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 \(\pi : Z\rightarrow M\) 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 counterclockwise by \(\pi /2\) with respect to [g]. Thus, we may think of a conformal structure as a section \([g] : M \rightarrow Z\). Denoting the area form of a Riemannian metric g by \(dA_g\), we show:

Theorem 1

An oriented projective surface \((M,{\mathfrak {p}})\) is metrisable by the metric g on M if and only if \([g] : M \rightarrow (Z,J_{{\mathfrak {p}}})\) is a holomorphic curve and \([g] : M \rightarrow (Z,{\mathfrak {J}}_{{\mathfrak {p}},dA_g})\) is a pseudo-holomorphic curve.

Applying a general existence result for pseudo-holomorphic curves [24, Theorem III] it follows that locally we can always find a Riemannian metric g so that \([g] : M \rightarrow (Z,J_{{\mathfrak {p}}})\) is a holomorphic curve or so that \([g] : M \rightarrow (Z,{\mathfrak {J}}_{{\mathfrak {p}},dA_g})\) is a pseudo-holomorphic curve. The geometric significance of the existence of such (pseudo-)holomorphic curves is given in Proposition 9 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 \(Y \rightarrow N\) 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 \(Y \rightarrow N\) 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.

2 Pseudo-holomorphic curves and metrisability

Recall that the set of torsion-free connections on the tangent bundle of a surface M is an affine space modelled on the smooth sections of the vector bundle \(V=S^2(T^*M)\otimes TM\). We have a natural trace mapping \({\text {tr}}: V \rightarrow T^*M\), given in abstract index notation by \(A^i_{jk} \mapsto A^{k}_{ik}\), as well as an inclusion \(\mathrm {Sym} : T^*M \rightarrow V\), given by \(b_i \mapsto \delta ^i_j b_k+\delta ^i_kb_j\). The bundle V thus decomposes as \(V=V_0\oplus T^*M\), where \(V_0\) denotes the trace-free part of V. We have (Cartan, Eisenhart, Weyl)—the reader may also consult [9] for a modern reference:

Lemma 2

Two torsion-free connections \(\nabla \) and \(\nabla ^{\prime }\) on TM are projectively equivalent if and only if there exists a 1-form \(\xi \) on M so that \(\nabla -\nabla ^{\prime }=\mathrm {Sym}(\xi )\).

This gives immediately:

Lemma 3

Let \((M,{\mathfrak {p}})\) be an oriented projective surface and \(\sigma \) a volume form on M. Then there exists a unique representative connection \({}^{\sigma }\nabla \in {\mathfrak {p}}\) preserving \(\sigma \).

Proof

Let \(\nabla \in {\mathfrak {p}}\) be a representative connection. Since \(\sigma \) is a volume form there exists a unique 1-form \(\alpha \) on M such that \(\nabla \sigma =\alpha \otimes \sigma \). An elementary computation shows that the connection \(\nabla +\mathrm {Sym}(\xi )\) satisfies

$$\begin{aligned} \left( \nabla +\mathrm {Sym}(\xi )\right) \sigma =\nabla \sigma -3\xi \otimes \sigma , \end{aligned}$$

for all \(\xi \in \Omega ^1(M)\). Thus the connection \({}^{\sigma }\nabla =\nabla +\frac{1}{3}\mathrm {Sym}(\alpha )\) preserves \(\sigma \) and clearly is the only connection in \({\mathfrak {p}}\) doing so. \(\square \)

We also have:

Lemma 4

Let \(\varphi \in \Gamma (V_0)\) and \(\nabla \) be a torsion-free connection on TM. Then \(\nabla +\varphi \) preserves a volume form \(\sigma \) on M if and only if \(\nabla \) preserves the volume form \(\sigma \).

Proof

Since \(\varphi \in \Gamma (V_0)\), an elementary computation shows that the connections \(\nabla \) and \(\nabla +\varphi \) induce the same connection on the bundle \(\Lambda ^2(T^*M)\) whose non-vanishing sections are the volume forms. \(\square \)

