Generic one-step bracket-generating distributions of rank four. (English) Zbl 1374.58002

de Zanet, Chiara

Arch. Math., Brno 51, No. 5, 257-264 (2015).

The short paper provides a beautiful illustration of the strength of the concept of Cartan geometries related to generic distributions. The author completely describes all generic four-dimensional distributions \(\mathcal D\subset TM\) which generate the entire tangent space via the Lie brackets \([X,Y]\) with vector fields \(X,Y\in\mathcal D\).
The dimension four is special due to algebraic properties of the space \(\Lambda^2\mathbb R^4\otimes \mathbb R^{n-k}\), \(5\leq n \leq 10\) allowing detailed understanding of the orbits of the action of the general linear groups. The complete classification yields the following dimensions \((k,n)\) of \(\mathcal D\) and \(M\): \((4,5)\), \((4,9)\), \((4,10)\) with one open orbit of the generic distributions, while \((4,6)\), \((4,7)\), and \((4,8)\) allow for two different generic orbits.
The work describes the basic geometric properties and invariants of all these cases. Particular attention is devoted to the case \((4,9)\) which was not treated before. All the other cases appear as instances of parabolic geometries, i.e., they allow local models of the type \(G/P\) with \(G\) semisimple and \(P\) parabolic.
The entire paper is very well written and is a nice contribution providing links to parabolic geometries which should be of interest for experts in the area of geometric control theory, too.

Reviewer: Jan Slovák (Brno)

MSC:

58A30	Vector distributions (subbundles of the tangent bundles)
53C15	General geometric structures on manifolds (almost complex, almost product structures, etc.)

Keywords:

generic distributions of rank four; canonical connection; parabolic geometry