Conformal Patterson-Walker metrics

Abstract

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.

v2: The article has been restructured: a section has been added and includes a characterisation of Patterson-Walker metrics. An error in Proposition 6.6 has been fixed v3: One reference clarified. v4: References added. Remark 6.9 added. Accepted for publication in The Asian Journal of Mathematics on 21 June 2018 v5: Minor changes, as published

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