Contact systems and corank one involutive subdistributions

TL;DR

This paper characterizes when a distribution is equivalent to the canonical contact system on jet spaces, studying the geometry of such systems, especially corank one involutive subdistributions, and introduces a new normal form for singular points.

Contribution

It provides necessary and sufficient geometric conditions for equivalence to the canonical contact system and introduces a generalized normal form for singular points.

Findings

Characterization of equivalence to canonical contact systems

Existence criteria for corank one involutive subdistributions

A new normal form for singular points

Abstract

We give necessary and sufficient geometric conditions for a distribution (or a Pfaffian system) to be locally equivalent to the canonical contact system on Jn(R,Rm), the space of n-jets of maps from R into Rm. We study the geometry of that class of systems, in particular, the existence of corank one involutive subdistributions. We also distinguish regular points, at which the system is equivalent to the canonical contact system, and singular points, at which we propose a new normal form that generalizes the canonical contact system on Jn(R,Rm) in a way analogous to that how Kumpera-Ruiz normal form generalizes the canonical contact system on Jn(R,R), which is also called Goursat normal form.