Contact geometry and isosystolic inequalities
  A long-standing open problem in systolic geometry asks whether a Riemannian metric on the real projective space whose volume equals that of the canonical metric, but is not isometric to it, must necessarily carry a periodic geodesic of length smaller than \pi. A contact-geometric reformulation of systolic geometry and the use of canonical perturbation theory allow us to solve a parametric version of this problem. Namely, we show that if g_s is a smooth volume-preserving deformation of the canonical metric and at s=0 the deformation is not tangent to all orders to trivial deformations (i.e., to deformations of the form \phi_s^* g_0 for some isotopy \phi_s), then the length of the shortest periodic geodesic of the metric g_s attains \pi 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. 
Juan-Carlos Alvarez Paiva and Florent Balacheff
math.SG math.DG