Smoothness of subRiemannian isometries

TL;DR

This paper proves that the isometry group of an equiregular subRiemannian manifold forms a finite-dimensional Lie group of smooth transformations, using a novel PDE approach and harmonic coordinate techniques.

Contribution

It introduces a new PDE-based method to show that all isometries are smooth on an open dense subset of subRiemannian manifolds, establishing their Lie group structure.

Findings

Isometry group is a finite-dimensional Lie group.

Existence of harmonic coordinates where all isometries are smooth.

Smoothness holds on an open dense subset of the manifold.

Abstract

We show that the group of isometries (i.e., distance-preserving homeomorphisms) of an equiregular subRiemannian manifold is a finite-dimensional Lie group of smooth transformations. The proof is based on a new PDE argument, in the spirit of harmonic coordinates, establishing that in an arbitrary subRiemannian manifold there exists an open dense subset where all isometries are smooth.