Myers–Steenrod Theorems for Metric and Singular Riemannian Foliations

Diego Corro^ab, Fernando Galaz-García<sup>c

a</sup> School of Mathematics, Cardiff University, UK
^b Fakultät für Mathematik, Karlsruher Institut für Technologie, Germany
^c Department of Mathematical Sciences, Durham University, UK

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.

Keywords: Alexandrov space, submetry, isometry group, singular Riemannian foliation, Lie group.

MSC: 53C12, 53C20, 53C21, 53C23, 53C24, 51K10

Received: November 4, 2024; in final form December 1, 2025; Published online December 16, 2025

Language: English

DOI: 10.3842/SIGMA.2025.106

ArXiv: 2407.03534