A few properties of Besse contact manifolds.
Joint work with Alberto Abbondandolo and Christian Lange.
A closed connected contact manifold is called Besse when all of its Reeb orbits are closed, and Zoll when furthermore all Reeb orbits have the same minimal period. In this talk, I will present a recollection of recent results/work in progress on the subject: - It is known that Besse contact 3-spheres are strictly contactomorphic to rational ellipsoids. In higher dimensions, the analogous statement is open. Nevertheless, I will show that at least those contact (2n-1)-spheres that are convex hypersurfaces in symplectic vector spaces still "resemble" a rational ellipsoid. This is joint work with Marco Radeschi. - Inspired by recent results on the systolic optimality of Zoll contact manifolds, I will show that Besse contact 3-manifolds are local maximizers of a suitable generalized systolic ratio.