Two constructions with parabolic geometries.

*(English)*Zbl 1120.53013
Čadek, Martin (ed.), The proceedings of the 25th winter school “Geometry and physics”, Srní, Czech Republic, January 15–22, 2006. Palermo: Circolo Matemático di Palermo. Supplemento ai Rendiconti del Circolo Matemático di Palermo. Serie II 79, 11-37 (2006).

This paper is an expanded version of the author’s lectures at the Winter School “Geometry and Physics” 2005 in Srni.

The extremely general concept of Cartan geometry, which expands and develops Klein’s Erlangen Program, associates to a homogeneous space \(\mathcal G/H\) a Cartan geometry \((\mathcal G, H)\). Here \(\mathcal G\) is a Lie group, and \(H\) is a closed subgroup of \(\mathcal G\). \((\mathcal G, H)\) can be considered as the curved analogue of \((\mathcal G, H)\). Parabolic geometries \((\mathcal G, P)\) are obtained if \(\mathcal G\) is a semisimple Lie group, and \(P\) is a parabolic subgroup of \(\mathcal G\). Under the conditions of regularity and normality parabolic geometries are the underlying structures. Some of the advantages of this unified approach are discussed. Constructions of different parabolic geometries are described. In the frame of Cartan geometries they are transparent, but not quite so in the frame of the underlying structures.

Then, correspondence spaces are considered. They are associated to nested parabolic subgroups. Characterization of the geometries obtained in this way leads to twistor spaces. A complete local characterization of correspondence spaces is given in terms of the harmonic curvature: The relation to the geometry of systems of second order ODE’s is discussed. Finally, Fefferman’s construction of conformal structures on the total space of a circle bundle over a \(CR\)-manifold is presented.

For the entire collection see [Zbl 1103.53001].

The extremely general concept of Cartan geometry, which expands and develops Klein’s Erlangen Program, associates to a homogeneous space \(\mathcal G/H\) a Cartan geometry \((\mathcal G, H)\). Here \(\mathcal G\) is a Lie group, and \(H\) is a closed subgroup of \(\mathcal G\). \((\mathcal G, H)\) can be considered as the curved analogue of \((\mathcal G, H)\). Parabolic geometries \((\mathcal G, P)\) are obtained if \(\mathcal G\) is a semisimple Lie group, and \(P\) is a parabolic subgroup of \(\mathcal G\). Under the conditions of regularity and normality parabolic geometries are the underlying structures. Some of the advantages of this unified approach are discussed. Constructions of different parabolic geometries are described. In the frame of Cartan geometries they are transparent, but not quite so in the frame of the underlying structures.

Then, correspondence spaces are considered. They are associated to nested parabolic subgroups. Characterization of the geometries obtained in this way leads to twistor spaces. A complete local characterization of correspondence spaces is given in terms of the harmonic curvature: The relation to the geometry of systems of second order ODE’s is discussed. Finally, Fefferman’s construction of conformal structures on the total space of a circle bundle over a \(CR\)-manifold is presented.

For the entire collection see [Zbl 1103.53001].

Reviewer: Lajos Tamássy (Debrecen)

##### MSC:

53C10 | \(G\)-structures |

53C28 | Twistor methods in differential geometry |

53D10 | Contact manifolds (general theory) |

32V05 | CR structures, CR operators, and generalizations |

53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |