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

## TL;DR

This paper proves that on a unit sphere, the maximum length of a shortest horizontal curve connecting any two points, with respect to certain special foliations, is always π, extending understanding of geometric structures on spheres.

## Contribution

It establishes that for polar foliations and infinitesimally polar actions on spheres, the horizontal diameter is exactly π, providing a new geometric bound for these foliations.

## Key findings

- Horizontal diameter of spheres with these foliations is π
- Any two points can be connected by a horizontal curve of length ≤ π
- Results apply to 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 $\leq\pi$.