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Fefferman-type metrics and the projective geometry of sprays in twodimensions

Published online by Cambridge University Press:  01 May 2007

M. CRAMPIN  and
D. J. SAUNDERS
M. CRAMPIN*
Affiliation:
Department of Mathematical Physics and Astronomy, Ghent University, Krijgslaan 281, B–9000 Ghent, Belgium and Department of Mathematics, King's College, Strand, London WC2R 2LS. e-mail: Crampin@btinternet.com
D. J. SAUNDERS
Affiliation:
Department of Algebra and Geometry, Palacky University, 779 00 Olomouc, Czech Republic.
*
†Address for correspondence:65 Mount Pleasant, AspleyGuise, Beds MK17 8JXUK.
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Abstract

A spray is a second-order differential equation field on the slit tangent bundleof a differentiable manifold, which is homogeneous of degree 1 in the fibrecoordinates in an appropriate sense; two sprays which are projectivelyequivalent have the same base-integral curves up to reparametrization. We showhow, when the base manifold is two-dimensional, to construct from any projectiveequivalence class of sprays a conformal class of metrics on a four-dimensionalmanifold, of signature (2, 2); the Weyl conformal curvature of these metrics issimply related to the projective curvature of the sprays, and the geodesics ofthe sprays determine null geodesics of the metrics. The metrics in question havepreviously been obtained by Nurowski and Sparling (Classical and Quantum Gravity20 (2003) 4995–5016), by a differentmethod involving the exploitation of a close analogy between the Cartan geometryof second-order ordinary differential equations and of three-dimensionalCauchy–Riemann structures. From this perspective the derived metricsare seen to be analoguous to those defined by Fefferman in the CR theory, andare therefore said to be of Fefferman type. Our version of the constructionreveals that the Fefferman-type metrics are derivable from the scalar tripleproduct, both directly in the flat case (which we discuss in some detail) and bya simple extension in general. There is accordingly in our formulation a verysimple expression for a representative metric of the class in suitablecoordinates.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 142 , Issue 3 , May 2007 , pp. 509 - 523
Copyright
Copyright © Cambridge Philosophical Society 2007

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References

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