A Projective-to-Conformal Fefferman-Type Construction

Abstract

We study a Fefferman-type construction based on the inclusion of Lie groups $\mathrm{SL}(n+1)$ into $\mathrm{Spin}(n+1,n+1)$. 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.

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Cited by in corpus (3)

• The Geometry of Almost Einstein (2,3,5) Distributions
• The ambient obstruction tensor and conformal holonomy
• Canonical curves and Kropina metrics in Lagrangian contact geometry