2 Preliminaries

Throughout the article \(\Sigma\) will denote an oriented smooth \(2\)-manifold without boundary. All manifolds and maps are assumed to be smooth and we adhere to the convention of summing over repeated indices.

2.1 The coframe bundle

We denote by \(\upsilon : F \to \Sigma\) the bundle of orientation preserving coframes whose fibre at \(p \in \Sigma\) consists of the linear isomorphisms \(f : T_p\Sigma \to \mathbb{R}^2\) that are orientation preserving with respect to the fixed orientation on \(\Sigma\) and the standard orientation on \(\mathbb{R}^2\). Recall that \(\upsilon : F \to \Sigma\) is a principal right \(\mathrm{GL}^+(2,\mathbb{R})\)-bundle with right action defined by the rule \(R_a(f)=f\cdot a=a^{-1}\circ f\) for all \(a \in \mathrm{GL}^+(2,\mathbb{R})\). The bundle \(F\) is equipped with a tautological \(\mathbb{R}^2\)-valued \(1\)-form \(\omega=(\omega^i)\) defined by \(\omega_f=f\circ \upsilon^{\prime}_f\), and this \(1\)-form satisfies the equivariance property \(R_a^*\omega=a^{-1}\omega\). A torsion-free connection \(\nabla\) on \(T\Sigma\) corresponds to a \(\mathfrak{gl}(2,\mathbb{R}\))-valued connection \(1\)-form \(\theta=(\theta^i_j)\) on \(F\) satisfying the structure equations \[\tag{2.1} \mathrm{d}\omega=-\theta\wedge \omega,\] \[\tag{2.2} \mathrm{d}\theta=-\theta\wedge\theta+\Theta,\] where \(\Theta\) denotes the curvature \(2\)-form of \(\theta\). The Ricci curvature of \(\nabla\) is the (not necessarily symmetric) covariant \(2\)-tensor field \(\mathrm{Ric}(\nabla)\) on \(\Sigma\) satisfying \[\mathrm{Ric}(\nabla)(X,Y)=\operatorname{tr}\left( Z\mapsto \nabla_Z\nabla_X Y-\nabla_X\nabla_Z Y-\nabla_{[Z,X]}Y\right), \quad Z\in \Gamma(TM),\] for all vector fields \(X,Y\) on \(\Sigma\). Denoting by \(\mathrm{Ric}^{\pm}(\nabla)\) the symmetric (respectively, the anti-symmetric) part of the Ricci curvature of \(\nabla\), so that \(\mathrm{Ric}(\nabla)=\mathrm{Ric}^{+}(\nabla)+\mathrm{Ric}^{-}(\nabla)\), the (projective) Schouten tensor of \(\nabla\) is defined as \[\mathrm{Schout}(\nabla)=\mathrm{Ric}^+(\nabla)-\frac{1}{3}\mathrm{Ric}^{-}(\nabla).\] Since the components of \(\omega\) are a basis for the \(\upsilon\)-semibasic forms on \(F\),¹ it follows that there exist real-valued functions \(S_{ij}\) on \(F\) such that \[\upsilon^*\mathrm{Schout}(\nabla)=\omega^tS\omega=S_{ij}\omega^i\otimes \omega^j,\] where \(S=(S_{ij})\). Note that \[R_a^*S=a^tSa\] for all \(a \in \mathrm{GL}^+(2,\mathbb{R})\), since \(\omega^tS\omega\) is invariant under \(R_a\). In terms of the functions \(S_{ij}\) the curvature \(2\)-form \(\Theta=(\Theta^i_j)\) can be written as² \[\tag{2.3} \Theta^i_j=\left(\delta^i_{[k}S_{l]j}-\delta^i_jS_{[kl]}\right)\omega^k\wedge\omega^l,\] or explicitly \[\tag{2.4} \Theta=\begin{pmatrix} 2S_{21}-S_{12} & S_{22}\\ -S_{11} & S_{21}-2S_{12}\end{pmatrix}\omega^1\wedge\omega^2.\]

2.2 The orthonormal coframe bundle

Recall that if \(g\) is a Riemannian metric on the oriented surface \(\Sigma\), the Levi-Civita connection \((\varphi^i_j)\) of \(g\) is the unique connection on the coframe bundle \(\upsilon : F \to \Sigma\) satisfying \[\begin{aligned} \mathrm{d}\omega^i&=-\varphi^i_j\wedge\omega^j,\\ \mathrm{d}g_{ij}&=g_{ik}\varphi^k_j+g_{kj}\varphi^k_i,\end{aligned}\] where we write \(\upsilon^*g=g_{ij}\omega^i\otimes\omega^j\) for real-valued functions \(g_{ij}=g_{ji}\) on \(F\). Differentiating these equations implies that there exists a unique function \(K_g\), the Gauss curvature of \(g\), so that \[\mathrm{d}\varphi^i_j+\varphi^i_k\wedge\varphi^k_j=g_{jk}K_g\omega^i\wedge\omega^k.\] We may reduce \(F\) to the \(\mathrm{SO}(2)\)-subbundle \(F_g\) consisting of orientation preserving coframes that are also orthonormal with respect to \(g\) , that is, the bundle defined by the equations \(g_{ij}=\delta_{ij}\). On \(F_g\) the identity \(\mathrm{d}g_{ij}= 0\) implies the identities \(\varphi^1_1=\varphi^2_2= 0\) as well as \(\varphi^1_2+\varphi^2_1= 0\). Therefore, writing \(\varphi:=\varphi^2_1\), we obtain the structure equations \[\tag{2.5} \begin{aligned} \mathrm{d}\omega_1&=-\omega_2\wedge\varphi,\\ \mathrm{d}\omega_2&=-\varphi\wedge\omega_1,\\ \mathrm{d}\varphi&=-K_g\omega_1\wedge\omega_2, \end{aligned}\] where \(\omega_i=\delta_{ij}\omega^j\). Continuing to denote the basepoint projection \(F_g \to \Sigma\) by \(\upsilon\), the area form \(dA_g\) of \(g\) satisfies \(\upsilon^*dA_g=\omega_1\wedge\omega_2\). Also, note that a complex-valued \(1\)-form \(\alpha\) on \(\Sigma\) is a \((1,\! 0)\)-form for the complex structure \(J\) induced on \(\Sigma\) by \(g\) and the orientation if and only if \(\upsilon^*\alpha\) is a complex multiple of the complex-valued form \(\omega=\omega_1+\mathrm{i}\omega_2\). In particular, denoting by \(K_{\Sigma}\) the canonical bundle of \(\Sigma\) with respect to \(J\), a section \(A\) of the \(\ell\)-th tensorial power of \(K_{\Sigma}\) satisfies \(\upsilon^*A=a\omega^{\ell}\) for some unique complex-valued function \(a\) on \(F_g\). Denote by \(S^3_0(T^*\Sigma)\) the trace-free part of \(S^3(T^*\Sigma)\) with respect to \([g]\), where \(S^3(T^*\Sigma)\) denotes the third symmetrical power of the cotangent bundle of \(\Sigma\). The proof of the following lemma is an elementary computation and thus omitted.

