A systolic inequality for geodesic flows on the two-sphere

TL;DR

This paper establishes sharp inequalities relating the lengths of shortest and longest simple closed geodesics to the area of positively curved, pinched metrics on the two-sphere, characterizing Zoll metrics as extremal cases.

Contribution

It proves new sharp systolic inequalities for positively curved metrics on the two-sphere, confirming a conjecture and characterizing Zoll metrics as equality cases.

Findings

The inequalities ll_{ m min}(g)^2 \u2264 ap ext{Area}(S^2,g)  ll_{ m max}(g)^2 hold under positive, pinched curvature.

Equality holds if and only if the metric is Zoll.

The proof combines Riemannian and symplectic geometry techniques.

Abstract

For a Riemannian metric $g$ on the two-sphere, let $\ell_{\min}(g)$ be the length of the shortest closed geodesic and $\ell_{\max}(g)$ be the length of the longest simple closed geodesic. We prove that if the curvature of $g$ is positive and sufficiently pinched, then the sharp systolic inequalities \[ \ell_{\rm min}(g)^2 \leq \pi \ {\rm Area}(S^2,g) \leq \ell_{\max}(g)^2, \] hold, and each of these two inequalities is an equality if and only if the metric $g$ is Zoll. The first inequality answers positively a conjecture of Babenko and Balacheff. The proof combines arguments from Riemannian and symplectic geometry.