The essence of the research project presented in this proposal is a further study of distinguished curves in Cartan geometries - parabolic geometries, in particular - their properties, and relations among Cartan geometries of different types which appear in this context. Supposed new results should follow the very recent construction which directly assigns the Cartan´s path geometry of Chern-Moser chains to a given CR manifold. This is a generalization of the classical Fefferman´s construction providing remarkable applications.

Results

Concerning the initial aims of the project, we were able to finish the study of geometry of chains for all parabolic contact geometries corresponding to contact gradings of classical simple Lie algebras, up to the for the present open case of contact-quaternionic geometries related to the algebra so*(2n). In all mentioned cases, the compatibility of the used construction with the normalization condition is satisfied only if the canonical Cartan connection corresponding to the parabolic contact structure is torsion free. Since the torsion freeness for Lie contact and contact-quaternionic geometries implies the flatness, the obtained results are less general then for CR, Lagrangean contact, and projective contact structures studied before. However, even the study of the homogeneous model of a parabolic contac geometry can be interesting, as we document by results on the field of special symplectic connections and Grassmannian symmetric spaces.

Publications