A Projective-to-Conformal Fefferman-Type Construction

作者：Hammerl Matthias^*; Sagerschnig Katja; Silhan Josef; Taghavi Chaber Arman; Zadnik Vojtech

来源：Symmetry Integrability and Geometry-Methods and Applications, 2017, 13: 081.

DOI：10.3842/SIGMA.2017.081

• parabolic geometry
• projective structure
• conformal structure
• Cartan connection
• Fefferman spaces
• twistor spinors

We study a Fefferman-type construction based on the inclusion of Lie groups SL(n + 1) into 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.

• 出版日期2017