Polar foliations on quaternionic projective spaces

TL;DR

This paper classifies irreducible polar foliations on quaternionic projective spaces, revealing conditions under which they are homogeneous and providing examples of inhomogeneous cases.

Contribution

It provides a complete classification of irreducible polar foliations on quaternionic projective spaces and characterizes their homogeneity based on the dimension.

Findings

All irreducible polar foliations are homogeneous if and only if n+1 is prime.

Irreducible polar foliations of codimension one are homogeneous if n is even or n=1.

Existence of inhomogeneous polar foliations in certain dimensions.

Abstract

We classify irreducible polar foliations of codimension $q$ on quaternionic projective spaces $\mathbb H P^n$, for all $(n,q)\neq(7,1)$. 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.