Split octonions and generic rank two distributions in dimension five. (English) Zbl 1164.53362

Sagerschnig, Katja

Arch. Math., Brno 42, No. 5, 329-339 (2006).

Summary: In his famous five variables paper Elie Cartan showed that one can canonically associate to a generic rank 2 distribution on a 5 dimensional manifold a Cartan geometry modeled on the homogeneous space \(\tilde {G}_2/P\), where \(P\) is one of the maximal parabolic subgroups of the exceptional Lie group \(\tilde {G}_2\). In this article, we use the algebra of split octonions to give an explicit global description of the distribution corresponding to the homogeneous model.

MSC:

53C30	Differential geometry of homogeneous manifolds
58A30	Vector distributions (subbundles of the tangent bundles)

Keywords:

octonions; Cartan geometry; homogeneous model