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We show that the germ of the contact structure surrounding a certain kind of convex hypersurfaces is overtwisted. We then find such hypersurfaces close to any plastikstufe with toric core so that these imply overtwistedness. All proofs in this article are explicit, and we hope that the methods used here might hint at a deeper understanding of the size of neighborhoods in contact manifolds. In the appendix we reprove in a concise way that the Legendrian unknot is loose if the ambient manifold contains a large enough neighborhood of a 2-dimensional overtwisted disk. Additionally we prove the folklore result that the singular distribution induced on a hypersurface $\Sigma$ of a contact manifold $(M, \xi)$ determines the germ of the contact structure around $\Sigma$., Comment: 13 pages, 2 figure

International audience; The Bourgeois construction associates to every contact open book on a manifold V a contact structure on V×T². We study in this article some of the properties of V that are inherited by V×T² and some that are not.Giroux has provided recently a suitable framework to work with contact open books. In the appendix of this article, we quickly review this formalism, and we work out a few classical examples of contact open books to illustrate how to use this language.

The main result in this thesis is the classification of SO(3)-actions on contact 5-manifolds. Using properties of the moment map, one can reduce the manifold to a 3-dimensional contact manifold with an S^1-action. This works everywhere outside of the singular orbits. For the singular orbits three models can be given that describe all possible cases. The 5-manifold is then obtained by gluing the singular set onto the 3-dimensional S^1-manifold in a compatible way. As it is well-known, S^1-bundles over a closed surface are classified by an integer called the Euler number. A similar invariant can be recovered in our 3-dimensional setting. We call it the Dehn-Euler number.