Asian Journal of Mathematics

Volume 23 (2019)

Number 5

Conformal Patterson–Walker metrics

Pages: 703 – 734



Matthias Hammerl (Faculty of Mathematics, University of Vienna, Austria)

Katja Sagerschnig (Center for Theoretical Physics PAS, Warsaw, Poland)

Josef Šilhan (Faculty of Science, Masaryk University, Brno, Czech Republic)

Arman Taghavi-Chabert (Department of Mathematics, Faculty of Arts and Sciences, American University of Beirut, Lebanon)

Vojtěch Žádník (Faculty of Education, Masaryk University, Brno, Czech Republic)


The classical Patterson–Walker construction of a split-signature (pseudo-)Riemannian structure from a given torsion-free affine connection is generalized to a construction of a split-signature conformal structure from a given projective class of connections. A characterization of the induced structures is obtained. We achieve a complete description of Einstein metrics in the conformal class formed by the Patterson–Walker metric. Finally, we describe all symmetries of the conformal Patterson–Walker metric. In both cases we obtain descriptions in terms of geometric data on the original structure.


differential geometry, parabolic geometry, projective structure, conformal structure, Einstein metrics, conformal killing field, twistor spinors

2010 Mathematics Subject Classification

53A20, 53A30, 53B30, 53C07

Received 2 March 2017

Accepted 21 June 2018

Published 30 April 2020