2 Preliminaries

Here we collect some standard facts about Riemann surfaces and the unit tangent bundle that will be needed throughout the paper.

2.1 The frame bundle

Throughout the article \(M\) will denote a connected oriented smooth surface with empty boundary. Unless stated otherwise, all maps are assumed to be smooth, i.e., \(C^{\infty}\). Let \(\pi : P \to M\) denote the oriented frame bundle of \(M\) whose fibre at a point \(x \in M\) consists of the linear isomorphisms \(f : \mathbb{R}^2 \to T_xM\) that are orientation preserving, where we equip \(\mathbb{R}^2\) with its standard orientation. The Lie group \(\mathrm{GL}^+(2,\mathbb{R})\) acts transitively from the right on each fibre by the rule \(R_h(f)=f\circ h\) and this action turns \(\pi : P \to M\) into a principal right \(\mathrm{GL}^+(2,\mathbb{R})\)-bundle. The bundle \(P\) is equipped with a tautological \(\mathbb{R}^2\)-valued \(1\)-form \(\omega=(\omega^i)\) defined by \(\omega_f=f^{-1}\circ d\pi_f\) and which satisfies the equivariance property \(R_h^*\omega=h^{-1}\omega\). The components of \(\omega\) are a basis for the \(1\)-forms on \(P\) that are semibasic for the projection \(\pi : P \to M\), i.e., those \(1\)-forms that vanish when evaluated on a vector field that is tangent to the fibres of \(\pi : P \to M\). Therefore, if \(g\) is a Riemannian metric on \(M\), there exist unique real-valued functions \(g_{ij}=g_{ji}\) on \(P\) so that \(\pi^*g=g_{ij}\omega^i\otimes \omega^j\). The Levi-Civita connection \({}^g\nabla\) of \(g\) corresponds to the unique connection form \(\psi=(\psi^i_j) \in \Omega^1(P,\mathfrak{gl}(2,\mathbb{R}))\) satisfying the structure equations \[\tag{2.1} \begin{aligned} d\omega^i&=-\psi^i_j\wedge\omega^j,\\ dg_{ij}&=g_{ik}\psi^k_j+g_{kj}\psi^k_i. \end{aligned}\] The curvature \(\Psi=(\Psi^i_j)\) of \(\psi\) is the \(2\)-form \[\Psi^i_j=d\psi^i_j+\psi^i_k\wedge\psi^k_j=K_gg_{jk}\omega^i\wedge\omega^k,\] where \(K_g\) denotes (the pullback to \(P\) of) the Gauss curvature of \(g\).

