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

Nicklas Day, Boris Doubrov, Igor Zelenko

TL;DR

The paper addresses local equivalence for bracket-generating rank $2$ distributions with a $5$-dimensional cube by integrating the Doubrov–Zelenko symplectification with Tanaka–Morimoto theory. It proves that the symplectified distribution $\mathrm{Symp}(D)$ carries a normal Cartan connection and is locally equivalent to the $(n-4)$th Cartan prolongation $\mathrm{pr}^{(n-4)}(D)$ at generic points for $n>5$, while the Tanaka symbols unify already at the $(n-5)$th prolongation. The $n=5$ case corresponds to a $G_2$-parabolic geometry, where normal Cartan connections exist for both $D$ and $\mathrm{Symp}(D)$. Moreover, lower prolongations fail to admit a linear invariant normalization condition, establishing that $(n-4)$ is the minimal prolongation supporting Tanaka–Morimoto normal Cartan geometry. Altogether, the results provide a uniform Cartan-theoretic framework for these distributions and clarify the role of Cartan prolongations in achieving canonical connections.

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.

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

TL;DR

The paper addresses local equivalence for bracket-generating rank distributions with a -dimensional cube by integrating the Doubrov–Zelenko symplectification with Tanaka–Morimoto theory. It proves that the symplectified distribution carries a normal Cartan connection and is locally equivalent to the th Cartan prolongation at generic points for , while the Tanaka symbols unify already at the th prolongation. The case corresponds to a -parabolic geometry, where normal Cartan connections exist for both and . Moreover, lower prolongations fail to admit a linear invariant normalization condition, establishing that is the minimal prolongation supporting Tanaka–Morimoto normal Cartan geometry. Altogether, the results provide a uniform Cartan-theoretic framework for these distributions and clarify the role of Cartan prolongations in achieving canonical connections.

Abstract

We study the Doubrov--Zelenko symplectification procedure for rank distributions with -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 , 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 th iterated Cartan prolongation at a generic point. This interpretation naturally leads to two questions for an arbitrary rank distribution with -dimensional cube: (1) Is the th iterated Cartan prolongation the minimal iteration where the Tanaka symbols become unified at generic points? (2) Is the th iterated Cartan prolongation the minimal iteration admitting a normal Cartan connection via Tanaka--Morimoto theory? Our main results demonstrate that: (a) For , the answer to the second question is positive (in contrast to the classical case from -parabolic geometries); (b) For , the answer to the first question is negative: unification occurs already at the th iterated Cartan prolongation.

Paper Structure

This paper contains 20 sections, 18 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tanaka Symbols and Tanaka Prolongations
  4. Regular Cartan Connections
  5. Invariant Normalization Conditions
  6. Cartan Prolongation
  7. The Symplectification Procedure
  8. The Characteristic Line Distribution
  9. Maximality of Class
  10. The Construction of $\mathop{\mathrm{Symp}}\nolimits(D)$
  11. Involutivity Conditions on $\mathop{\mathrm{Symp}}\nolimits(D)$
  12. The Tanaka Symbol of $\mathop{\mathrm{Symp}}\nolimits(D)$
  13. Existence of Invariant Normalization Condition on $\mathop{\mathrm{Symp}}\nolimits(D)$
  14. The Symplectification Procedure via Cartan Prolongations
  15. Unparameterized Jet Spaces
  16. ...and 5 more sections

Key Result

Proposition 1.1

Let $D$ be a $(2,n)$ distribution with $5$-dimensional cube. Then at a generic point, the $(n-5)$th iterated Cartan prolongation $\mathop{\mathrm{pr}}\nolimits^{(n-5)}(D)$ has Tanaka symbol which is independent of $D$, but this symbol does not admit a linear invariant normalization condition. On the

Theorems & Definitions (36)

  • Proposition 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Remark 2.6
  • Theorem 3.1
  • Proposition 3.2
  • Proposition 3.3
  • ...and 26 more