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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?

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A complete topological classification is due to Geiges, and can be found in this 2001 paper by Guilfoyle. (the first Theorem in the paper).

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  • $\begingroup$ @BenSmith: With the caveat that these manifolds are not necessarily circle bundles, they are Seifert fiber spaces. $\endgroup$
    – Misha
    Commented Dec 2, 2015 at 17:21
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Not quite, but it is true up to deformations and is essentially due to Belgun. See Section 10.1 of the my book Sasakian Geometry, C.P. Boyer and K. Galicki.

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