A Projective-to-Conformal Fefferman-Type Construction

TL;DR

This paper introduces a new Fefferman-type construction linking projective and conformal structures via Lie group inclusions, providing a geometric framework for conformal Patterson-Walker metrics using parabolic geometry.

Contribution

It establishes a novel construction connecting projective and conformal geometries through Lie group embeddings, with explicit characterizations and solutions in the context of parabolic geometry.

Findings

Existence of a canonical pure twistor spinor

Presence of a light-like conformal Killing field

Complete characterization of conformal spaces via solutions to overdetermined equations

Abstract

We study a Fefferman-type construction based on the inclusion of Lie groups ${\rm SL}(n+1)$ into ${\rm Spin}(n+1,n+1)$. The construction associates a split-signature $(n,n)$-conformal spin structure to a projective structure of dimension $n$. We prove the existence of a canonical pure twistor spinor and a light-like conformal Killing field on the constructed conformal space. We obtain a complete characterisation of the constructed conformal spaces in terms of these solutions to overdetermined equations and an integrability condition on the Weyl curvature. The Fefferman-type construction presented here can be understood as an alternative approach to study a conformal version of classical Patterson-Walker metrics as discussed in recent works by Dunajski-Tod and by the authors. The present work therefore gives a complete exposition of conformal Patterson-Walker metrics from the viewpoint of parabolic geometry.