Contact projective structures and chains

TL;DR

This paper explores the relationship between contact projective structures and chains, showing how a Fefferman-type construction relates to Cartan geometries and characterizing chain-preserving contactomorphisms.

Contribution

It interprets Fox's construction within parabolic geometries, linking it to Fefferman's construction and characterizing chains as added paths to contact geodesics.

Findings

Fefferman-type construction is compatible with normality if contact torsion vanishes.

Chains are the added paths to contact geodesics to form subordinate projective structures.

Chain-preserving contactomorphisms are morphisms of contact projective structures.

Abstract

Contact projective structures have been profoundly studied by D.J.F. Fox. He associated to a contact projective structure a canonical projective structure on the same manifold. We interpret Fox' 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.