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arXiv · 0910.2618

Weyl metrisability of two-dimensional projective structures

Thomas Mettler
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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.

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Thomas Mettler. 2013-08-14. Weyl metrisability of two-dimensional projective structures. https://doi.org/10.1017/s0305004113000522

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