For our purposes it is convenient to construct the almost complex structures \((J,{\mathfrak {J}})\) associated to \(\nabla \) in terms of the connection form \(\theta \) on the oriented frame bundle of M. The oriented frame bundle \(F\) of the oriented surface M is the bundle \(\upsilon : F\rightarrow M\) whose fibre at \(x \in M\) consists of the linear isomorphisms \(u : \mathbb {R}^2 \rightarrow T_xM\) that are orientation preserving with respect to the standard orientation on \(\mathbb {R}^2\) and the given orientation on \(T_xM\). The group \(\mathrm {GL}^+(2,\mathbb {R})\) acts transitively from the right on each fibre by the rule \(R_a(u)=u\circ a\) for all \(a \in \mathrm {GL}^+(2,\mathbb {R}), u \in F\) and this action turns \(\upsilon : F\rightarrow M\) into a principal right \(\mathrm {GL}^+(2,\mathbb {R})\)-bundle. The total space \(F\) carries a tautological \(\mathbb {R}^2\)-valued 1-form \(\omega \) defined by \(\omega _u=u^{-1}\circ \upsilon ^{\prime }_{u}\) and \(\omega \) satisfies the equivariance property

$$\begin{aligned} R_a^*\omega =a^{-1}\omega \end{aligned}$$
(1)

for all \(a \in \mathrm {GL}^+(2,\mathbb {R})\). We may embed \(\mathrm {GL}(1,\mathbb {C})\) as the subgroup of \(\mathrm {GL}^+(2,\mathbb {R})\) consisting of matrices that commute with the standard linear complex structure on \(\mathbb {R}^2\). Note that may think of the oriented frame bundle \(\upsilon : F\rightarrow M\) as a principal \(\mathrm {GL}(1,\mathbb {C})\)-bundle over \(Z=F/\mathrm {GL}(1,\mathbb {C})\). We may describe an almost complex structure on \(Z\) by describing the pullback of its \((1,\!0)\)-forms to \(F\). The pullback of a 1-form on \(Z\) to \(F\) is semi-basic for the projection \(\nu : F\rightarrow Z\), that is, it vanishes when evaluated on vector fields that are tangent to the fibres of \(\nu \). For \(y \in {{\mathfrak {g}}}{{\mathfrak {l}}}(2,\mathbb {R})\) we denote by \(Y_y\) the vector field on \(F\) that is generated by the flow \(R_{\exp (ty)}\). Clearly, the vector fields \(Y_y\) for \(y \in {{\mathfrak {g}}}{{\mathfrak {l}}}(1,\mathbb {C})\) span the vector fields on \(F\) that are tangent to the fibres of \(\nu \).

Let \(\nabla \) be a torsion-free connection on TM with connection form \(\theta =(\theta ^i_j)\) on \(F\). Recall that \(\theta \) satisfies the equivariance property

$$\begin{aligned} R_a^*\theta =a^{-1}\theta a \end{aligned}$$
(2)

for all \(a \in \mathrm {GL}^+(2,\mathbb {R})\) and the structure equations

$$\begin{aligned} \begin{aligned} \mathrm {d}\omega ^i&=-\theta ^i_j\wedge \omega ^j,\\ \mathrm {d}\theta ^i_j&=-\theta ^i_k\wedge \theta ^k_j+\Theta ^i_j, \end{aligned} \end{aligned}$$
(3)

where \(\Theta =(\Theta ^i_j)\) denotes the curvature form of \(\theta \). Since \(\theta \) is a principal connection on \(F\) it also satisfies \(\theta (Y_y)=y\) for all \(y \in {{\mathfrak {g}}}{{\mathfrak {l}}}(2,\mathbb {R})\). Since the Lie algebra of \(\mathrm {GL}(1,\mathbb {C})\) is spanned by the matrices of the form

$$\begin{aligned} \begin{pmatrix} z &{}\quad -w \\ w &{}\quad z\end{pmatrix} \end{aligned}$$

for \((z,w)\in \mathbb {R}^2\), the complex-valued 1-forms on \(F\) that are semi-basic for the projection \(\nu : F\rightarrow Z\) are spanned by the forms \(\omega =\omega ^1+\mathrm {i}\omega ^2\) and

$$\begin{aligned} \zeta =(\theta ^1_1-\theta ^2_2)+\mathrm {i}\left( \theta ^1_2+\theta ^2_1\right) \end{aligned}$$

and their complex conjugates. We now have:

Proposition 5

Let \(\nabla \) be a torsion-free connection on TM with connection form \(\theta =(\theta ^i_j)\) on \(F\). Then there exists a unique pair \((J,{\mathfrak {J}})\) of almost complex structures on \(Z\) whose \((1,\! 0)\)-forms pull back to become linear combinations of the forms \((\omega ,\zeta )\) in the case of J and to \((\omega ,\overline{\zeta })\) in the case of \({\mathfrak {J}}\). Moreover, the almost complex structure J is always integrable, whereas \({\mathfrak {J}}\) is never integrable.

