Parabolic (3,5,6)-distributions and Gl(2)-structures

TL;DR

This paper classifies parabolic (3,5,6)-distributions and establishes their connection with Gl(2)-structures, showing that any Gl(2)-structure on 3- or 4-dimensional manifolds can be viewed as such a distribution.

Contribution

It provides a classification of parabolic (3,5,6)-distributions and demonstrates their equivalence to Gl(2)-structures on low-dimensional manifolds.

Findings

Classification of parabolic (3,5,6)-distributions

Connection between distributions and Gl(2)-structures

Any Gl(2)-structure on 3- or 4-manifolds corresponds to a parabolic (3,5,6)-distribution

Abstract

We consider rank 3 distributions with growth vector (3,5,6). The class of such distributions splits into three subclasses: parabolic, hyperbolic and elliptic. In the present paper, we deal with the parabolic case. We provide a classification of such distributions and exhibit connections between them and Gl(2)-structures. We prove that any Gl(2)-structure on three and four dimensional manifold can be interpreted as a parabolic (3,5,6)-distribution.