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.