3 A complex geometry solution for Weyl metrisability
In this section we will show that the Weyl metrisability problem for an oriented projective surface is globally equivalent to finding a section of the bundle of conformal inner products with holomorphic image.
3.1 The bundle of conformal inner products
Recall that a conformal inner product on a real vector space \(V\) is an equivalence class \([b]\) of inner products on \(V\), where two inner products are called equivalent if one is a positive multiple of the other.
Let \(N\) be a manifold of even dimension \(2n\) and let \(\mathcal{F}(N) \to N\) be the right principal \(\mathrm{GL}(2n,\mathbb{R})\)-bundle of \(1\)-frames over \(N\). We embed \(\mathrm{GL}(n,\mathbb{C})\) as a closed subgroup of \(\mathrm{GL}(2n,\mathbb{R})\). Let \(\mathcal{F}(N)/\mathrm{GL}(n,\mathbb{C}) \to N\) be the bundle whose fibre at \(p\in N\) consists of the complex structures on \(T_pN\). It was observed in [7, 24] that the choice of an affine connection \(\nabla\) on \(N\) induces an almost complex structure \(\mathfrak{J}\) on \(\mathcal{F}(N)/\mathrm{GL}(n,\mathbb{C})\). If \(\nabla\) is torsion-free, then \(\mathfrak{J}\) is integrable if and only if the Weyl projective curvature tensor of \(\nabla\) vanishes. In fact, \(\mathfrak{J}\) only depends on the projective equivalence class of \(\nabla\). In the case where \(N\) is oriented, this almost complex structure \(\mathfrak{J}\) restricts to become an almost complex structure on the subbundle \(\mathcal{F}^+(N)/\mathrm{GL}(n,\mathbb{C})\) where \(\mathcal{F}^+(N) \to N\) denotes bundle of positively oriented frames.
For the case of an oriented surface \(M\), the fibre of the bundle \(\rho : \mathcal{C}(M)=\mathcal{F}^+(M)/\mathrm{GL}(1,\mathbb{C})\to M\) at \(p\in M\) may be identified with the space of conformal inner products on \(T_pM\). Consequently, a conformal structure on \(M\) may also be thought of as a section of the bundle of conformal inner products \(\rho : \mathcal{C}(M) \to M\). Note that in two dimensions the Weyl projective curvature tensors vanishes identically for every projective structure \([\nabla]\). It follows that the almost complex structure \(\mathfrak{J}\) is always integrable.
3.2 The complex surface \(\mathcal{C}(M)\) and Cartan’s connection
In this subsection we will characterise the complex structure on \(\mathcal{C}(M)\) in terms of the Cartan geometry \((\pi : B \to M, \theta)\) associated to \([\nabla]\). To this end let \(C\subset H\) be the closed Lie subgroup consisting of elements \(h_{a,b}\) with \(a \in \mathrm{GL}(1,\mathbb{C})\) where we identify \(\mathrm{GL}(1,\mathbb{C})\) with the non-zero \(2\)-by-\(2\) matrices of the form \[\left(\begin{array}{rr} x & -y \\ y & x \end{array}\right).\] Consider the smooth map \[\nu : B \to \mathcal{C}(M), \; j^2_0\varphi \mapsto \left[(\varphi_*g_E)_{\varphi(0)}\right].\]
The map \(\nu : B \to \mathcal{C}(M)\) makes \(B\) into a right principal \(C\)-bundle over \(\mathcal{C}(M)\).
Proof. Since \(C\) is a closed Lie subgroup of \(H\), it is sufficient to show that \(\nu\) is a smooth surjection whose fibres are the \(C\)-orbits. Clearly \(\nu\) is smooth and surjective. Suppose \(\nu(j^2_0\varphi)=\nu(j^2_0\tilde{\varphi})\) for some elements \(j^2_0\varphi, j^2_0\tilde{\varphi} \in B\). Then these two elements are in the same fibre of \(\pi : B \to M\), hence there exists \(h_{a,b} \in H\) such that \(j^2_0\tilde{\varphi}=j^2_0\varphi \cdot h_{a,b}\) and \[\tag{3.1} c \left(\varphi_*g_E\right)_{\varphi(0)}=\left(\left(\varphi \circ f_{a,b} \right)_*g_E\right)_{\varphi(0)}\] which is equivalent to \[c(g_E)_0=(\tilde{a}_*g_E)_0,\] where \(\tilde{a} \in \mathrm{GL}^+(2,\mathbb{R})\) is the linear map \(x \mapsto (\det a)\, a\cdot x\) and \(c\in \mathbb{R}^+\). This is equivalent to \(\tilde{a}\) being in \(\mathrm{GL}(1,\mathbb{C})\) or \(h_{a,b} \in C\). In other words, the \(\nu\) fibres are the \(C\)-orbits.
