Parabolic Symmetric Spaces

TL;DR

This paper investigates systems of symmetries in |1|-graded parabolic geometries, focusing on smooth, non-flat homogeneous cases, and demonstrates the existence of an invariant affine connection under certain conditions.

Contribution

It introduces the study of symmetry systems in |1|-graded parabolic geometries and proves the existence of invariant affine connections under weak assumptions.

Findings

Existence of invariant affine connection under weak conditions

Analysis of smooth systems of symmetries in |1|-graded geometries

Characterization of non-flat homogeneous geometries

Abstract

We study here systems of symmetries on $|1|$--graded parabolic geometries. We are interested in smooth systems of symmetries and we discuss non--flat homogeneous $|1|$--graded geometries. We show the existence of an invariant admissible affine connection under quite weak condition on the system.