On metrics on 2-orbifolds all of whose geodesics are closed

TL;DR

This paper investigates the spectrum of geodesic periods on 2-orbifolds with all geodesics closed, showing it depends only on topology and extending known results from manifolds to orbifolds.

Contribution

It computes the geodesic period spectrum for such orbifolds, generalizes the constant length property of prime geodesics to orbifolds, and discusses rigidity results related to contact forms and systolic inequalities.

Findings

The geodesic period spectrum depends only on orbifold topology.

All prime geodesics have the same length in these orbifolds.

The results extend known manifold properties to orbifolds.

Abstract

We show that the geodesic period spectrum of a Riemannian 2-orbifold all of whose geodesics are closed depends, up to a constant, only on its orbifold topology and compute it. In the manifold case we recover the fact proved by Gromoll, Grove and Pries that all prime geodesics have the same length. In the appendix we partly strengthen our result in terms of conjugacy of contact forms and explain how to deduce rigidity on the real projective plane based on a systolic inequality due to Pu. (We do not use a Lusternik-Schnirelmann type theorem on the existence of at least three simple closed geodesics.)