Jet spaces on Carnot groups

TL;DR

This paper develops jet spaces over stratified Lie groups, demonstrating they are themselves stratified Lie groups and establishing a prolongation theory for contact maps, including an embedding theorem for stratified Lie groups.

Contribution

It introduces jet spaces adapted to stratified Lie groups, proves they are stratified Lie groups themselves, and establishes a prolongation theory for contact maps with an embedding theorem.

Findings

Jet spaces over stratified Lie groups are themselves stratified Lie groups.

A prolongation theory for contact maps on these jet spaces is developed.

An embedding theorem shows stratified Lie groups of higher step embed into jet spaces of lower step.

Abstract

Jet spaces on $\mathbb R^n$ have been shown to have a canonical structure of stratified Lie groups (also known as Carnot groups). We construct jet spaces over stratified Lie groups adapted to horizontal differentiation and show that these jet spaces are themselves stratified Lie groups. Furthermore, we show that these jet spaces support a prolongation theory for contact maps, and in particular, a B\"acklund type theorem holds. A byproduct of these results is an embedding theorem that shows that every stratified Lie group of step $s+1$ can be embedded in a jet space over a stratified Lie group of step $s$.