The concept of chains extends to all parabolic contact structures. The family of chains defines a path geometry on an open subset of the projectivized tangent bundle. Since path geometries are another instance of parabolic geometries, there is the natural question of constructing the Cartan connection determined by the path geometry directly from the Cartan connection determined by the parabolic contact structure.

I will discuss this question for Lagrangean contact structures. It turns out that, for torsion free geometries, the Cartan connection determined by the path geometry can be obtained by a generalization of the Fefferman construction. Studying the relations between the harmonic curvatures of the two Cartan connections leads to a nice conceptual proof that a diffeomorphism which maps chains to chains must essentially be an morphism of the Lagrangean contact structures.

This is joint work in progress with V. Zadnik (Brno)