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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.

For the geometry of oriented $(2,3,5)$ distributions $(M,)$, which correspond to regular, normal parabolic geometries of type $(G_2,P)$ for a particular parabolic subgroup $P<G_2$, we develop the corresponding tractor calculus and use it to analyze the first BGG operator ${\Theta }_0$ associated to the $7$-dimensional irreducible representation of $G_2$. We give an explicit formula for the normal connection on the corresponding tractor bundle and use it to derive explicit expressions for this operator. We also show that solutions of this operator...

Summary: Let $𝔤$ be a real semisimple $\left|k\right|$-graded Lie algebra such that the Lie algebra cohomology group $H^1({𝔤}_{-},𝔤)$ is contained in negative homogeneous degrees. We show that if we choose $G=Aut\left(𝔤\right)$ and denote by $P$ the parabolic subgroup determined by the grading, there is an equivalence between regular, normal parabolic geometries of type $(G,P)$ and filtrations of the tangent bundle, such that each symbol algebra $\text{gr}\left(T_{x}M\right)$ is isomorphic to the graded Lie algebra ${𝔤}_{-}$. Examples of parabolic geometries determined by filtrations of the...