A Projective-to-Conformal Fefferman-Type Construction

Symmetry Integrability and Geometry-Methods and Applications

article

Matthias Hammerl¹  Katja Sagerschnig²  Josef Šilhan³  Arman Taghavi-Chabert⁴  Vojtĕch Zádník⁵ 

[1] University of Vienna, Faculty of Mathematics;Dipartimento di Scienze Matematiche;Masaryk University, Faculty of Science;Università di Torino, Dipartimento di Matematica ''G. Peano'';Masaryk University, Faculty of Education

关键词: parabolic geometry;    projective structure;    conformal structure;    Cartan connection;    Fef ferman spaces;    twistor spinors;   

DOI  :  10.3842/SIGMA.2017.081

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown