Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 081, 33 pages      arXiv:1510.03337      http://dx.doi.org/10.3842/SIGMA.2017.081

A Projective-to-Conformal Fefferman-Type Construction

Matthias Hammerl a, Katja Sagerschnig b, Josef Šilhan c, Arman Taghavi-Chabert d and Vojtĕch Zádník e
a) University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1010 Vienna, Austria
b) INdAM-Politecnico di Torino, Dipartimento di Scienze Matematiche, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
c) Masaryk University, Faculty of Science, Kotlářská 2, 61137 Brno, Czech Republic
d) Università di Torino, Dipartimento di Matematica ''G. Peano'', Via Carlo Alberto 10, 10123 Torino, Italy
e) Masaryk University, Faculty of Education, Poříčí 31, 60300 Brno, Czech Republic

Received February 09, 2017, in final form October 09, 2017; Published online October 21, 2017

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.

Key words: parabolic geometry; projective structure; conformal structure; Cartan connection; Fefferman spaces; twistor spinors.

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