Duality of singular paths for (2,3,5)-distributions

TL;DR

This paper explores a duality in geometric control theory related to (2,3,5)-distributions, revealing a natural identification between the space of singular paths and the original space, with an observed asymmetry.

Contribution

It introduces a duality framework for (2,3,5)-distributions using singular paths and cone structures, highlighting an asymmetry in the duality.

Findings

The space of singular paths forms a cone structure.

The space of singular paths is naturally identified with the original space.

An asymmetry exists in the duality of singular paths.

Abstract

We show a duality which arises from distributions of Cartan type, having growth (2, 3, 5), from the view point of geometric control theory. In fact we consider the space of singular (or abnormal) paths on a given five dimensional space endowed with a Cartan distribution, which form another five dimensional space with a cone structure. We regard the cone structure as a control system and show that the space of singular paths of the cone structure is naturally identified with the original space. Moreover we observe an asymmetry on this duality in terms of singular paths.