2.2 Conformal connections

The conformal frame bundle of the conformal equivalence class \([g]\) of \(g\) is the principal right \(\mathrm{CO}(2)\)-subbundle \(\pi : P_{[g]} \to M\) defined by \[P_{[g]}=\left\{f \in P : g_{11}(f)=g_{22}(f)\;\land \;g_{12}(f)=0\right\}.\] Here \(\mathrm{CO}(2)=\mathbb{R}^+\times \mathrm{SO}(2)\) denotes the linear conformal group whose Lie algebra \(\mathfrak{co}(2)\) is spanned by the matrices \[\begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}\quad \text{and}\quad \begin{pmatrix} 0 & -1 \\ 1 & 0\end{pmatrix}.\] A conformal connection for \([g]\) is principal \(\mathrm{CO}(2)\) connection \[\kappa=\begin{pmatrix} \kappa_1 & -\kappa_2 \\ \kappa_2 & \kappa_1\end{pmatrix}, \quad \kappa_i \in \Omega^1(P_{[g]})\] on \(P_{[g]}\) which is torsion-free, that is, satisfies \[\tag{2.2} d\begin{pmatrix}\omega^1 \\ \omega^2\end{pmatrix}=-\begin{pmatrix} \kappa_1 & -\kappa_2 \\ \kappa_2 & \kappa_1\end{pmatrix}\wedge\begin{pmatrix}\omega^1 \\ \omega^2\end{pmatrix}.\] The standard identification \(\mathbb{R}^2\simeq {\mathbb C}\) gives an identification \(\mathrm{CO}(2)\simeq \mathrm{GL}(1,{\mathbb C})\) and consequently, \(\mathfrak{co}(2)\simeq {\mathbb C}\). In particular, (2.2) takes the form \(d\omega=-\kappa\wedge\omega\) where we think of \(\kappa\) and \(\omega\) as being complex-valued. Writing \(r\mathrm{e}^{i\phi}\) for the elements of \(\mathrm{CO}(2)\), the equivariance property for \(\omega\) implies \((R_{r\\e^{i\phi}})^*\omega=\frac{1}{r}\mathrm{e}^{-i\phi}\omega\). In particular, we see that the \(\pi\)-semibasic complex-valued \(1\)-form \(\omega\) is well-defined on \(M\) up to complex scale. It follows that there exists a unique complex-structure \(J\) on \(M\) whose \((1, \! 0)\)-forms are represented by smooth complex-valued functions \(u\) on \(P_{[g]}\) satisfying the equivariance property \((R_{r\\e^{i\phi}})^*u=r\mathrm{e}^{i\phi}u\), that is, so that \(u\omega\) is invariant under the \(\mathrm{CO}(2)\)-right action. Of course, this is the standard complex structure on \(M\) obtained by rotation of a tangent vector \(v\) counter-clockwise by \(\pi/2\) with respect to \([g]\). Denoting the canonical bundle of \(M\) with respect to \(J\) by \(K\), it follows that the sections of \(L_{m,\ell}:=K^{m}\otimes \overline{K^{\ell}}\) are in one-to-one correspondence with the smooth complex-valued functions \(u\) on \(P_{[g]}\) satisfying the equivariance property \((R_{r\mathrm{e}^{i\phi}})^*u=r^{m+\ell}\mathrm{e}^{i(m-\ell)\phi}u\). Infinitesimally, this translates to the existence of unique smooth complex-valued functions \(u^{\prime}\) and \(u^{\prime\prime}\) on \(P_{[g]}\) so that \[\tag{2.3} du=u^{\prime}\omega+u^{\prime\prime}\overline{\omega}+mu\kappa+\ell u\overline{\kappa}.\]

Recall, if \(\alpha\) is a \(1\)-form on \(M\) taking values in some complex vector bundle over \(M\), the decomposition \(\alpha=\alpha^{\prime}+\alpha^{\prime\prime}\) of \(\alpha\) into its \((1,\! 0)\) part \(\alpha^{\prime}\) and \((0, \! 1)\) part \(\alpha^{\prime\prime}\) is given by \[\alpha^{\prime}=\frac{1}{2}(\alpha-i J\alpha)\quad \text{and}\quad \alpha^{\prime\prime}=\frac{1}{2}(\alpha+i J\alpha)\] where we define \((J\alpha)(v):=\alpha(Jv)\) for all tangent vectors \(v\in TM\). The principal \(\mathrm{CO}(2)\)-connection \(\kappa\) induces a connection on all (real or complex) vector bundles associated to \(P_{[g]}\) and – by standard abuse of notation – we use the same letter \(\mathrm{D}\) to denote the induced connection on the various bundles. If \(s\) is the section of \(L_{m,\ell}\) represented by the function \(u\) satisfying (2.3), then \(\mathrm{D}^{\prime}s:=(\mathrm{D}s)^{\prime}\) is represented by \(u^{\prime}\) and \(\mathrm{D}^{\prime\prime}s:=(\mathrm{D}s)^{\prime\prime}\) is represented by \(u^{\prime\prime}\).

