On the horizontal diameter of the unit sphere |
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Authors: | Yi Shi Zhiqi Xie |
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Institution: | 1.School of Mathematical Sciences,Shanxi University,Taiyuan,China;2.School of Mathematics and Statistics,Yulin University,Yulin,China |
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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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