Notes on symmetric conformal geometries

TL;DR

This paper reviews symmetric conformal geometries, proves new results about their structure, and constructs examples of locally flat geometries that are not pseudo-Riemannian symmetric spaces.

Contribution

It establishes that symmetric conformal geometries are either locally flat or covered by pseudo-Riemannian symmetric spaces, and provides new examples of locally flat geometries.

Findings

Symmetric conformal geometries are either locally flat or covered by pseudo-Riemannian symmetric spaces.

Constructed examples of locally flat symmetric conformal geometries that are not pseudo-Riemannian symmetric spaces.

Proved several new results for symmetric conformal geometries.

Abstract

In this article, we summarize the results on symmetric conformal geometries. We review the results following from the general theory of symmetric parabolic geometries and prove several new results for symmetric conformal geometries. In particular, we show that each symmetric conformal geometry is either locally flat or covered by a pseudo-Riemannian symmetric space, where the covering is a conformal map. We construct examples of locally flat symmetric conformal geometries that are not pseudo-Riemannian symmetric spaces.