Parabolic geometries, CR-tractors, and the Fefferman construction.
This is a survey article on the recent joint work of the author with A. R. Gover on the standard tractor bundle and its canonical linear connections in the geometry of nondegenerate CR structures of hyperbolic type. The author mainly treats three classical constructions in CR-geometry: (I) The standard tractor bundles and its linear connections, which are equivalent to the Cartan connection on a CR-manifold. (II) The construction of Fefferman, which associates to any CR-manifold \(M\) the standard tractor bundle and indefinite conformal structure on the total space of a certain circle bundle \(\widetilde M\) over \(M\). (III) The ambient metric construction for embedded CR-manifold. The author tries to clarify the importance of the standard tractor bundles and connections in these three constructions. The paper consists of two sections, the first of which begins with the relevant definitions and treats CR-manifolds, the flat model of CR-structures, canonical Cartan connections of CR-manifolds, parabolic geometries, tractor bundles, tractor calculus and Bernstein-Gelfand-Gelfand sequences. The second section with title `The ambient metric and Feffeyman space' treats the ambient metric constructions, the Fefferman spaces and standard tractors as well as the relations between the ambient metrics and standard tractors.
scientific article; zbMATH DE number 1959896
scientific article