Conformal Patterson-Walker metrics

TL;DR

This paper generalizes the Patterson-Walker construction to produce split-signature conformal structures from projective classes, characterizes the resulting structures, and describes Einstein metrics and symmetries within this framework.

Contribution

It introduces a new method to construct conformal structures from projective classes and provides a complete description of Einstein metrics and symmetries in this setting.

Findings

Complete characterization of Einstein metrics in the conformal class

Description of all symmetries of the conformal Patterson-Walker metric

Geometric data-based descriptions of the structures

Abstract

The classical Patterson-Walker construction of a split-signature (pseudo-)Riemannian structure from a given torsion-free affine connection is generalized to a construction of a split-signature conformal structure from a given projective class of connections. A characterization of the induced structures is obtained. We achieve a complete description of Einstein metrics in the conformal class formed by the Patterson-Walker metric. Finally, we describe all symmetries of the conformal Patterson-Walker metric. In both cases we obtain descriptions in terms of geometric data on the original structure.