Abstract
For a singular Riemannian foliation \(\mathcal {F}\) on a Riemannian manifold M, a curve is called horizontal if it meets the leaves of \(\mathcal {F}\) perpendicularly. For a singular Riemannian foliation \(\mathcal {F}\) on a unit sphere \(\mathbb {S}^{n}\), we show that if \(\mathcal {F}\) satisfies some properties, then the horizontal diameter of \(\mathbb {S}^{n}\) is \(\pi \), i.e., any two points in \(\mathbb {S}^{n}\) can be connected by a horizontal curve of length \(\le \pi \).
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Shi, Y., Xie, Z. On the horizontal diameter of the unit sphere. Arch. Math. 110, 91–97 (2018). https://doi.org/10.1007/s00013-017-1118-0
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DOI: https://doi.org/10.1007/s00013-017-1118-0