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Weyl metrisability of two-dimensional projective structures

Published online by Cambridge University Press:  19 September 2013

THOMAS METTLER
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA. e-mail: t.mettler@damtp.cam.ac.uk
Corresponding
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Abstract

We show that on a surface locally every affine torsion-free connection is projectively equivalent to a Weyl connection. First, this is done using exterior differential system theory. Second, this is done by showing that the solutions of the relevant PDE are in one-to-one correspondence with the sections of the ‘twistor’ bundle of conformal inner products having holomorphic image. The second solution allows to use standard results in algebraic geometry to show that the Weyl connections on the two-sphere whose geodesics are the great circles are in one-to-one correspondence with the smooth quadrics without real points in the complex projective plane.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2013 

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