Contact geometry and isosystolic inequalities

A long-standing open problem asks whether a Riemannian metric on the real projective space with the same volume as the canonical metric carries a periodic geodesic whose length is at most π. A contact-geometric reformulation of systolic geometry and the use of canonical perturbation theory allow us to solve a parametric version of this problem: if g _s is a smooth, constant-volume deformation of the canonical metric that is not formally trivial, the length of the shortest periodic geodesic of the metric g _s attains π as a strict local maximum at s = 0. This result still holds for complex and quaternionic projective spaces as well as for the Cayley plane. Moreover, the same techniques can be applied to show that Zoll Finsler manifolds are the unique smooth critical points of the systolic volume. Pour résoudre un problème nouveau, nous cherchons toujours à le simplifier par une série de transformations; mais cette simplification a un terme, car il y a dans tout problème quelque chose d'essentiel, pour ainsi dire, que toute transformation est impuissante à modifier.