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

TL;DR

This paper explores the symplectification of rank 2 distributions with 5-dimensional cubes, establishing the existence of normal Cartan connections and analyzing the minimal iterated Cartan prolongations needed for unification of Tanaka symbols.

Contribution

It demonstrates the existence of normal Cartan connections for symplectified distributions in higher dimensions and clarifies the minimal iterated Cartan prolongations for symbol unification.

Findings

Normal Cartan connections exist for distributions with dimension n ≥ 5.

Unification of Tanaka symbols occurs at the (n-5)th iterated Cartan prolongation.

For n > 5, the (n-4)th prolongation admits a normal Cartan connection.

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 \geq 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 $n \geq 5$, the answer to the first question is negative: unification occurs already at the $(n-5)$th iterated Cartan prolongation.