Contact geometry and isosystolic inequalities

By J. '. Paiva, Florent Balacheff. 2011

48 citations recorded by the source corpus.

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 gs 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 gs 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.Henri Poincaré

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/accd099637171006b66adad4c98c6f8dd974886d

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

The Restricted Three-Body Problem and Holomorphic Curves (2018, methodology): “The introduction of systolic question to the contact world is rather new and started with the work of ´Alvarez Paiva and Balacheﬀ [13].”