Contact geometry and isosystolic inequalities
Alvarez Paiva, Juan Carlos (Université de Lille. Laboratoire Paul Painlevé)
Balacheff, Florent Nicolas (Université de Lille. Laboratoire Paul Painlevé)

Date:	2014
Abstract:	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.
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Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	37G05 ; 53C23 ; 53C60 ; 53D10 ; Lie transforms ; Method of averaging ; Normal forms ; Regular contact manifold ; Systolic inequalities ; Zoll manifolds
Published in:	Geometric and Functional Analysis, Vol. 24, Issue 2 (April 2014) , p. 648-669, ISSN 1420-8970

DOI: 10.1007/s00039-014-0250-2

Postprint 22 p, 506.0 KB

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