A Projective-to-Conformal Fefferman-Type Construction

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Year of publication 2017
Type Article in Periodical
Magazine / Source Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
MU Faculty or unit

Faculty of Science

Web https://www.emis.de/journals/SIGMA/2017/081/
Doi http://dx.doi.org/10.3842/SIGMA.2017.081
Field General mathematics
Keywords parabolic geometry; projective structure; conformal structure; Cartan connection; Fefferman spaces; twistor spinors
Description 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.
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