Two Fefferman-Type Constructions Involving Almost Grassmann Structures and Path Geometries.

Abstract

We introduce a Fefferman-type construction that associates an almost Grassmann 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 Grassmann 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 Grassmann structure of type $(2,n+1)$ to every almost Grassmann structure of type (2, n). We prove that this construction is non-normal and characterize all almost Grassmann structures which locally arise in this way in Cartan geometric terms.