiv.org/pdf/math/0102015v1.pdf). (the first Theorem in the paper).


