Sharp systolic inequalities for Riemannian and Finsler spheres of revolution

Abstract

We prove that the systolic ratio of a sphere of revolution S S does not exceed π \pi and equals π \pi if and only if S S is Zoll. More generally, we consider the rotationally symmetric Finsler metrics on a sphere of revolution which are defined by shifting the tangent unit circles by a Killing vector field. We prove that in this class of metrics the systolic ratio does not exceed π \pi and equals π \pi if and only if the metric is Riemannian and Zoll.