Geometric resolution of singular Riemannian foliations

TL;DR

This paper characterizes when an isometric Lie group action on a Riemannian manifold can be geometrically resolved while maintaining its transverse structure, focusing on infinitesimally polar actions and extending results to singular Riemannian foliations.

Contribution

It establishes a criterion for the existence of geometric resolutions of singular Riemannian foliations based on infinitesimal polarity, generalizing previous results to broader classes.

Findings

Resolution exists if and only if the action is infinitesimally polar.

Applications include topological simplicity of polar and variationally complete actions.

Results are applicable to the broader setting of singular Riemannian foliations.

Abstract

We prove that an isometric action of a Lie group on a Riemannian manifold admits a resolution preserving the transverse geometry if and only if the action is infinitesimally polar. We provide applications concerning topological simplicity of several classes of isometric actions, including polar and variationally complete ones. All results are proven in the more general case of singular Riemannian foliations.