Horizontal diameter of unit spheres with polar foliations and infinitesimally polar actions

Abstract

For a singular Riemannian foliation $\mathcal{F}$ on a Riemannian manifold, 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 $\mathcal{F}$ is a polar foliation or if $\mathcal{F}$ is given by the orbits of an infinitesimally polar action, 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 $\le \pi$.