Contact projective structures and chains

Abstract

Contact projective structures have been thoroughly studied by D. Fox. He associated to a contact projective structure a canonical projective structure on the same manifold. We interpret Fox’s construction in terms of the equivalent parabolic (Cartan) geometries, showing that it is an analog of Fefferman’s construction of a conformal structure associated to a CR structure. We show that, on the level of Cartan connections, this Fefferman-type construction is compatible with normality if and only if the initial structure has vanishing contact torsion. This leads to a geometric description of the paths that have to be added to the contact geodesics of a contact projective structure in order to obtain the subordinate projective structure. They are exactly the chains associated to the contact projective structure, which are analogs of the Chern–Moser chains in CR geometry. Finally, we analyze the consequences for the geometry of chains and prove that a chain–preserving contactomorphism must be a morphism of contact projective structures.