On Polar Foliations and the Fundamental Group

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 alternative proofs of two results. The first one, due to the author and Toben, implies that a polar action does not admit exceptional orbits, if M is simply connected. The second result, due to Lytchak, implies that the orbits are closed and embedded if M is simply connected. All results are proved in the more general case of polar foliations.