Tractor bundles for irreducible parabolic geometries (Q2765846)

Parabolic geometries may be viewed as curved analogs of homogeneous spaces of the form \(G/P\), where \(G\) is a real or complex simple Lie group and \(P\subset G\) is a parabolic subgroup. In general, a parabolic geometry of type \((G,P)\) on a smooth manifold \(M\) is defined as a principal \(P\)-bundle over \(M\), which is endowed with a Cartan connection. Tractor bundles and connections are an alternative approach to these structures, which do not require knowledge of the Cartan connection but may be constructed directly. Irreducible parabolic geometries correspond to certain maximal parabolics. Important examples are conformal and classical projective structures.NEWLINENEWLINENEWLINEIn the present paper a simple and effective characterization is given for arbitrary normal tractor bundles on manifolds equipped with an irreducible parabolic geometry. The explicit formulae for the normal tractor connections are deduced as well as the fundamental \(D\)-operators on such bundles. For such structures, part of this information is equivalent to giving the canonical Cartan connection. As an application, a new simple construction of the standard tractor bundle in conformal geometry is given.NEWLINENEWLINEFor the entire collection see [Zbl 0973.00041].