A characterisation of the tight three-sphere (Q1913580)

Let \(M\) be a compact oriented 3-manifold and assume that \(M\) carries a positive co-orientable contact structure with plane distribution \(\xi\subset TM\). The contact structure \(\xi\) is said to be tight if there is no disk \(F\) imbedded in \(M\) such that \(T_z(\partial F)\subset\xi_z\) and \(T_zF\not\subset\xi_z\) for all \(z\in\partial F\). The authors' main result gives a characterization of the positive tight contact structure on \(S^3\): \(M\) as above is contact isomorphic to \(S^3\) with its positive tight contact structure iff there exists a contact form \(\lambda\) defining \(\xi\) with the following property. The Reeb vector field \(X\) associated to \(\lambda\) admits a nondegenerate periodic orbit \(P_0\) spanning an imbedded disk \(F\) whose interior is traversal to \(X\) and whose index is \(\mu(P_0,F)=3\).