Geodesic rigidity of conformal connections on surfaces

Abstract

We show that a conformal connection on a closed oriented surface $\Sigma$ of negative Euler characteristic preserves precisely one conformal structure and is furthermore uniquely determined by its unparametrised geodesics. As a corollary it follows that the unparametrised geodesics of a Riemannian metric on $\Sigma$ determine the metric up to constant rescaling. It is also shown that every conformal connection on the $2$-sphere lies in a complex $5$-manifold of conformal connections, all of which share the same unparametrised geodesics.

16 pages, exposition improved, references added

References in corpus (1)

• On Kähler metrisability of two-dimensional complex projective structures

Cited by in corpus (5)

• Convex projective surfaces with compatible Weyl connection are hyperbolic
• Geometric Theory of Weyl Structures
• Metrisability of projective surfaces and pseudo-holomorphic curves
• Minimal Lagrangian connections on compact surfaces
• Deformations of the Veronese embedding and Finsler 2-spheres of constant curvature