Fefferman-type constructions and overdetermined systems

In 1976 Charles Fefferman constructed a pseudo-Riemannian conformal structure on a circle bundle over a non- degenerate CR manifold of hypersurface type, in order to understand the CR geometry by means of the associated conformal geometry. Fefferman`s construction has been intensively studied in recent years, and it has been observed that it admits a natural generalization in the framework of parabolic geometry. The Fefferman-type constructions one obtains in this way establish relationships between quite diverse geometric structures, whose unifying property is an equivalent description as a Cartan geometry associated with a semisimple Lie group and a parabolic subgroup. These so-called parabolic geometries include conformal structures, projective structures, CR structures, some types of generic distributions and geometries of differential equations. Fefferman-type constructions have recently attracted a lot of interest due to their relation with holonomy reductions of Cartan connections and with geometric overdetermined systems of PDEs (e.g. the conformally invariant twistor and conformal Killing equations). The objective of the project is to develop a better understanding of various Fefferman-type constructions and overdetermined systems that arise in the context of such constructions. It seems that there are several constructions between different types of geometric structures that have not been explored yet, and where a treatment via parabolic geometry will be fruitful. Furthermore, I intend to investigate interesting open questions concerning known relationships, specifically about Nurowski`s conformal structures attached to generic rank two distributions in dimension five. For that purpose I aim to visit Warsaw. I expect that discussions with Professor Pawel Nurowski will also lead to new ideas for constructions relating different geometries. The subsequent investigation of the relationships by means of parabolic geometry and methods developed for the study of geometric overdetermined PDEs should be carried out at the Australian National University in Canberra under the guidance of Professor Michael Eastwood. I intend to complete the project in Vienna, where close collaboration with research groups in Vienna, Brno and Prague is planned.

This research project was concerned with the study of a diverse class of geometries through a construction, called Fefferman-type construction that establishes relations between different geometries within the considered class. The project revealed previously unknown links between geometric structures, and as an application, established the existence of geometric structures with special properties.Although being structurally similar, the geometries under consideration can have very di- verse appearances and occur in various contexts in mathematics and physics. One of the geometries studied, the geometry of (2, 3, 5)-distributions, arises from certain mechanical systems. For example, two balls of different radii rolling on each other without slipping or twisting give rise to a (2, 3, 5)-distribution. Other geometries that were studied, such as conformal and projective structures, play an important role in mathematical physics. In general, the geometries are interesting from a purely mathematical point of view. For example, the symmetries of the most symmetric (2, 3, 5)-distribution form interesting algebraic structure, namely the exceptional Lie group G2.Fefferman-type constructions were chosen as a tool to gain new insights into these geometries since (a) they typically give rise to interesting examples of geometric structures with special properties, (b) they allow to derive properties of one geometry from known properties of another geometry. The properties of interest in this context can be formulated by means of overdetermined systems of differential equations. They include, for example, the existence of Einstein metrics in a conformal class, or the existence of symmetries of a given geometric structure.Particular results obtained during the course of the project include a relationship between generic distributions in dimension 5 and 6 and twistor spinors, a Fefferman-type construction of G2-reduced Lie contact structures, a relationship between Nurowski conformal structures with Einstein scales, (para-) SasakiEinstein metrics and Fefferman conformal structures on the singularity sets, a local characterisation of split-signature conformal structures coming from projective structures.Since several of the geometries that were studied are relevant in physics, the new geometrical constructions may find applications in mathematical physics.