Proof

Writing

$$\begin{aligned} r\mathrm {e}^{\mathrm {i}\phi }\simeq \begin{pmatrix} r\cos \phi &{}\quad -r\sin \phi \\ r\sin \phi &{}\quad r\cos \phi \end{pmatrix} \end{aligned}$$

for the elements of \(\mathrm {GL}(1,\mathbb {C})\), the equivariance property (1) of \(\omega \) and (2) of \(\theta \) implies

$$\begin{aligned} (R_{r\mathrm {e}^{\mathrm {i}\phi }})^*\omega =\frac{1}{r}\mathrm {e}^{\mathrm {i}\phi }\omega \quad \text {and}\quad (R_{r\mathrm {e}^{\mathrm {i}\phi }})^*\zeta =\mathrm {e}^{-2\mathrm {i}\phi }\zeta . \end{aligned}$$
(4)

It follows that there exists a unique almost complex structure J on \(Z\) whose \((1,\! 0)\)-forms pull back to \(F\) to become linear combinations of the forms \(\omega ,\zeta \). Likewise there exists a unique almost complex structure \({\mathfrak {J}}\) on \(Z\) whose \((1,\! 0)\)-forms pull back to F to become linear combinations of the forms \(\omega ,\overline{\zeta }\). Furthermore, simple computations using the structure equations (3) imply that

$$\begin{aligned} 0=\mathrm {d}\zeta \wedge \omega \wedge \zeta =\mathrm {d}\omega \wedge \omega \wedge \zeta . \end{aligned}$$

Consequently, the Newlander–Nirenberg theorem [23] implies that J is integrable. On the other hand, we get

$$\begin{aligned} \mathrm {d}\omega \wedge \omega \wedge \overline{\zeta }=\frac{1}{2}\omega \wedge \overline{\omega }\wedge \zeta \wedge \overline{\zeta } \end{aligned}$$

so that \({\mathfrak {J}}\) is never integrable. \(\square \)

Remark 6

The equivariance properties (4) imply that the bundles

$$\begin{aligned} H=\nu ^{\prime }\left\{ {\text {Re}}(\zeta )=0,{\text {Im}}(\zeta )=0\right\} \quad \text {and}\quad V=\nu ^{\prime } \{{\text {Re}}(\omega )=0,{\text {Im}}(\omega )=0\} \end{aligned}$$

are well-defined distributions on \(Z\) that are invariant with respect to J (and \({\mathfrak {J}}\)). Hence we have \(TZ=H\oplus V\).

For the convenience of the reader, we also show [7, 26]:

Proposition 7

Suppose the torsion-free connections \(\nabla \) and \(\nabla ^{\prime }\) on TM are projectively equivalent, then they induce the same integrable almost complex structure J on \(Z\).

Proof

The connections \(\nabla \) and \(\nabla ^{\prime }\) are projectively equivalent if and only if there exists a 1-form \(\xi \) on M such that \(\nabla ^{\prime }=\nabla +\mathrm {Sym}(\xi )\). Writing \(\theta =(\theta ^i_j)\) for the connection form of \(\nabla \) on \(F\) and \(\upsilon ^*\xi =x_i\omega ^i\) for real-valued functions \(x_i\) on \(F\), the connection form \(\theta ^{\prime }\) of \(\nabla ^{\prime }\) becomes

$$\begin{aligned} \theta ^{\prime }=\theta +\begin{pmatrix} 2x_1\omega ^1+x_2\omega ^2 &{}\quad x_2\omega ^1 \\ x_1\omega ^2 &{}\quad x_1\omega ^1+2x_2\omega ^2\end{pmatrix}. \end{aligned}$$

Consequently, we obtain

$$\begin{aligned} \zeta ^{\prime }=\zeta +(x_1\omega ^1-x_2\omega ^2)+\mathrm {i}(x_2\omega ^1+x_1\omega ^2)=\zeta +(x_1+\mathrm {i}x_2)\omega \end{aligned}$$

which shows that the complex span of \(\omega ,\zeta \) is the same as the one of \(\omega ,\zeta ^{\prime }\) and hence the two integrable almost complex structures are the same. \(\square \)

Remark 8

For a projective structure \({\mathfrak {p}}\) on M we will write \(J_{{\mathfrak {p}}}\) for the integrable almost complex structure defined by any representative connection \(\nabla \in {\mathfrak {p}}\). For a projective structure \({\mathfrak {p}}\) and a volume form \(\sigma \) on M we will write \({\mathfrak {J}}_{{\mathfrak {p}},\sigma }\) for the non-integrable almost complex structure defined by the representative connection \({}^{\sigma }\nabla \in {\mathfrak {p}}\). Note that the non-integrable almost complex structure is not projectively invariant.

