On automorphism groups of some types of generic distributions

TL;DR

This paper demonstrates how canonical Cartan connections impose restrictions on the sizes of automorphism groups of certain generic distributions, with applications to specific low-dimensional cases.

Contribution

It shows that properties of canonical Cartan connections can restrict automorphism group sizes without detailed construction knowledge.

Findings

Restrictions on automorphism group sizes derived from Cartan connections

Application to specific low-dimensional generic distributions

No detailed construction of Cartan connections needed for these restrictions

Abstract

To certain types of generic distributions (subbundles in a tangent bundle) one can associate canonical Cartan connections. Many of these constructions fall into the class of parabolic geometries. The aim of this article is to show how strong restrictions on the possibles sizes of automorphism groups of such distributions can be deduced from the existence of canonical Cartan connections. This needs no information on how the Cartan connections are actually constructed and only very basic information on their properties. In particular, we discuss the examples of generic distributions of rank two in dimension five, rank three in dimension six, and rank four in dimension seven.