1 Introduction

A projective structure on a smooth manifold consists of an equivalence class \(\mathfrak{p}\) of torsion-free connections on its tangent bundle, where two such connections are called equivalent if they have the same geodesics up to parametrisation. A projective structure \(\mathfrak{p}\) is called metrisable if it contains the Levi-Civita connection of some Riemannian metric. The problem of (locally) characterising the projective structures that are metrisable was first studied in the work of R. Liouville [17] in 1889, but was solved only relatively recently by Bryant, Dunajski and Eastwood for the case of two dimensions [2]. Since then, there has been renewed interest in the problem, see [5, 6, 8, 10, 11, 13, 14, 25, 27] for related recent work.

The purpose of this short note is to show that in the case of an oriented projective surface \((M,\mathfrak{p}),\) the metrisability of \(\mathfrak{p}\) is equivalent to the existence of certain pseudo-holomorphic curves.

An orientation compatible complex structure on \(M\) corresponds to a section of the bundle \(\pi : Z\to M\) whose fibre at \(x \in M\) consists of the orientation compatible linear complex structures on \(T_xM.\) The choice of a torsion-free connection \(\nabla\) on \(TM\) equips \(Z\) with an almost complex structure \(J\) [7, 26]. Namely, at \(j \in Z\) we lift \(j\) horizontally and take a natural complex structure on each fibre vertically. It turns out that \(J\) is always integrable and does only depend on the projective equivalence class \(\mathfrak{p}\) of \(\nabla,\) we thus denote it by \(J_{\mathfrak{p}}.\) Reversing the orientation on each fibre yields another almost complex structure \(\mathfrak{J}\) which is however never integrable and is not projectively invariant. Fixing a volume form \(\sigma\) on the projective surface \((M,\mathfrak{p})\) determines a unique representative connection \({}^{\sigma}\nabla \in \mathfrak{p}\) which preserves \(\sigma.\) We will write \(\mathfrak{J}_{\mathfrak{p},\sigma}\) for the non-integrable almost complex structure arising from \({}^{\sigma}\nabla \in \mathfrak{p}.\)

The choice of a conformal structure \([g]\) on an oriented surface \(M\) defines an orientation compatible complex structure by rotating a tangent vector counter-clockwise by \(\pi/2\) with respect to \([g].\) Thus, we may think of a conformal structure as a section \([g] : M \to Z.\) Denoting the area form of a Riemannian metric \(g\) by \(dA_g,\) we show:

Applying a general existence result for pseudo-holomorphic curves [24] it follows that locally we can always find a Riemannian metric \(g\) so that \([g] : M \to (Z,J_{\mathfrak{p}})\) is a holomorphic curve or so that \([g] : M \to (Z,\mathfrak{J}_{\mathfrak{p},dA_g})\) is a pseudo-holomorphic curve. The geometric significance of the existence of such (pseudo-)holomorphic curves is given in Proposition 2.8 below.

The construction of the (integrable) almost complex structure \(J_{\mathfrak{p}}\) on \(Z\) given in [7, 26] is adapted from the construction of an almost complex structure \(J\) on the twistor space \(Y \to N\) of an oriented Riemannian \(4\)-manifold \((N,g),\) see [1]. In the Riemannian setting the almost complex structure \(J\) is integrable if and only if \(g\) is self-dual. In [12], Eells–Salamon observe that reversing the orientation on each fibre of \(Y \to N\) associates another almost complex structure \(\mathfrak{J}\) on \(Y\) to \((N,g)\) which is never integrable. Thus, the non-integrable almost complex structure \(\mathfrak{J}\) used here may be thought of as the affine analogue of the non-integrable almost complex structure in oriented Riemannian \(4\)-manifold geometry.

Acknowledgements

The author is grateful to Maciej Dunajski and Gabriel Paternain for helpful conversations and correspondence. A part of the research for this article was carried out while the author was visiting FIM at ETH Zürich. The author would like to thank FIM for its hospitality and DFG for partial support via the priority programme SPP 2026 “Geometry at Infinity”.