Recall that a Weyl connection for a conformal structure [g] is a torsion-free connection \({}^{[g]}\nabla \) on TM which preserves [g]. Fixing a Riemannian metric \(g \in [g]\), the Weyl connections for [g] can be written as \({}^{[g]}\nabla ={}^g\nabla +g\otimes B-\mathrm {Sym}(\beta )\) for some 1-form \(\beta \) on M and where \(B\) denotes the g-dual vector field to \(\beta \). In [20] and in the language of thermostats in [22], it was observed that for every choice of a conformal structure [g] on a projective surface \((M,{\mathfrak {p}})\), there exists a unique Weyl connection \({}^{[g]}\nabla \) for [g] and a unique 1-form \(\varphi \in \Gamma (V_0)\) so that \({}^{[g]}\nabla +\varphi \) is a representative connection of \({\mathfrak {p}}\). Moreover the endomorphism \(\varphi (X)\) is symmetric with respect to [g] for every vector field X on M. We call \({}^{[g]}\nabla \) the Weyl connection determined by [g]. Explicitly, if \(\nabla \) is any representative connection of \({\mathfrak {p}}\), \(g \in [g]\) and if we define a vector field \(B=\frac{3}{4}\mathrm {tr}\left( g^{\sharp }\otimes (\nabla -{}^g\nabla )_0\right) \), then

$$\begin{aligned} \varphi =\left( \nabla -{}^g\nabla -g \otimes B\right) _0\qquad \text {and}\qquad {}^{[g]}\nabla ={}^g\nabla +g\otimes B-\mathrm {Sym}(\beta ), \end{aligned}$$

where \(A_0\) denotes the trace-free part of a tensor field \(A \in \Gamma (S^2(T^*M)\otimes TM)\). We refer the reader to [20, 22] for a proof that \({}^{[g]}\nabla \) and \(\varphi \) do satisfy the claimed properties.

Proposition 9

Let \((M,{\mathfrak {p}})\) be an oriented projective surface and g a Riemannian metric on M. Then we have:

  1. (i)

    \({\mathfrak {p}}\) contains a Weyl connection for [g] if and only if \([g] : M \rightarrow (Z,J_{\mathfrak {p}})\) is a holomorphic curve;

  2. (ii)

    the Weyl connection determined by [g] is the Levi-Civita connection of g if and only if \([g] : M \rightarrow (Z,{\mathfrak {J}}_{{\mathfrak {p}},dA_g})\) is a pseudo-holomorphic curve.

Remark 10

Here we say \([g] : M \rightarrow (Z,{\mathfrak {J}})\) is a (pseudo-)holomorphic curve if the image \(\Sigma =[g](M)\subset Z\) admits the structure of a (pseudo-)holomorphic curve. By admitting the structure of (pseudo-)holomorphic curve, we mean that \(\Sigma \) can be equipped with a complex structure J, so that the inclusion \(\iota : \Sigma \rightarrow Z\) is \((J,{\mathfrak {J}})\)-linear, that is, satisfies \({\mathfrak {J}}\circ \iota ^{\prime }=\iota ^{\prime } \circ J\).

As an immediate consequence, we obtain the Theorem 1:

Proof of Theorem 1

The projective structure \({\mathfrak {p}}\) is metrisable by g if and only if the Weyl connection determined by [g] is the Levi-Civita connection of g and the 1-form \(\varphi \) vanishes identically. The claim follows by applying Proposition 9. \(\square \)

For the proof of Proposition 9 we also need the following Lemma:

Lemma 11

Let \((Z,{\mathfrak {J}})\) be an almost complex four-manifold and \(\omega ,\chi \in \Omega ^1(Z,\mathbb {C})\) a basis for the \((1,\! 0)\)-forms of Z. Suppose \(\iota : \Sigma \rightarrow Z\) is an immersed surface so that \(\iota ^*(\omega \wedge \overline{\omega })\) is non-vanishing on \(\Sigma \). Then \(\Sigma \) admits the structure of a pseudo-holomorphic curve if and only if \(\iota ^*(\omega \wedge \chi )\) vanishes identically on \(\Sigma \).

Proof

Since \(\iota ^*(\omega \wedge \overline{\omega })\) is non-vanishing on \(\Sigma \), the forms \(\iota ^*\omega \) and \(\iota ^*\overline{\omega }\) span the complex-valued 1-forms on \(\Sigma \). Recall that \(\iota : \Sigma \rightarrow Z\) is \((j,{\mathfrak {J}})\)-linear if and only if the pullback of every \((1,\! 0)\)-form on Z is a \((1,\! 0)\)-form on \(\Sigma \), the claim follows. \(\square \)

Proof of Proposition 9

Let g be a Riemannian metric on the oriented projective surface \((M,{\mathfrak {p}})\). Without losing generality we can assume that the projective structure \({\mathfrak {p}}\) arises from a connection of the form \({}^{[g]}\nabla +\varphi \). The Weyl connection \({}^{[g]}\nabla \) satisfies

$$\begin{aligned} {}^{[g]}\nabla dA_g=2\beta \otimes dA_g \end{aligned}$$

for some 1-form \(\beta \) on M and hence can be written as \({}^{[g]}\nabla ={}^g\nabla +g\otimes \beta ^{\sharp }-\mathrm {Sym}(\beta )\).

