Prolongations of $(3, 6)$-distributions by singular curves

TL;DR

This paper studies how (3, 6)-distributions on 6-manifolds can be extended using singular curves into more complex distributions, revealing new equivalences and structures in geometric control theory.

Contribution

It introduces a method to prolong (3, 6)-distributions via singular curves into higher distributions, generalizing known classification correspondences.

Findings

Prolongation to (3,5,7,8) and (3,5,7,8,9)-distributions with pseudo-product structures.

Equivalence of classification problems among four classes of distributions derived from (3,6)-distributions.

Generalization of B_3-SO(3,4)-homogeneous model correspondences.

Abstract

A subbundle of rank 3 in the tangent bundle over a 6-dimensional manifold is called a (3, 6)-distribution if its local sections generate the whole tangent bundle by taking their Lie brackets once. An integral curve of a distribution, whose velocity vectors belong to the distribution, can be a singular curve or an abnormal extremal in the sense of geometric control theory. In this paper, given a (3, 6)-distribution, we prolong it, using the data of singular curves, to a (3,5,7,8)-distribution, to a (3, 5, 7, 8, 9)-distribution which possesses additional pseudo-product structure respectively. Regarding also another prolongation to a (4, 6, 8)-distribution, we show the equivalence of the classification problems of those four classes of distributions obtained from (3, 6)-distributions, generalising the correspondences of those in B_3-SO(3,4)-homogeneous models.