Weyl metrisability of two-dimensional projective structures

TL;DR

This paper demonstrates that on a surface, every affine torsion-free connection can be locally represented as a Weyl connection, linking geometric structures with algebraic geometry through PDE solutions and twistor theory.

Contribution

It introduces two methods to establish the Weyl metrisability of two-dimensional projective structures, connecting differential geometry with algebraic geometry.

Findings

Local affine torsion-free connections are projectively equivalent to Weyl connections.

Weyl connections on the two-sphere correspond to smooth quadrics without real points.

Solutions of the PDE are in one-to-one correspondence with sections of a twistor bundle.

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.