  The chains studied in this paper generalize Chern-Moser chains for CR
structures. They form a distinguished family of one dimensional submanifolds in
manifolds endowed with a parabolic contact structure. Both the parabolic
contact structure and the system of chains can be equivalently encoded as
Cartan geometries (of different types). The aim of this paper is to study the
relation between these two Cartan geometries for Lagrangean contact structures
and partially integrable almost CR structures.
  We develop a general method for extending Cartan geometries which generalizes
the Cartan geometry interpretation of Fefferman's construction of a conformal
structure associated to a CR structure. For the two structures in question, we
show that the Cartan geometry associated to the family of chains can be
obtained in that way if and only if the original parabolic contact structure is
torsion free. In particular, the procedure works exactly on the subclass of
(integrable) CR structures.
  This tight relation between the two Cartan geometries leads to an explicit
description of the Cartan curvature associated to the family of chains. On the
one hand, this shows that the homogeneous models for the two parabolic contact
structures give rise to examples of non-flat path geometries with large
automorphism groups. On the other hand, we show that one may (almost)
reconstruct the underlying torsion free parabolic contact structure from the
Cartan curvature associated to the chains. In particular, this leads to a very
conceptual proof of the fact that chain preserving contact diffeomorphisms are
either isomorphisms or anti-isomorphisms of parabolic contact structures.
