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

Abstract

For a singular Riemannian foliation [Formula: see text] on a Riemannian manifold, a curve is called horizontal if it meets the leaves of [Formula: see text] perpendicularly. For a singular Riemannian foliation [Formula: see text] on a unit sphere [Formula: see text], we show that if [Formula: see text] is a polar foliation or if [Formula: see text] is given by the orbits of an infinitesimally polar action, then the horizontal diameter of [Formula: see text] is [Formula: see text], i.e. any two points in [Formula: see text] can be connected by a horizontal curve of length [Formula: see text].