[Paper Review] Isometries between leaf spaces

Marcos M. Alexandrino, Marco Radeschi|arXiv (Cornell University)|Nov 26, 2011

Geometric and Algebraic Topology9 references4 citations

TL;DR

This paper establishes that isometries between leaf spaces of singular Riemannian foliations are smooth under two conditions: when they preserve the codimension of leaves or when the leaf space has no boundary. The key result generalizes the Myers–Steenrod theorem to orbit spaces and singular Riemannian foliations by showing that metric isometries lift to smooth maps via control of mean curvature and Laplacian structure on regular strata.

ABSTRACT

In this paper we prove that an isometry between orbit spaces of two proper isometric actions is smooth if it preserves the codimension of the orbits or if the orbit spaces have no boundary. In other words, we generalize Myers-Steenrod's theorem for orbit spaces. These results are proved in the more general context of singular Riemannian foliations.

Motivation & Objective

• To determine under what conditions an isometry between leaf spaces of singular Riemannian foliations is smooth.
• To extend the classical Myers–Steenrod theorem, which relates isometries and smooth structures on Riemannian manifolds, to quotient spaces that are not manifolds.
• To address the open problem of whether metric structure on orbit spaces uniquely determines smooth structure, particularly in the context of proper isometric group actions and singular foliations.
• To provide sufficient conditions under which isometries between such quotient spaces are smooth, even when the spaces are not orbifolds or manifolds.

Proposed method

• Use the framework of singular Riemannian foliations (SRFs) with closed leaves to generalize the setting beyond group actions.
• Define smoothness of maps between leaf spaces via pullback of smooth basic functions, aligning with Schwarz’s smooth structure on quotients.
• Prove that an isometry preserving codimension or acting on a boundaryless leaf space lifts to a smooth map by analyzing the Laplacian and mean curvature vector fields on the regular stratum.
• Leverage the fact that the pullback of a smooth basic function under the isometry satisfies a weak Laplace equation, which is preserved across singular sets of codimension ≥2.
• Apply regularity theory for elliptic PDEs to conclude that the pullback function is smooth, even across lower-codimension singularities.
• Use Green’s identity and approximation via neighborhoods of singular sets to extend pointwise Laplacian identities to weak solutions on the full manifold.

Experimental results

Research questions

• RQ1 Under what conditions is an isometry between leaf spaces of singular Riemannian foliations smooth?
• RQ2 Can the Myers–Steenrod theorem be extended to orbit spaces that are not manifolds or orbifolds?
• RQ3 Does preserving the codimension of leaves guarantee smoothness of an isometry between leaf spaces?
• RQ4 What role does the absence of boundary in the leaf space play in ensuring smoothness of isometries?
• RQ5 How do mean curvature vector fields on regular strata influence the smoothness of isometries between leaf spaces?

Key findings

• An isometry between leaf spaces of two singular Riemannian foliations with closed leaves is smooth if it preserves the codimension of the leaves.
• If the leaf space has no boundary, then any isometry between such leaf spaces is smooth, even without codimension preservation.
• The proof relies on showing that the pullback of a smooth basic function under the isometry satisfies a weak Laplace equation, which implies smoothness via elliptic regularity.
• The mean curvature vector fields on the regular strata must be preserved by the isometry for the smoothness result to hold, ensuring compatibility in the Laplacian structure.
• The complement of the regular stratum has codimension at least 2, allowing the extension of pointwise identities to weak solutions via integration and approximation.
• The result implies that isometric orbifolds are diffeomorphic in the sense of Schwarz, recovering classical notions of diffeomorphism for orbifolds.