General construction of symmetric parabolic structures

TL;DR

This paper generalizes symmetric spaces to parabolic geometries, providing a construction method from classical symmetric spaces, classifying certain contact geometries, and explicitly constructing examples with specific symmetry groups.

Contribution

It introduces a framework for symmetric parabolic structures, classifies contact geometries with semisimple symmetry groups, and constructs explicit examples with non-complex simple groups.

Findings

All regular parabolic geometries with smooth involutive symmetries can be constructed from classical symmetric spaces.

Complete classification of parabolic contact geometries with semisimple symmetry groups without complex factors.

Explicit constructions of non-trivial contact geometries with non-complex simple symmetry groups.

Abstract

First we introduce a generalization of symmetric spaces to parabolic geometries. We provide construction of such parabolic geometries starting with classical symmetric spaces and we show that all regular parabolic geometries with smooth systems of involutive symmetries can be obtained this way. Further, we investigate the case of parabolic contact geometries in great detail and we provide the full classification of those with semisimple groups of symmetries without complex factors. Finally, we explicitly construct all non-trivial contact geometries with non-complex simple groups of symmetries. We also indicate geometric interpretations of some of them.