Now suppose \(\nabla \in {\mathfrak {p}}\) preserves the volume form \(dA_g\) of g. Then, by Lemma 4 it must be of the form

$$\begin{aligned} \nabla ={}^{[g]}\nabla +\varphi +\frac{2}{3}\mathrm {Sym}(\beta )={}^g\nabla +g\otimes \beta ^{\sharp }-\frac{1}{3}\mathrm {Sym}(\beta )+\varphi . \end{aligned}$$
(5)

Proposition 5 and Lemma 11 imply that the condition that \([g] : M \rightarrow Z\) defines a pseudo-holomorphic curve with respect to \(J_{{\mathfrak {p}}}\) respectively \({\mathfrak {J}}_{{\mathfrak {p}},dA_g}\) is equivalent to the condition that on the pullback bundle \([g]^*F \rightarrow M\) the form \(\omega \wedge \zeta \), respectively \(\omega \wedge \overline{\zeta }\) vanishes identically, where \(\zeta \) is computed from the connection form of \(\nabla \) and where we think of F as fibering over Z. Keeping this in mind we now compute the pullback of the forms \(\zeta \) and \(\overline{\zeta }\) to \([g]^*F\). Recall that the semi-basic 1-forms on F are spanned by the components of \(\omega \), hence there exist unique real-valued functions \(g_{ij}=g_{ji}\) on F so that \(\upsilon ^*g=g_{ij}\omega ^i\otimes \omega ^j\). Likewise, there exist unique real-valued functions \(b_i\) on F so that \(\upsilon ^*\beta =b_i\omega ^i\) and unique real-valued function \(A^i_{jk}=A^i_{kj}\) on F so that \((\upsilon ^*\varphi )^i_j=A^i_{jk}\omega ^k\). The functions \(A^i_{jk}\) satisfy furthermore \(A^k_{ki}=0\) and \(g_{ik}A^k_{jl}=g_{jk}A^k_{il}\) since \(\varphi \) takes values in the endomorphisms of TM that are trace-free and symmetric with respect to g. The Levi-Civita connection \((\psi ^i_j)\) of g is the unique principal \(\mathrm {GL}^+(2,\mathbb {R})\)-connection on F that satisfies

$$\begin{aligned} \mathrm {d}\omega ^i&=-\psi ^i_j\wedge \omega ^j,\\ \mathrm {d}g_{ij}&=g_{ik}\psi ^k_j+g_{kj}\psi ^k_i. \end{aligned}$$

The pullback bundle \(P:=[g]^*F\) is cut out by the equations \(g_{11}=g_{22}\) and \(g_{12}=0\). On P we have

$$\begin{aligned} 0=\mathrm {d}g_{12}&=g_{11}\psi ^1_2+g_{22}\psi ^2_1=g_{11}(\psi ^1_2+\psi ^2_1),\\ 0=\mathrm {d}g_{11}-\mathrm {d}g_{22}&=2g_{11}\psi ^1_1-2g_{22}\psi ^2_2=g_{11}(\psi ^1_1-\psi ^2_2) \end{aligned}$$

On P the condition \(g_{ik}A^k_{jl}=g_{jk}A^k_{il}\) implies \(A^2_{11}=-A^2_{22}\) and \(A^1_{22}=-A^1_{11}\). Writing \(A^1_{11}=a_1\) and \(A^2_{22}=a_2\) and using (5), the connection form \(\theta \) of \(\nabla \) thus becomes

$$\begin{aligned} \theta= & {} \begin{pmatrix} \psi ^1_1 &{}\quad -\psi ^2_1 \\ \psi ^2_1 &{}\quad \psi ^1_1\end{pmatrix}+\begin{pmatrix} b_1\omega ^1 &{}\quad b_1\omega ^2 \\ b_2\omega ^1 &{}\quad b_2\omega ^2\end{pmatrix} -\frac{1}{3}\begin{pmatrix} 2b_1\omega ^1+b_2\omega ^2 &{}\quad b_2\omega ^1 \\ b_1\omega ^2 &{}\quad b_1\omega ^1+2b_2\omega ^2\end{pmatrix}\\&+\begin{pmatrix} a_1\omega ^1-a_2\omega ^2 &{}\quad -a_2\omega ^1-a_1\omega ^2 \\ -a_2\omega ^1-a_1\omega ^2 &{}\quad -a_1\omega ^1+a_2\omega ^2\end{pmatrix} \end{aligned}$$

