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

Abstract

Abstract We introduce a Fefferman-type construction that associates an almost Grassmann structure of type $$(2,n+1)$$ ( 2 , n + 1 ) to every $$(n+1)$$ ( 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 $$(2,n+1)$$ ( 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.