CR-Tractors and the Fefferman Space

We develop the natural tractor calculi associated to conformal and CR structures as a fundamental tool for the study of Fefferman's construction of a canonical conformal class on the total space of a circle bundle over a non--degenerate CR manifold of hypersurface type. In particular we construct and treat the basic objects that relate the natural bundles and natural operators on the two spaces. This is illustrated with several applications: We prove that a number of conformally invariant overdetermined systems admit non--trivial solutions on any Fefferman space. We show that the space of conformal Killing fields on a Fefferman space naturally decomposes into a direct sum of subspaces, which admit an interpretaion as solutions of certain CR invariant PDE's. Finally we explicitly analyze the relation between tractor calculus on a CR manifold and the complexified conformal tractor calculus on its Fefferman space, thus obtaining a powerful computational tool for working with the Fefferman construction.