The contact mappings of a flat $(2,3,5)$-distribution

TL;DR

This paper proves that contact mappings between open subsets of flat (2,3,5)-distributions are inherently smooth, leveraging the rigidity of associated stratified Lie groups and differential identities of the Pansu-derivative.

Contribution

It establishes the smoothness of contact mappings in flat (2,3,5)-distributions, showing they are automatically infinitely differentiable due to Lie group rigidity.

Findings

Contact mappings are $C^ abla$-smooth if $C^1$-smooth.

Rigidity of stratified Lie groups underpins smoothness.

Differential identities of Pansu-derivative are key to proof.

Abstract

Let $\Omega$ and $\Omega'$ be open subsets of a flat $(2,3,5)$-distribution. We show that a $C^1$-smooth contact mapping $f : \Omega \to \Omega'$ is a $C^\infty$-smooth contact mapping. Ultimately, this is a consequence of the rigidity of the associated stratified Lie group (the Tanaka prolongation of the Lie algebra is of finite-type). The conclusion is reached through a careful study of some differential identities satisfied by components of the Pansu-derivative of a $C^1$-smooth contact mapping.