1 Introduction

A projective structure \(\mathfrak{p}\) on a surface \(\Sigma\) is an equivalence class of affine torsion-free connections on \(\Sigma\) where two connections are declared to be projectively equivalent if they share the same geodesics up to parametrisation. A surface equipped with a projective structure will be called a projective surface. In [12] it was shown that an oriented projective surface \((\Sigma,\mathfrak{p})\) defines a complex surface \(Z\) together with a projection to \(\Sigma\) whose fibres are holomorphically embedded disks. Moreover, a conformal connection in the projective equivalence class corresponds to a section whose image is a holomorphic curve in \(Z\). Locally such sections always exist and hence every affine torsion-free connection on a surface is locally projectively equivalent to a conformal connection. The problem of characterising the affine torsion-free connections on surfaces that are locally projectively equivalent to a Levi-Civita connection was recently solved in [3].

Here we show that if a closed holomorphic curve \(D\subset Z\) is the image of a section of \(Z \to \Sigma\), then its normal bundle \(N \to D\) has degree twice the Euler characteristic of \(\Sigma\). This is achieved by observing that the projective structure on \(\Sigma\) canonically equips the co-normal bundle of \(D\) with a Hermitian bundle metric whose Chern connection can be computed explicitly. Using the fact that the normal bundle \(N \to D\) has degree \(2\chi(\Sigma)\) and that the bundle \(Z \to \Sigma\) has a contractible fibre, we prove that on a closed surface \(\Sigma\) with \(\chi(\Sigma)<0\) there is at most one section of \(Z \to \Sigma\) whose image is a holomorphic curve. It follows that a conformal connection on \(\Sigma\) preserves precisely one conformal structure and is furthermore uniquely determined by its unparametrised geodesics. In particular, as a corollary one obtains that the unparametrised geodesics of a Riemannian metric on \(\Sigma\) determine the metric up to constant rescaling, a result previously proved in [11].

In the case where \(\Sigma\) is the \(2\)-sphere, it follows that the normal bundle of a holomorphic curve \(D\simeq \mathbb{CP}^1\subset Z\), arising as the image of a section of \(Z \to S^2\), is isomorphic to \(\mathcal{O}(4)\). Consequently, Kodaira’s deformation theorem can be applied to show that every conformal connection on \(S^2\) lies in a complex \(5\)-manifold of conformal connections, all of which share the same unparametrised geodesics.

Acknowledgments

This paper would not have come into existence without several very helpful discussions with Nigel Hitchin. I would like to warmly thank him here. I also wish to thank Vladimir Matveev for references and the anonymous referee for her/his careful reading and useful suggestions. Research for this article was carried out while the author was visiting the Mathematical Institute at the University of Oxford as a postdoctoral fellow of the Swiss NSF, PA00P2_142053. The author would like to thank the Mathematical Institute for its hospitality.