On symmetric CR geometries of hypersurface type

TL;DR

This paper investigates symmetric non-degenerate CR geometries of hypersurface type, revealing they are either flat or homogeneous, and explores their metric properties and examples.

Contribution

It classifies symmetric CR geometries of hypersurface type, showing their flatness or homogeneity, and constructs examples with specific metric properties.

Findings

Symmetric CR geometries are either flat or homogeneous.

Non-flat symmetric CR geometries admit compatible pseudo-Riemannian metrics.

Examples of flat symmetric CR geometries without compatible metrics are provided.

Abstract

We study non-degenerate CR geometries of hypersurface type that are symmetric in the sense that, at each point, there is a CR transformation reversing the CR distribution at that point. We show that such geometries are either flat or homogeneous. We show that non-flat non-degenerate symmetric CR geometries of hypersurface type are covered by CR geometries with a compatible pseudo-Riemannian metric preserved by all symmetries. We construct examples of simply connected flat non-degenerate symmetric CR geometries of hypersurface type that do not carry a pseudo-Riemannian metric compatible with the symmetries.