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.

我们表明，适当定义的收缩比在旋转对称纺锤体轨道空间上从上方全局界定，并且上限精确地达到所谓的贝斯度量，即所有测地线都闭合的黎曼轨道度量。我们考虑的第一种类型的收缩率是根据封闭测地线定义的，该封闭测地线提升到单位球束某些覆盖层上的可收缩环。第二种类型的收缩率是根据第k个最短闭合测地线定义的，其中数字k取决于底层轨道。我们的结果推广了 Abbondandolo、Bramham、Hryniewicz 和 Salomão 对于革命领域的相应结果，即使在流形情况下也是如此。此外，它们补充了 Abbondandolo、Mazzucchelli 和第一作者关于封闭 3 流形上 Besse Reeb 流的局部收缩不等式的最新结果。