Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 106, 23 pages      arXiv:2407.03534      https://doi.org/10.3842/SIGMA.2025.106

Myers-Steenrod Theorems for Metric and Singular Riemannian Foliations

Diego Corro ab and Fernando Galaz-García c
a) Fakultät für Mathematik, Karlsruher Institut für Technologie, Germany
b) School of Mathematics, Cardiff University, UK
c) Department of Mathematical Sciences, Durham University, UK

Received November 04, 2024, in final form December 01, 2025; Published online December 16, 2025

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.

Key words: Alexandrov space; submetry; isometry group; singular Riemannian foliation; Lie group.

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