We will now use the forms \(\alpha_1=\theta^1_0+\mathrm{i}\theta^2_0\) and \(\alpha_2=(\theta^1_2+\theta^2_1)+\mathrm{i}(\theta^2_2-\theta^1_1)\) to define an almost complex structure on \(\mathcal{C}(M)\). Note that the forms \(\alpha_1, \alpha_2\) are \(\nu\)-semibasic, i.e. \(\alpha_i(X)\) vanishes for every vector field \(X\in\mathfrak{X}(B)\) which is tangent to the fibres of \(\nu\).
There exists a unique complex structure \(J\) on \(\mathcal{C}(M)\) such that a complex valued \(1\)-form \(\mu \in \mathcal{A}^1(\mathcal{C}(M),\mathbb{C})\) is of type \((1,\! 0)\) if and only if \(\nu^*\mu\) is a linear combination of \(\left\{\alpha_1,\alpha_2\right\}\) with coefficients in \(C^{\infty}(B,\mathbb{C})\).
Proof. Let \(T^i_j\) denote the vector fields dual to the coframing \(\theta^i_j\). For \(\xi \in T\mathcal{C}(M)\) define \[\begin{gathered} J(\xi)=\nu^{\prime}\left(-\theta^2_0(\tilde{\xi})T^1_0+\theta^1_0(\tilde{\xi})T^2_0-\frac{1}{2}(\theta^2_2-\theta^1_1)(\tilde{\xi})(T^1_2+T^2_1)+\right.\\\left.+\frac{1}{2}(\theta^1_2+\theta^2_1)(\tilde{\xi})(T^2_2-T^1_1)\right),\end{gathered}\] where \(\tilde{\xi} \in TB\) satisfies \(\nu^{\prime}(\tilde{\xi})=\xi\). Any other vector in \(TB\) which is mapped to \(\xi\) under \(\nu^{\prime}\) is of the form \((R_c)^{\prime}(\tilde{\xi})+\chi\) for some \(c\in C\) and \(\chi \in \ker\nu^{\prime}\). Using the identities \[(R_c)^*\theta=c^{-1}\,\theta\, c, \quad \nu \circ R_c=\nu, \quad c \in C\] and the fact that \(\alpha_1,\alpha_2\) are \(\nu\)-semibasic, it follows from straightforward computations that \(J\) is a well defined almost complex structure on \(\mathcal{C}(M)\) which has all the desired properties. Moreover the structure equations (2.2) imply \[\begin{aligned} \mathrm{d}\alpha_1=&\left(-3\theta^2_2+i\left(2\theta^1_2+\theta^2_1\right)\right)\wedge\alpha_1+\left(-\theta^2_0+2i\theta^1_0\right)\wedge\alpha_2,\\ \mathrm{d}\alpha_2=&\left(\theta^0_2-i\theta^0_1\right)\wedge\alpha_1+i\left(\theta^1_2-\theta^2_1\right)\wedge\alpha_2,\\ \end{aligned}\] and hence, by Newlander-Nirenberg [21], \(J\) is integrable. Clearly such a complex structure is unique.
In fact it is not hard to show that every \(\rho\)-fibre admits the structure of a Riemann surface biholomorphic to the unit disk \(D^2\) such that the canonical inclusion into \(\mathcal{C}(M)\) is a holomorphic embedding.
3.3 The compatibility problem and holomorphic curves
In this subsection we will relate holomorphic curves in \(\mathcal{C}(M)\) to the Weyl metrisability problem. We will use the following lemma whose prove is elementary and thus omitted.
Let \((X,\! J)\) be a complex surface, \(\mu_1,\mu_2 \in \mathcal{A}^1(X,\mathbb{C})\) a basis for the \((1,\! 0)\)-forms of \(J\) and \(f : \Sigma \to X\) a \(2\)-submanifold with \[f^*(\operatorname{\mathrm{Re}}(\mu_1)\wedge\operatorname{\mathrm{Im}}(\mu_1))\neq 0.\] Then \(f : \Sigma \to X\) is a holomorphic curve if and only if \(f^*(\mu_1\wedge\mu_2)=0\). Moreover through every point \(p \in X\) passes such a holomorphic curve.
