On the smoothness of $C^1$-contact maps in $C^{\infty}$-rigid Carnot groups

TL;DR

This paper proves that in certain highly structured mathematical groups called $C^ ext{infty}$-rigid Carnot groups, any contact map that is once differentiable is necessarily infinitely differentiable, revealing a strong regularity property.

Contribution

It establishes the automatic smoothness of $C^1$-contact maps in $C^ ext{infty}$-rigid Carnot groups, a significant advancement in understanding their geometric structure.

Findings

$C^1$-contact maps are automatically smooth in these groups

The result applies specifically to $C^ ext{infty}$-rigid Carnot groups

This enhances the understanding of regularity in geometric group theory

Abstract

We show that in any $C^\infty$-rigid Carnot group in the sense of Ottazzi - Warhurst, $C^1$-contact maps are automatically smooth.