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2009

Weyl metrisability of two-dimensional projective structures

Mettler, Thomas

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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.

Weyl metrisability of two-dimensional projective structures · Around

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1922
Earlier work this paper cites.
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1957
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1986
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