Here a \(2\)-submanifold \(f : \Sigma \to X\) is called a holomorphic curve if \((J\circ f^{\prime})(T_p\Sigma)=f^{\prime}(T_p\Sigma)\) for every \(p \in \Sigma\).
Let \([g]\) be a conformal structure on \((M,[\nabla])\) and \(x : U \to \mathbb{R}^2\) local orientation preserving coordinates which are isothermal for \([g]\). Then it is easy to check that the coordinate section \(\sigma_x : U \to B\) satisfies \(\nu \circ \sigma_x= [g]\vert_U\).
We will now relate conformal structures \([g] : M \to \mathcal{C}(M)\), which are holomorphic curves in \(\mathcal{C}(M)\) to the eds with independence condition \((\mathcal{I},\Omega)\) on \(B\) given by \[\mathcal{I}=\left\langle \theta^0_0, \mathrm{d}\theta^1_0,\mathrm{d}\theta^2_0,\operatorname{\mathrm{Re}}(\alpha_1\wedge\alpha_2),\operatorname{\mathrm{Im}}(\alpha_1\wedge\alpha_2) \right\rangle, \quad \Omega=\operatorname{\mathrm{Re}}(\alpha_1)\wedge\operatorname{\mathrm{Im}}(\alpha_1).\] Note that we may write \[\left(\theta^1_0\wedge(3\theta^2_1+\theta^1_2)\right)+\mathrm{i}\left(\theta^2_0\wedge(\theta^2_1+3\theta^1_2)\right)=\alpha_1\wedge\alpha_2+3\mathrm{i}\bar{\alpha}_1\wedge\theta^0_0+2\mathrm{i}\mathrm{d}\bar{\alpha}_1\] where \(\bar{\alpha}_1=\theta^1_0-i\theta^2_0\). It follows that the eds \((\mathcal{I},\Omega)\) equals the eds (2.8).
Let \([g]\) be a conformal structure on \((M,[\nabla])\) and \(x=(x^1,x^2) : U \to \mathbb{R}^2\) local orientation preserving \([g]\)-isothermal coordinates. Then the coordinate section \(\sigma_x : U \to B\) is an integral manifold of \((\mathcal{I},\Omega)\) if and only if \([g]\vert_U : U \to \mathcal{C}(M)\) is a holomorphic curve.
Proof. Let \(s : \rho^{-1}(U) \to B\) be a local section of the bundle \(\nu : B\to \mathcal{C}(M)\) and let \(\mu_1=s^*\alpha_1, \mu_2=s^*\alpha_2\) be a local basis for the \((1,\! 0)\)-forms on \(\rho^{-1}(U)\). Note that such sections exist, since the principal bundle \(H \to H/C\) is trivial. Now \[\nu^*\mu_k=(s \circ \nu)^*\alpha_k=R_t^*\alpha_k,\] for some smooth function \(t : \pi^{-1}(U) \to C\). Write the elements of \(C\subset H\) in the form \[c_{z,w}=\left(\begin{array}{ccr} \vert z \vert^{-2} & \operatorname{\mathrm{Re}}(w) & \operatorname{\mathrm{Im}}(w) \\ 0 & \operatorname{\mathrm{Re}}(z) & -\operatorname{\mathrm{Im}}(z) \\ 0 & \operatorname{\mathrm{Im}}(z) & \operatorname{\mathrm{Re}}(z) \end{array}\right),\] for some complex numbers \(z \neq 0\) and \(w\). Since \(\theta\) is a Cartan connection, we have \(R_h^*\theta=h^{-1}\theta h,\) for every \(h \in H\) which yields together with a short computation \[\tag{3.2} R_{c_{z,w}}^*\left(\begin{array}{c} \alpha_1 \\ \alpha_2\end{array}\right)=\left(\begin{array}{cc} \bar{z}/\vert z \vert^4 & 0 \\ \mathrm{i}\bar{w}/\bar{z} & \bar{z}/z \end{array}\right)\left(\begin{array}{c} \alpha_1 \\ \alpha_2\end{array}\right).