Polar foliations on quaternionic projective spaces

Abstract

We classify irreducible polar foliations of codimension $q$ on quaternionic projective spaces $\mathbb{H}{P}^n$, for all $(n,q)\ne (7,1)$. We prove that all irreducible polar foliations of any codimension (resp. of codimension one) on $\mathbb{H}{P}^n$ are homogeneous if and only if $n+1$ is a prime number (resp. $n$ is even or $n=1$). This shows the existence of inhomogeneous examples of codimension one and higher.