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.