For a singular Riemannian foliation ℱ on a Riemannian manifold, a curve is called horizontal if it meets the leaves of ℱ perpendicularly. For a singular Riemannian foliation ℱ on a unit sphere 𝕊n, we show that if ℱ is a polar foliation or if ℱ is given by the orbits of an infinitesimally polar action, then the horizontal diameter of 𝕊n is π, i.e. any two points in 𝕊n can be connected by a horizontal curve of length ≤ π.

中文翻译：

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具有极叶和极小极作用的单位球体的水平直径

对于一个奇异的黎曼叶ℱ在黎曼流形上，如果一条曲线与ℱ垂直。对于一个奇异的黎曼叶ℱ在单位球面上𝕊n, 我们证明如果ℱ是极叶，或者如果ℱ由极小极作用的轨道给出，然后水平直径𝕊n是π，即中的任意两点𝕊n可以通过一条长度的水平曲线连接 ≤ π.