Since \(dg_{11}=dg_{22}\) and \(dg_{12}=0\) on \(P_{[g]}\), it follows from (2.1) that the pullback of the Levi-Civita connection \(\psi\) of \(g\) to \(P_{[g]}\) is a conformal connection. The difference of any two principal \(\mathrm{CO}(2)\)-connections is \(\pi\)-semibasic. Therefore, any other torsion-free principal \(\mathrm{CO}(2)\)-connection \(\kappa\) on \(P_{[g]}\) is of the form \(\kappa=\psi-2\theta_{1}\omega\) for a unique complex-valued function \(\theta_1\) on \(P_{[g]}\). Since \(\kappa\) is a connection, it satisfies the equivariance property \((R_{r\mathrm{e}^{i\phi}})^*\kappa=\frac{1}{r}\mathrm{e}^{-i\phi}\kappa r\mathrm{e}^{i\phi}=\kappa\) and so does \(\psi\). Therefore, \(2\theta_1\omega\) is invariant under the \(\mathrm{CO}(2)\)-right action as well and hence twice the pullback of a \((1,\! 0)\)-form on \(M\) which we denote by \(\theta^{\prime}\). From (2.3) we see that we may think of \(\kappa\) as being the connection form of the induced connection on the anti-canonical bundle \(K^*\). In particular, \(\psi\) may be thought of as being the connection form of the Chern connection induced by \(g\) on \(K^*\). By the definition of the Chern connection, it induces the complex structure of \(K^*\). Since \(\psi\) and \(\kappa\) differ by a \((1,\! 0)\)-form, \(\kappa\) also induces the complex structure of \(K^*\). Consequently, the conformal connections on \(P_{[g]}\) are in one-to-one correspondence with the connections \(\mathrm{D}\) on \(K^*\) inducing the complex structure, that is, \(\mathrm{D}^{\prime\prime}=\overline{\partial}_{K^*}\).

2.3 The unit tangent bundle

For what follows it will be necessary to further reduce \(P_{[g]}\). The unit tangent bundle \[SM=\left\{(x,v) \in TM : g(v,v)=1\right\}\] of \(g\) may be interpreted as the principal right \(\mathrm{SO}(2)\)-subbundle of \(P\) defined by \[SM=\left\{f \in P : g_{ij}(f)=\delta_{ij}\right\}\] On \(SM\) the identities \(dg_{ij}\equiv 0\) imply the identities \(\psi^1_1\equiv\psi^2_2\equiv 0\) and \(\psi^1_2\equiv -\psi^2_1\), so that \(\psi\) is purely imaginary.

Abusing notation by henceforth writing \(\psi\) instead of \(\psi^2_1\), the structure equations thus take the form \[\tag{2.4} d\begin{pmatrix}\omega_1\\ \omega_2\end{pmatrix}=-\begin{pmatrix} 0 & -\psi \\ \psi & 0 \end{pmatrix}\wedge\begin{pmatrix}\omega_1\\ \omega_2\end{pmatrix} \quad \text{and} \quad d\psi=-K_g\,\omega_1\wedge\omega_2,\] where we write \(\omega_i=\delta_{ij}\omega^j\). Note that on \(SM\) the \(1\)-forms \(\omega_1,\omega_2\) take the explicit form \[\tag{2.5} \omega_1(\xi)=g(v,d\pi(\xi))\quad\text{and}\quad \omega_2(\xi)=g(Jv,d\pi(\xi)), \quad \xi \in T_{(x,v)}SM.\] Furthermore, the \(1\)-form \(\psi\) becomes \[\tag{2.6} \psi(\xi)=g\left(\gamma^{\prime\prime}(0),Jv\right)\] where \(\xi \in T_{(x,v)}SM\) and \(\gamma : (-\varepsilon,\varepsilon) \to SM\) is any curve with \(\gamma(0)=(x,v)\), \(\dot{\gamma}(0)=\xi\) and \(\gamma^{\prime\prime}\) denotes the covariant derivative of \(\gamma\) along \(\pi\circ \gamma\).

