Polar foliations and isoparametric maps

TL;DR

This paper proves that leaves of polar foliations on simply connected manifolds are level sets of smooth maps, extending previous results and linking polar actions to isoparametric maps.

Contribution

It establishes that polar foliations on simply connected manifolds are given by level sets of smooth maps, generalizing earlier work on polar actions and isoparametric maps.

Findings

Leaves of polar foliations are level sets of smooth maps on simply connected manifolds.

Polar actions on simply connected spaces have orbits as level sets of isoparametric maps.

Extends previous results connecting polar foliations and isoparametric maps.

Abstract

A singular Riemannian foliation $F$ on a complete Riemannian manifold $M$ is called a polar foliation if, for each regular point $p$, there is an immersed submanifold $\Sigma$, called section, that passes through $p$ and that meets all the leaves and always perpendicularly. A typical example of a polar foliation is the partition of $M$ into the orbits of a polar action, i.e., an isometric action with sections. In this work we prove that the leaves of $F$ coincide with the level sets of a smooth map $H: M\to \Sigma$ if $M$ is simply connected. In particular, we have that the orbits of a polar action on a simply connected space are level sets of an isoparametric map. This result extends previous results due to the author and Gorodski, Heintze, Liu and Olmos, Carter and West, and Terng.