Geodesic rigidity of conformal connections on surfaces

TL;DR

This paper proves that on surfaces with negative Euler characteristic, conformal connections are uniquely determined by their unparametrised geodesics, and explores the structure of conformal connections on the sphere.

Contribution

It establishes the uniqueness of conformal structures preserved by conformal connections on certain surfaces and characterizes the space of conformal connections on the sphere.

Findings

Conformal connections on surfaces of negative Euler characteristic preserve a unique conformal structure.

Unparametrised geodesics determine the conformal connection up to rescaling.

On the sphere, all conformal connections sharing the same unparametrised geodesics form a 5-dimensional complex manifold.

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.