Sharp systolic inequalities for Riemannian and Finsler spheres of revolution

TL;DR

This paper establishes sharp bounds on the systolic ratio for Riemannian and Finsler spheres of revolution, showing it is maximized at Zoll metrics, thus extending classical systolic inequalities to broader geometric contexts.

Contribution

It proves the systolic ratio bound of π for spheres of revolution and extends this result to certain Finsler metrics, characterizing equality cases as Zoll metrics.

Findings

Systolic ratio of spheres of revolution does not exceed π.

Equality holds if and only if the sphere is Zoll.

Extension of systolic inequalities to Finsler metrics with Killing fields.

Abstract

We prove that the systolic ratio of a sphere of revolution $S$ does not exceed $\pi$ and equals $\pi$ if and only if $S$ 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 $\pi$ and equals $\pi$ if and only if the metric is Riemannian and Zoll.