Two constructions with parabolic geometries

TL;DR

This paper explores the theory of parabolic geometries, detailing their equivalence with underlying structures, and introduces constructions like correspondence, twistor, and Fefferman spaces to relate different geometries.

Contribution

It provides a comprehensive description of constructions relating various parabolic geometries, including new insights into their equivalences and applications in twistor theory and second order ODEs.

Findings

Detailed description of the equivalence between parabolic geometries and underlying structures.

Construction of correspondence and twistor spaces related to nested parabolic subgroups.

Discussion of Fefferman-type constructions linking different semisimple Lie group 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.