Introducing the complex notation \(a=a_1+\mathrm {i}a_2\) and \(b=\frac{1}{2}(b_1-\mathrm {i}b_2)\), we obtain from a simple calculation

$$\begin{aligned} \zeta =(\theta ^1_1-\theta ^2_2)+\mathrm {i}(\theta ^1_2+\theta ^2_1)=\frac{4}{3}\overline{b}\omega +2\overline{a}\overline{\omega }, \end{aligned}$$

where we write \(\omega =\omega ^1+\mathrm {i}\omega ^2\).

Finally, since \([g] : M \rightarrow (Z,J_{{\mathfrak {p}}})\) is a holomorphic curve if and only if \(\omega \wedge \zeta \) vanishes identically on P, it follows that \([g] : M \rightarrow (Z,J_{{\mathfrak {p}}})\) is a holomorphic curve if and only if

$$\begin{aligned} 0=\omega \wedge \zeta =2\overline{a}\omega \wedge \overline{\omega } \end{aligned}$$

which is equivalent to \(\varphi \) vanishing identically. This shows (i).

Likewise \([g] : M \rightarrow (Z,{\mathfrak {J}}_{{\mathfrak {p}},dA_g})\) is a pseudo-holomorphic curve if and only if

$$\begin{aligned} 0=\omega \wedge \overline{\zeta }=\frac{4}{3}b\omega \wedge \overline{\omega } \end{aligned}$$

on P. This is equivalent to \(\beta \) vanishing identically. This shows (ii). \(\square \)

As a corollary we obtain:

Corollary 12

Let \((M,{\mathfrak {p}})\) be a projective surface. Then locally \({\mathfrak {p}}\) contains

  1. (i)

    a Weyl connection \({}^{[g]}\nabla \) for some conformal structure [g];

  2. (ii)

    a connection of the form \({}^{\tilde{g}}\nabla +\varphi \) for some Riemannian metric \(\tilde{g}\) and some \(\varphi \in \Gamma (V_0)\) with \(\varphi \) taking values in the endomorphisms that are \(\tilde{g}\)-symmetric.

Remark 13

The first statement of Proposition 9 and Corollary 12 was previously obtained in [19].

Proof of Corollary 12

We first consider the case (ii). We fix a volume form \(\sigma \) on M. We need to show that in a neighbourhood \(U_x\) of every point \(x \in M\) there exists a conformal structure [g] which is a pseudo-holomorphic curve into the total space of the bundle \(\pi : Z \rightarrow M\), where we equip Z with the almost complex structure \({\mathfrak {J}}_{{\mathfrak {p}},\sigma }\). Choose \(j \in Z\) with \(\pi (j)=x\). Recall from Remark 6 that the subspace \(H_j \subset T_jZ\) is invariant under \({\mathfrak {J}}_{{\mathfrak {p}},\sigma }\). Now [24, Theorem III] implies that there exists a pseudo-holomorphic curve \(\Sigma \subset (Z,{\mathfrak {J}}_{{\mathfrak {p}},\sigma })\) which contains j and has \(H_j\) as its tangent space at j. Since \(H_j\subset T_jZ\) is horizontal, the restriction \(\pi ^{\prime }_j|_{H_j} : H_j \rightarrow T_xM\) is an isomorphism. Therefore, the restriction of \(\pi \) to \(\Sigma \) is a local diffeomorphism in some neighbourhood of j. Hence there exists a neighbourhood \(U_x\) of \(x \in M\) and a section \([g] : U_x \rightarrow Z\) so that \([g](U_x)\subset \Sigma \). Thus, \([g] : U_x \rightarrow (Z,{\mathfrak {J}}_{{\mathfrak {p}},\sigma })\) is a pseudo-holomorphic curve in the sense of Remark 10. Taking \(\tilde{g}\) to be the unique metric in [g] with volume form \(\sigma \) and applying Proposition 9 shows the claim. The case (i) follows in the same fashion, except that [24] is not needed, as \(J_{{\mathfrak {p}}}\) is integrable and hence the construction of a holomorphic curve realising a prescribed \(J_{{\mathfrak {p}}}\)-invariant tangent plane is an elementary exercise. \(\square \)

Remark 14

Locally we can always find a holomorphic curve \([g] : M \rightarrow (Z,J_{{\mathfrak {p}}})\), but globally this is not always possible. A properly convex projective structure \({\mathfrak {p}}\) on a closed surface M with \(\chi (M)<0\) admits a holomorphic curve \([g] : M \rightarrow (Z,J_{{\mathfrak {p}}})\) if and only if \({\mathfrak {p}}\) is hyperbolic [22]. One would expect that a corresponding global non-existence result should also hold in the pseudo-holomorphic setting for a suitable class of projective surfaces.