\] Using (3.2) it follows \[\nu^*\mu_1=\lambda_1 \alpha_1, \quad \nu^*\mu_2=\lambda_2 \alpha_1 + \lambda_3 \alpha_2,\] for some smooth functions \(\lambda_k : \pi^{-1}(U) \to \mathbb{C}\) with \(\lambda_1\lambda_3 \neq 0\). Then (2.5) and the identity \([g]\vert_U=\nu \circ \sigma_x\) yield \[\begin{aligned} \sigma_x^*\left(\operatorname{\mathrm{Re}}(\alpha_1)\wedge\operatorname{\mathrm{Im}}(\alpha_1)\right)=\mathrm{d}x^1\wedge \mathrm{d}x^2&\neq 0,\\\sigma_x^*\mathrm{d}\alpha_1=\mathrm{d}(\mathrm{d}x^1+\mathrm{i}\mathrm{d}x^2)&=0,\\ \sigma_x^*\theta^0_0&=0,\\ \end{aligned}\] and \[\tag{3.3} ([g]\vert_U)^*\mu_1=(\nu\circ\sigma_x)^*\mu_1=\sigma_x^*(\lambda_1\alpha_1)=(\lambda_1\circ \sigma_x)(\mathrm{d}x^1+\mathrm{i}\mathrm{d}x^2)\neq 0,\] which shows that \(([g]\vert_U)^*(\operatorname{\mathrm{Re}}(\mu_1)\wedge\operatorname{\mathrm{Im}}(\mu_1))\neq 0\). Therefore according to Lemma 3.3, \([g]\vert_U : U \to \mathcal{C}(M)\) is a holomorphic curve if and only if \[([g]\vert_U)^*(\mu_1\wedge\mu_2)=(\nu\circ\sigma_x)^*(\mu_1\wedge\mu_2)=(\lambda_1\lambda_3\circ \sigma_x)\,\sigma_x^*(\alpha_1\wedge\alpha_2)=0,\] which finishes the proof.
The eds \((\mathcal{I},\Omega)\) precisely governs the Weyl metrisability problem for an oriented projective surface.
Let \([g]\) be a conformal structure on \((M,[\nabla])\). Then the following two statements are equivalent:
There exists a Weyl connection for \([g]\) on \(M\) which is projectively equivalent to \(\nabla\).
The coordinate section \(\sigma_x : U \to B\) associated to any local orientation preserving \([g]\)-isothermal coordinate chart \(x=(x^1,x^2) : U \to \mathbb{R}^2\) is an integral manifold of \((\mathcal{I},\Omega)\).
Proof. (i) \(\Rightarrow\) (ii): This direction is an immediate consequence of Lemma 2.2.
(ii)\(\Rightarrow\) (i): Let \(x : U\to\mathbb{R}^2\), be local orientation preserving isothermal coordinates for \([g]\) and \(\sigma_{x} : U \to B\) the corresponding coordinate section which is an integral manifold of \((\mathcal{I},\Omega)\). Fix a \([g]-\)representative \(g\), then \[g\vert_{U}=e^{2f}x^*g_E\] for some smooth \(f : U \to \mathbb{R}\). Since \(\sigma_x\) is an integral manifold, the projective invariants \(\kappa_i : U \to \mathbb{R}\) with respect to \(x\) satisfy \(3\kappa_1=\kappa_3\) and \(3\kappa_2=\kappa_0\). On \(U\) define the \(1\)-form \[\beta=-\kappa_3 \mathrm{d}x^1+\kappa_0 \mathrm{d}x^2+\mathrm{d}f,\] then the Weyl connection \(\mathrm{D}^{g,\beta}\) on \(U\) associated to the pair \((g\vert_U,\beta)\) is projectively equivalent to \(\nabla\). Let \(\tilde{x} : \tilde{U} \to \mathbb{R}^2\) be another local orientation preserving isothermal coordinate chart for \([g]\) overlapping with \(U\). Writing \(g\vert_{\tilde{U}}=e^{2\tilde{f}}\tilde{x}^*g_E\) for some smooth \(\tilde{f} : \tilde{U} \to \mathbb{R}\) and \(\tilde{\kappa}_i : \tilde{U} \to\mathbb{R}\) for the projective invariants of \([\nabla]\) with respect to \(\tilde{x}\), then again the Weyl connection \(\mathrm{D}^{g,\tilde{\beta}}\) on \(\tilde{U}\) associated to the pair \((g\vert_{\tilde{U}},\tilde{\beta})\) with \[\tilde{\beta}=-\tilde{\kappa}_3\mathrm{d}\tilde{x}^1+\tilde{\kappa}_0\mathrm{d}\tilde{x}^2+\mathrm{d}\tilde{f}\] is projectively equivalent to \(\nabla\). On \(U\cap\tilde{U}\) we have \[\left[\mathrm{D}^g+g\otimes \beta^{\sharp}\right]=[\nabla]=\left[\mathrm{D}^g+g\otimes\tilde{\beta}^{\sharp}\right],\] using Weyl’s result (2.1), this is equivalent to the existence of a \(1\)-form \(\varepsilon\) on \(U\cap\tilde{U}\) such that \[\mathrm{D}^g_{X}Y+g(X,Y)\beta^{\#}=\mathrm{D}^g_{X}Y+g(X,Y)\tilde{\beta}^{\#}+\varepsilon(X)Y+\varepsilon(Y)X\] for every pair of vector fields \(X,Y\) on \(U\cap \tilde{U}\). In particular the choice of a basis of \(g\)-orthonormal vector fields implies \(\varepsilon=0\). Thus \(\beta=\tilde{\beta}\) on \(U \cap \tilde{U}\) and therefore, using a coordinate cover, \(\beta\) extends to a well defined global \(1\)-form which proves the existence of a smooth Weyl connection on \(M\) which is projectively equivalent to \(\nabla\).
