Two Fefferman-type constructions involving almost Grassmann structures and path geometries

Abstract

We introduce a Fefferman-type construction that associates an almost Grassmannian structure of type $(2,n+1)$ to every $(n+1)$-dimensional path geometry. We prove that the construction is normal and provide two equivalent characterizing conditions for all almost Grassmannian structures which locally arise from this construction: one in terms of certain parallel tractors and the other in terms of a Weyl connection of an almost Grassmann structure. We prove that the latter condition is independent of the choice of Weyl connection. We then introduce a related Fefferman-type construction associating an almost Grassmannian structure of type $(2,n+1)$ to every almost Grassmannian structure of type $(2,n)$. We prove that this construction is non-normal and characterize all almost Grassmannian structures which locally arise in this way in Cartan geometric terms.