Canonical Frames for Bracket Generating Rank 2 Distributions which are not Goursat

TL;DR

This paper completes a uniform construction of canonical absolute parallelism for rank 2 distributions with a 5-dimensional cube, showing maximality of class holds generically, and explores implications for control theory and distribution classification.

Contribution

It establishes that the maximality of class condition is automatic at generic points for these distributions, extending classical Goursat distribution theory to non-Goursat cases.

Findings

Maximality of class holds generically for these distributions.

Classifies maximally symmetric germs among rank 2 distributions.

Shows abundance of abnormal extremals in control problems with such distributions.

Abstract

We complete a uniform construction of canonical absolute parallelism for bracket generating rank $2$ distributions with $5$-dimensional cube on $n$-dimensional manifold with $n\geq 5$ by showing that the condition of maximality of class that was assumed previously by Doubrov-Zelenko for such a construction holds automatically at generic points. This also gives analogous constructions in the case when the cube is not $5$-dimensional but the distribution is not Goursat through the procedure of iterative Cartan deprolongation. This together with the classical theory of Goursat distributions covers in principle the local geometry of all bracket generating rank 2 distributions in a neighborhood of generic points. As a byproduct, for any $n\geq 5$ we describe the maximally symmetric germs among bracket generating rank $2$ distributions with $5$-dimensional cube, as well as among those which reduce to such a distribution under a fixed number of Cartan deprolongations. Another consequence of our results on maximality of class is for optimal control problems with constraint given by a rank $2$ distribution with $5$-dimensional cube: it implies that for a generic point $q_0$ of $M$, there are plenty abnormal extremal trajectories of corank $1$ (which is the minimal possible corank) starting at $q_0$. The set of such points contains all points where the distribution is equiregular.