Contact geometry and isosystolic inequalities

Details Contact geometry and isosystolic inequalities Contact geometry and isosystolic inequalities

2011-09-20

arxiv.org/abs/1109.4253v1

Mathematics

Symplectic Geometry

23 pages

Scientific paper

Let $g_s$ be a smooth volume-preserving deformation of the canonical metric in a projective space (real, complex, quaternionic, or Cayley). If at $s=0$ the deformation does not agree to first order with a (trivial) deformation 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 is one of a number of results that follow from a contact-geometric reformulation of systolic geometry, and the use of canonical perturbation theory to exploit the large symmetry group that the theory inherits. As other applications, we show that Zoll manifolds are critical points of the systolic volume, and give a partial answer to a question of Viterbo relating the volume and the symplectic capacity of convex bodies.

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