Two constructions with parabolic geometries

Abstract

This is an expanded version of a series of lectures delivered at the 25th Winter School ``Geometry and Physics'' in Srni. After a short introduction to Cartan geometries and parabolic geometries, we give a detailed description of the equivalence between parabolic geometries and underlying geometric structures. The second part of the paper is devoted to constructions which relate parabolic geometries of different type. First we discuss the construction of correspondence spaces and twistor spaces, which is related to nested parabolic subgroups in the same semisimple Lie group. An example related to twistor theory for Grassmannian structures and the geometry of second order ODE's is discussed in detail. In the last part, we discuss analogs of the Fefferman construction, which relate geometries corresponding different semisimple Lie groups.