Myers-Steenrod Theorems for Metric and Singular Riemannian Foliations

TL;DR

This paper extends Myers-Steenrod theorems to metric and singular Riemannian foliations on Alexandrov spaces, establishing isometry group properties, bounds, and geometric structures of foliations.

Contribution

It proves that isometry groups preserving such foliations form closed subgroups, provides sharp dimension bounds, and characterizes foliations achieving these bounds as fiber bundles with spherical or projective space fibers.

Findings

Isometry groups form closed subgroups of the isometry group.

Sharp upper bounds for the dimension of these subgroups.

Foliations achieving bounds are induced by fiber bundles with spherical or projective fibers.

Abstract

We prove that the group of isometries preserving a metric foliation on a closed Alexandrov space $X$ is a closed subgroup of the isometry group of $X$. We obtain a sharp upper bound for the dimension of this subgroup and show that, when equality holds, the foliations that realize this upper bound are induced by fiber bundles whose fibers are round spheres or projective spaces. As a corollary, singular Riemannian foliations that realize the upper bound are induced by smooth fiber bundles whose fibers are round spheres or projective spaces.