Geometric structures invariant to symmetries

TL;DR

This paper explores generalized local symmetries in geometric structures using Cartan geometries, classifies symmetric structures on semisimple spaces, and provides detailed analysis of symmetric parabolic geometries.

Contribution

It introduces a framework for understanding symmetries in diverse geometric structures and classifies symmetric geometries, including parabolic types, on semisimple spaces.

Findings

Classification of symmetric AHS-structures

Classification of symmetric parabolic contact geometries

Explicit descriptions of symmetric geometric structures

Abstract

We introduce and discuss (local) symmetries of geometric structures. These symmetries generalize the classical (locally) symmetric spaces to various other geometries. Our main tools are homogeneous Cartan geometries and their explicit description. This allows us to describe the structure of symmetric geometric structures and to provide a general construction of such structures. Since we can view the classical (locally) symmetric spaces as special case, this allows us to classify various geometric structures on semisimple symmetric spaces. Then we investigate the case of symmetric parabolic geometries in detail and obtain classification of symmetric AHS-structures and symmetric parabolic contact geometries in the semisimple cases.