Remark 15

If \((M,{\mathfrak {p}})\) is a closed oriented projective surface of with \(\chi (M)<0\), then there exists at most one holomorphic curve \([g] : M \rightarrow (Z,J_{{\mathfrak {p}}})\), see [21].

Remark 16

Hitchin [15] gave a twistorial construction of (complex) two-dimensional holomorphic projective structures. In the holomorphic category such a projective structure corresponds to a complex surface Z having a family of rational curves with self-intersection number one. Denoting the canonical bundle of Z by \(K_Z\), such a holomorphic projective surface is metrisable if and only if \(K_{Z}^{-2/3}\) admits a holomorphic section which intersects each rational curve in Z at two points [2, 3, 16].

Remark 17

The notion of a projective structure also makes sense in the complex setting and such structures are referred to as c-projective, see [4]. Correspondingly, there is a Kähler metrisability problem of c-projective structures. Some obstructions to Kähler metrisability of a (complex) two-dimensional c-projective structure have been obtained in [18].

We conclude by describing the holomorphic curves for the standard projective structure \({\mathfrak {p}}_0\) on the 2-sphere whose geodesics are the great circles.

Example 18

Let \(S^2\) denote the sphere of radius 1 centered at the origin in \(\mathbb {R}^3\) and g its induced round metric of constant Gauss curvature 1 whose geodesics are the great circles. We equip \(S^2\) with its standard orientation.

Recall that the unit tangent bundle \(\lambda : T_1S^2 \rightarrow S^2\) of \((S^2,g)\) carries a canonical coframing \((\omega _1,\omega _2,\psi )\), where \(\omega _1,\omega _2\) span the 1-forms on \(T_1S^2\) that are semi-basic for the projection \(\lambda \) and \(\psi \) denotes the Levi-Civita connection form of g. The 1-forms \((\omega _1,\omega _2,\psi )\) satisfy the structure equations

$$\begin{aligned} \mathrm {d}\omega _1=-\omega _2\wedge \psi \quad \text {and}\quad \mathrm {d}\omega _2=-\psi \wedge \omega _1\quad \text {and}\quad \mathrm {d}\psi =-\omega _1\wedge \omega _2. \end{aligned}$$
(6)

Let \(\hat{g}\) be a Riemannian metric on \(S^2\) and write \(\lambda ^*\hat{g}=\hat{g}_{ij}\omega _i\otimes \omega _j\) for unique real-valued functions \(\hat{g}_{ij}=\hat{g}_{ji}\) on \(T_1S^2\). Phrased in modern language (c.f. [2]) and applied to the case of the 2-sphere, Liouville’s result [17] implies that if the metrics \(\hat{g}\) and g have the same unparametrised geodesics then the functions \(h_{ij}:=\hat{g}_{ij}(\hat{g}_{11}\hat{g}_{22}-\hat{g}_{12}^2)^{-2/3}\) satisfy the linear differential equations

$$\begin{aligned} \begin{aligned} \mathrm {d}h_{11}&=-2h_1\omega _2+2h_{12}\psi ,\\ \mathrm {d}h_{12}&=h_1\omega _1-h_2\omega _2-(h_{11}-h_{22})\psi ,\\ \mathrm {d}h_{22}&=2h_2\omega _1-2h_{12}\psi , \end{aligned} \end{aligned}$$
(7)

for some smooth real-valued functions \(h_i\) on \(T_1S^2\). Conversely, a solution to (7) on \(T_1S^2\) satisfying \(h_{11}h_{22}-h_{12}^2\ne 0\) gives a Riemannian metric \(\hat{g}\) on \(S^2 \) with \(\lambda ^*\hat{g}=(h_{ij}(h_{11}h_{22}-h_{12}^2)^{-2})\omega _i\otimes \omega _j\) and that has the same unparametrised geodesics as g.

