Sharp systolic inequalities for rotationally symmetric 2-orbifolds

Abstract

We show that suitably defined systolic ratios are globally bounded from above on the space of rotationally symmetric spindle orbifolds and that the upper bound is attained precisely at so-called Besse metrics, i.e. Riemannian orbifold metrics all of whose geodesics are closed. The first type of systolic ratios that we consider are defined in terms of closed geodesics that lift to contractible loops on certain covers of the unit sphere bundle. The second type of systolic ratios are defined in terms of the kth shortest closed geodesic, where the number k depends on the underlying orbifold. Our results generalize a corresponding result of Abbondandolo, Bramham, Hryniewicz and Salomão for spheres of revolution, even in the manifold case. Moreover, they complement recent results by Abbondandolo, Mazzucchelli and the first named author on local systolic inequalities for Besse Reeb flows on closed 3-manifolds.