On the horizontal diameter of the unit sphere

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 $\le \pi$.