Symmetries of parabolic contact structures

TL;DR

This paper extends the concept of symmetric spaces to parabolic contact structures, showing that symmetric normal structures are torsion-free, some are flat, and analyzing the properties and limitations of symmetries with non-zero harmonic curvature.

Contribution

It introduces a generalization of symmetric spaces to parabolic contact structures and analyzes their torsion, flatness, and symmetry properties.

Findings

Symmetric normal parabolic contact structures are torsion-free.

Some symmetric structures must be locally flat.

Symmetries with non-zero harmonic curvature are involutive.

Abstract

We generalize the concept of locally symmetric spaces to parabolic contact structures. We show that symmetric normal parabolic contact structures are torsion--free and some types of them have to be locally flat. We prove that each symmetry given at a point with non--zero harmonic curvature is involutive. Finally we give restrictions on number of different symmetries which can exist at such a point.