On polar foliations and fundamental group

TL;DR

This paper explores the relationship between the fundamental group of a complete Riemannian manifold and the structure of polar actions, providing new proofs and generalizations within the framework of polar foliations.

Contribution

It offers new insights into the connection between fundamental groups and Weyl/reflection groups in polar actions, extending results to polar foliations.

Findings

Non-exceptional orbits in simply connected manifolds.

Orbits are closed and embedded in simply connected manifolds.

Provides alternative proofs of existing theorems.

Abstract

In this work we investigate the relation between the fundamental group of a complete Riemannian manifold $M$ and the quotient between the Weyl group and reflection group of a polar action on $M$, as well as the relation between the fundamental group of $M$ and the quotient between the lifted Weyl group and lifted reflection group. As applications we give short alternative proofs of two results. The first one, due to the author and T\"{o}ben, states that there is non-exceptional orbit, if $M$ is simply connected. The second result, due to Lytchak, states that the orbits are closed and embedded if $M$ is simply connected. All results are proved in the more general case of polar foliations.