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

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].

Octonions, triality, the exceptional Lie algebra F4 and polar actions on the Cayley hyperbolic plane

Using octonions and the triality property of Spin(8), we find explicit formulae for the Lie brackets of the exceptional simple real Lie algebras [Formula: see text] and [Formula: see text], i.e. the Lie algebras of the isometry groups of the Cayley projective plane and the Cayley hyperbolic plane. As an application, we determine all polar actions on the Cayley hyperbolic plane which leave a totally geodesic subspace invariant.