Symplectification of Rank 2 Distributions, Normal Cartan Connections, and Cartan Prolongations

Abstract

We study the Doubrov--Zelenko symplectification procedure for rank $2$ distributions with $5$-dimensional cube -- originally motivated by optimal control theory -- through the lens of Tanaka--Morimoto theory for normal Cartan connections. In this way, for ambient manifolds of dimension $n\ge 5$, we prove the existence of the normal Cartan connection associated with the symplectified distribution. Furthermore, we show that this symplectification can be interpreted as the $(n-4)$th iterated Cartan prolongation at a generic point. This interpretation naturally leads to two questions for an arbitrary rank $2$ distribution with $5$-dimensional cube: (1) Is the $(n-4)$th iterated Cartan prolongation the minimal iteration where the Tanaka symbols become unified at generic points? (2) Is the $(n-4)$th iterated Cartan prolongation the minimal iteration admitting a normal Cartan connection via Tanaka--Morimoto theory? Our main results demonstrate that: (a) For $n>5$, the answer to the second question is positive (in contrast to the classical $n=5$ case from $G_2$-parabolic geometries); (b) For $n\ge 5$, the answer to the first question is negative: unification occurs already at the $(n-5)$th iterated Cartan prolongation.

22 pages. In view of a recent result by the first and third authors (arXiv:2508.09307, accepted for publication in Proceedings of the AMS), the assumption of maximality of class is no longer required. The present version reflects this refinement throughout