Canonical Frames for Bracket Generating Rank 2 Distributions which are not Goursat
Nicklas Day, Igor Zelenko
TL;DR
This work constructs a canonical absolute parallelism for bracket-generating rank $2$ distributions on $n$-manifolds with generic $ ext{rank}\,D^3=5$, yielding a frame on a $(2n-1)$-dimensional bundle and an upper bound of $(2n-1)$ on the symmetry algebra. It shows that non-Goursat cases reduce to a Monge-model via Cartan deprolongations, while Goursat cases align with Cartan's jet-space framework; together these results describe the local geometry near generic points for all bracket-generating rank $2$ distributions. The maximally symmetric germs are identified with the Monge equation model $z'(x)=(y^{(n-3)}(x))^2$, and the theory extends to cases with different cube dimensions through iterative deprolongation. Additionally, the paper proves that at generic points with maximal class, there exist abundant corank-$1$ regular abnormal extremals, with implications for optimal control constrained by rank-$2$ distributions and the structure of the equiregular locus.
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.