Summarising the results found so far we have the main
A conformal structure \([g]\) on an oriented projective surface \((M,[\nabla])\) is preserved by a \([\nabla]\)-representative if and only if \([g] : M\to \mathcal{C}(M)\) is a holomorphic curve.
Proof. This follows immediately from Lemma 3.5 and Proposition 3.6.
It is easy to check that for a given projective structure \([\nabla]\) on \(M\) every holomorphic curve \([g] : M \to \mathcal{C}(M)\) determines a unique Weyl connection which is projectively equivalent to \(\nabla\). Theorem 3.7 therefore gives a one-to-one correspondence between the Weyl connections on an oriented surface whose unparametrised geodesics are prescribed by a projective structure \([\nabla]\) and sections of the fibre bundle \(\rho : \mathcal{C}(M) \to M\) which are holomorphic curves. Note also that a conformal structure \([g]\) on an oriented surface determines a unique complex structure whose holomorphic coordinates are given by orientation preserving isothermal coordinates for \([g]\). It follows that a conformal structure \([g]\) on an oriented projective surface \((M,[\nabla])\) is preserved by a \([\nabla]\)-representative if and only if \([g] : M\to \mathcal{C}(M)\) is holomorphic with respect to the complex structure on \(\mathcal{C}(M)\) and the complex structure on \(M\) induced by \([g]\) and the orientation.
An affine torsion-free connection \(\nabla\) on a surface \(M\) is locally projectively equivalent to a Weyl connection.
Proof. Since the statement is local we may assume that \(M\) is oriented. Let \((\pi : B \to M,\theta)\) be the Cartan geometry associated to \([\nabla]\). For a given point \(p \in M\), choose \(q \in \mathcal{C}(M)\) with \(\rho(q)=p\) and a coordinate neighbourhood \(U_p\). Let \(\mu_1,\mu_2\) be a basis for the \((1,\! 0)\)-forms on \(\rho^{-1}(U_p)\) as constructed in Lemma 3.5. Using Lemma 3.3 there exists a complex \(2\)-submanifold \(f : \Sigma \to \rho^{-1}(U_p)\) passing through \(q\) for which \(f^*(\operatorname{\mathrm{Re}}(\mu_1)\wedge\operatorname{\mathrm{Im}}(\mu_1))\neq 0\). Since the \(\pi : B \to M\) pullback of a volume form on \(M\) is a nowhere vanishing multiple of \(\theta^1_0\wedge\theta^2_0\), the \(\rho\) pullback of a volume form on \(U_p\) is a nowhere vanishing multiple of \(\operatorname{\mathrm{Re}}(\mu_1)\wedge\operatorname{\mathrm{Im}}(\mu_1)\) and hence \(\rho \circ f : \Sigma \to U_p\) is a local diffeomorphism. Composing \(f\) with the locally available inverse of this local diffeomorphism one gets a local section of the bundle of conformal inner products which is defined in a neighbourhood of \(p\) and which is a holomorphic curve. Using Theorem 3.7 it follows that \(\nabla\) is locally projectively equivalent to a Weyl connection.