The three \(1\)-forms \((\omega_1,\omega_2,\psi)\) trivialise the cotangent bundle of \(SM\) and we denote by \((X,H,V)\) the corresponding dual vector fields. The vector field \(X\) is the geodesic vector field of \(g\), \(V\) is the infinitesimal generator of the \(\mathrm{SO}(2)\)-action and \(H\) is the horizontal vector field satisfying \(H=[V,X]\). The structure equations (2.4) imply the additional commutation relations \[[V,H]=-X\quad \text{and}\quad [X,H]=K_gV.\]

Following [7], we use the volume form \(\Theta=\omega_1\wedge\omega_2\wedge\psi\) on \(SM\) to define an inner product \[\langle u,v\rangle=\int_{SM}u \bar v\,\Theta\] for complex-valued functions \(u,v\) on \(SM\) and we denote by \(L^2(SM)\) the corresponding space of square integrable complex-valued functions on \(SM\). The structure equations (2.4) and Cartan’s formula imply that all vector fields \(X,H,V\) preserve \(\Theta\). In particular, \(-i V\) is densely defined and self-adjoint with respect to \(\langle \cdot,\cdot\rangle\). Consequently, we have an orthogonal direct sum decomposition into the kernels \(\mathcal{H}_m\) of the operators \(m\mathrm{Id}+i V\) \[\tag{2.7} L^2(SM)=\bigoplus_{m \in {\mathbb Z}}\mathcal{H}_{m}.\]

2.4 Weyl connections

If \(\theta\) is a \(1\)-form on \(M\), we may write \(\pi^*\theta=\theta\omega_1+V(\theta)\omega_2\), where on the right hand side we think of \(\theta\) as being a real-valued function on \(SM\). Therefore, \(\pi^*\theta^{\prime}=\theta_1\omega\), where \(\theta_1=\frac{1}{2}(\theta-iV\theta)\) and likewise \(\pi^*\theta^{\prime\prime}=\theta_{-1}\overline{\omega}\), where \(\theta_{-1}=\frac{1}{2}(\theta+iV\theta)\). On \(SM\) the connection form \(\kappa\) of a conformal connection thus becomes \(\kappa=i\psi-2\theta_1\omega\) or in matrix notation \[\tag{2.8} \kappa=\begin{pmatrix} 0 & -\psi \\ \psi & 0\end{pmatrix}+\begin{pmatrix} -\theta\omega_1-V(\theta)\omega_2 & -V(\theta)\omega_1+\theta\omega_2 \\ V(\theta)\omega_1-\theta\omega_2 & -\theta\omega_1-V(\theta)\omega_2\end{pmatrix}.\] Finally, without the identification \(\mathbb{R}^2\simeq {\mathbb C}\), we may equivalently think of the connection form \(\kappa\) as the connection form of a torsion-free connection on \(TM\). Writing \(\kappa\) as \[\kappa=\begin{pmatrix} 0 & -\psi \\ \psi & 0 \end{pmatrix}+\begin{pmatrix} \theta\omega_1 & \theta\omega_2 \\ V(\theta)\omega_1 & V(\theta)\omega_2\end{pmatrix}-\begin{pmatrix} 2\theta\omega_1+V(\theta)\omega_2 & V(\theta)\omega_1 \\ \theta\omega_2 & \theta\omega_1+2V(\theta)\omega_2 \end{pmatrix},\] the reader may easily check that \(\kappa\) is the connection form of \[\tag{2.9} \mathrm{D}={}^g\nabla+g\otimes \theta^{\sharp}-\mathrm{Sym}(\theta),\] where the section \(\mathrm{Sym}(\theta)\) of \(S^2(T^*M)\otimes TM\) is defined by the rule \[\mathrm{Sym}(\theta)(v_1,v_2)=\theta(v_1)v_2+\theta(v_2)v_1\] for all tangent vectors \(v_1,v_2 \in TM\). Connections of the form (2.9) for \(g \in [g]\) and \(\theta \in \Omega^1(M)\) are known as Weyl connections for the conformal structure \([g]\). By construction, they preserve \([g]\), that is, the parallel transport maps are angle preserving with respect to \([g]\). Conversely, every torsion-free connection on \(TM\) preserving \([g]\) is of the form (2.9) for some \(g \in [g]\) and \(1\)-form \(\theta\). Summarising, we have the following folklore result: