Weyl metrisability of two-dimensional projective structures

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.

15 pages. Final version

References in corpus (4)

• Four--Dimensional Metrics Conformal to Kahler
• Geodesically equivalent metrics in general relativity
• Projective vs metric structures
• Reduction of beta-integrable 2-Segre structures

Cited by in corpus (8)

• Light cone and Weyl compatibility of conformal and projective structures
• Convex projective surfaces with compatible Weyl connection are hyperbolic
• Geodesic rigidity of conformal connections on surfaces
• GL(2)-geometry and complex structures
• Metrisability of projective surfaces and pseudo-holomorphic curves
• Projectively equivalent 2-dimensional superintegrable systems with projective symmetries
• Deformations of the Veronese embedding and Finsler 2-spheres of constant curvature
• Weyl metrizability of 3-dimensional projective structures and CR submanifolds