Applying the exterior derivative to the above system of equations implies the existence of a unique real-valued function h on \(T_1S^2\) such that

$$\begin{aligned} \mathrm {d}h_1&=-h_{12}\omega _1+(h_{11}+h)\omega _2+h_2\psi ,\\ \mathrm {d}h_2&=-(h_{22}+h)\omega _1+h_{12}\omega _2-h_1\psi . \end{aligned}$$

Taking yet another exterior derivative gives that

$$\begin{aligned} \mathrm {d}h=-2h_1\omega _1+2h_2\omega _2. \end{aligned}$$

Writing

$$\begin{aligned} \vartheta =\begin{pmatrix} 0 &{}\quad -\omega _1 &{}\quad -\omega _2 \\ \omega _1 &{} 0 &{}\quad -\psi \\ \omega _2 &{}\quad \psi &{}\quad 0 \end{pmatrix}\quad \text {and}\quad H=\begin{pmatrix} h &{}\quad h_2 &{}\quad -h_1 \\ h_2 &{}\quad -h_{22} &{}\quad \quad h_{12} \\ -h_1 &{}\quad h_{12} &{}\quad -h_{11} \end{pmatrix} \end{aligned}$$

the above system of differential equations can be expressed as

$$\begin{aligned} \mathrm {d}H+\vartheta H+H \vartheta ^t=0. \end{aligned}$$

The structure equations (6) imply that \(\mathrm {d}\vartheta +\vartheta \wedge \vartheta =0\), hence we may write \(\vartheta =\Xi ^{-1} \mathrm {d}\Xi \) for some diffeomorphism \(\Xi : T_1S^2 \rightarrow \mathrm {SO}(3)\). It follows that the solutions are of the form \(H=\Xi ^{-1}C(\Xi ^{-1})^t\) for some constant symmetric 3-by-3 matrix C. In particular, taking \(C=AA^t\) for some \(A \in \mathrm {SL}(3,\mathbb {R})\), we obtain a solution \(H_A\) providing a metric \(\hat{g}_A\) on \(S^2\) having the great circles as its geodesics.

Finally, in order to construct the holomorphic curve \([\hat{g}_A] : S^2 \rightarrow Z\) from \(H_A\), we interpret Z as an associated bundle to \(T_1S^2\). We will only give a sketch of the construction and refer the reader to [22, §4] for additional details. The orientation and metric turn \(S^2\) into a Riemann surface and hence a conformal structure on \(S^2\) is given in terms of a Beltrami differential. Denoting the canonical bundle of \(S^2\) by \(K_{S^2}\), a Beltrami differential is a section \(\mu \) of \(\overline{K_{S^2}}\otimes K_{S^2}^{-1}\) satisfying \(|\mu (x)|<1\) for all \(x \in S^2\), where \(|\cdot |\) denotes the norm induced by the natural Hermitian bundle metric on \(\overline{K_{S^2}}\otimes K_{S^2}^{-1}\). The Riemannian metric g gives an isomorphism \(\overline{K_{S^2}}\otimes K_{S^2}^{-1}\simeq K_{S^2}^{-2}\) and thus Z may be identified with \(T_1S^2\times _{S^1} \mathbb {D}\), where \(S^1\) acts by usual rotation on \(T_1S^2\) and by \(z \cdot \mathrm {e}^{\mathrm {i}\phi }=z\mathrm {e}^{-2\mathrm {i}\phi }\) on the open unit disk \(\mathbb {D}\subset \mathbb {C}\). A holomorphic curve \([\hat{g}] : S^2 \rightarrow Z\) is therefore represented by a map \(\mu : T_1S^2 \rightarrow \mathbb {D}\). Explicitly, the conformal structure arising from a Riemannian metric \(\hat{g}\) on \(S^2\) is represented by the map

$$\begin{aligned} \mu =\frac{p-q+2\mathrm {i}r}{p+q+2\sqrt{pq-r^2}}, \end{aligned}$$

where we write \(\lambda ^*\hat{g}=p\omega _1\otimes \omega _1+2r\omega _1\circ \omega _2+q\omega _2\otimes \omega _2\) for unique real-valued functions p, q, r on \(T_1S^2\). In our case, the holomorphic curve \([\hat{g}_A] : S^2 \rightarrow Z\) is thus represented by \(\mu \) with

$$\begin{aligned} p=\frac{h_{11}}{(h_{11}h_{22}-h_{12}^2)^2},\qquad r=\frac{h_{12}}{(h_{11}h_{22}-h_{12}^2)^2}, \qquad q=\frac{h_{22}}{(h_{11}h_{22}-h_{12}^2)^2} \end{aligned}$$

and where the functions \(h_{ij}\) arise from \(H_A\) as above.

Remark 19

In the case of the standard projective structure on \(S^2\) the complex surface \((Z,J_{{\mathfrak {p}}_0})\) is biholomorphic to \(\mathbb {CP}^2\setminus \mathbb {RP}^2\) and moreover, the image of a holomorphic curve \([g] : S^2 \rightarrow Z\) is a smooth quadric, see [19]. Trying to explicitly relate the holomorphic curve \([\hat{g}_A]\) to its image quadric does in general however not seem to give manageable expressions.