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

Yutaka, Kanda

Commun. Anal. Geom. 5, No. 3, 413-438 (1997).

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.

Reviewer: Stephan Klaus (Mainz)

MSC:

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

Keywords:

tight contact structure; characteristic foliation; 3-torus; convex surface