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.

Cited by in corpus (23)

• Conformal Structures Associated to Generic Rank 2 Distributions on 5-Manifolds - Characterization and Killing-Field Decomposition
• The twistor spinors of generic 2- and 3-distributions
• On Nurowski's conformal structure associated to a generic rank two distribution in dimension five
• Curved Casimir Operators and the BGG Machinery
• Geometry of third-order ODEs
• Geometry of Third-Order Ordinary Differential Equations and Its Applications in General Relativity
• Extension Phenomena for Holomorphic Geometric Structures
• About complex structures in conformal tractor calculus
• Polyakov soldering and second order frames : the role of the Cartan connection
• A Ferrand-Obata theorem for rank one parabolic geometries
• New relations between $G_2$-geometries in dimensions 5 and 7
• Symmetries of Parabolic Geometries
• Contact projective structures and chains
• Contact Path Geometries
• Morphisms of Cartan connections
• Free 3-distributions: holonomy, Fefferman constructions and dual distributions
• A Projective-to-Conformal Fefferman-Type Construction
• Remarks on Grassmannian Symmetric Spaces
• Canonical curves and Kropina metrics in Lagrangian contact geometry
• On quaternionic contact Fefferman spaces
• Free $n$-distributions: holonomy, sub-Riemannian structures, Fefferman constructions and dual distributions
• On the geometry of conformal geodesics equations
• Rigid geometry on projective varieties