## The classification of tight contact structures on the 3-torus.(English)Zbl 0899.53028

A contact structure $$\zeta$$ on a 3-manifold is a completely non-integrable 2-plane field. It is called tight if the characteristic foliation of any embedded disc has no limit cycle. In the other case, $$\zeta$$ is called overtwisted, and its classification is then given by the homotopy classes of 2-plane fields. In contrast, the classification of tight contact structures is much more complicated.
Here, the author classifies orientable tight contact structures on the 3-torus. A complete list up to contact diffeomorphism is given by $$\zeta_n$$, $$n$$ a positive integer, where $$\zeta_n$$ is defined by the 1-form $$\cos 2\pi nzdx +\sin 2\pi nzdy$$. The proof uses the theory of characteristic foliations and convex surfaces of Giroux and a cut-and-paste method.

### MSC:

 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C12 Foliations (differential geometric aspects)
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