In complex notation, the structure equations of a cubic differential \(C \in \Gamma(K_{\Sigma}^3)\) can be written as follows. Writing \(\upsilon^*C=c\omega^3\) for a complex-valued function \(c\) on \(F_g\), it follows from the \(\mathrm{SO}(2)\)-equivariance of \(c\omega^3\) that there exist complex-valued functions \(c^{\prime}\) and \(c^{\prime\prime}\) on \(F_g\) such that \[\mathrm{d}c=c^{\prime}\omega+c^{\prime\prime}\overline{\omega}+3\mathrm{i}c\varphi,\] where we write \(\overline{\omega}=\omega_1-\mathrm{i}\omega_2\). Note that the Hermitian metric induced by \(g\) on \(K_{\Sigma}^3\) has Chern connection \(\mathrm{D}\) given by \[c \mapsto \mathrm{d}c-3\mathrm{i}c\varphi.\] In particular, the \((0,\! 1)\)-derivative of \(C\) with respect to \(\mathrm{D}\) is represented by \(c^{\prime\prime}\), that is, \(\upsilon^*(\mathrm{D}^{0,1}C)=c^{\prime\prime}\omega^3\otimes \overline{\omega}\). Since \(\overline{\partial}=\mathrm{D}^{0,1}\), we obtain \[\tag{2.6} \upsilon^*\left(\overline{\partial}C\right)=c^{\prime\prime}\omega^3\otimes\overline{\omega}.\] Also, we record the identity \[\upsilon^*|C|_g^2=|c|^2.\]

Moreover, recall that for \(u \in C^{\infty}(\Sigma)\) we have the following standard identity for the change of the Gauss curvature of a metric \(g\) under conformal rescaling \[K_{\mathrm{e}^{2u}g}=\mathrm{e}^{-2u}\left(K_g-\Delta_g u\right),\] where \(\Delta_g=-\left(\delta_g \mathrm{d}+\mathrm{d}\delta_g\right)\) is the negative of the Laplace–Beltrami operator with respect to \(g\). Also, \[dA_{\mathrm{e}^{2u}g}=\mathrm{e}^{2u}dA_g\] for the change of the area form \(dA_g\), \[\Delta_{\mathrm{e}^{2u}g}=\mathrm{e}^{-2u}\Delta_g\] for \(\Delta_g\) acting on functions and \[\delta_{\mathrm{e}^{2u}g}=\mathrm{e}^{-2u}\delta_g\] for the co-differential acting on \(1\)-forms. Finally, the norm of \(C\) changes as \[|C|^2_{\mathrm{e}^{2u}g}=\mathrm{e}^{-6u}|C|^2_{g}.\]

2.3 The cotangent bundle and induced structures

Recall that we have a \(\mathrm{GL}^+(2,\mathbb{R})\) representation \(\varrho\) on \(\mathbb{R}_2\) – the real vector space of row vectors of length two with real entries – defined by the rule \(\varrho(a) \xi=\xi a^{-1}\) for all \(\xi \in \mathbb{R}_2\) and \(a \in \mathrm{GL}^+(2,\mathbb{R})\). The cotangent bundle of \(\Sigma\) is the vector bundle associated to the coframe bundle \(F\) via the representation \(\varrho\), that is, the bundle obtained by taking the quotient of \(F\times \mathbb{R}_2\) by the \(\mathrm{GL}^+(2,\mathbb{R})\)-right action induced by \(\varrho\). Consequently, a \(1\)-form on \(\Sigma\) is represented by an \(\mathbb{R}_2\)-valued function \(\xi\) on \(F\) which is \(\mathrm{GL}^+(2,\mathbb{R})\)-equivariant, that is, \(\xi\) satisfies \(R_a^*\xi=\xi a\) for all \(a \in \mathrm{GL}^+(2,\mathbb{R})\).

Using \(\theta\) we may define a Riemannian metric \(h_{\nabla}\) as well as a symplectic form \(\Omega_{\nabla}\) on \(T^*\Sigma\) as follows. Let \[\pi : F\times \mathbb{R}_2 \to \left(F\times \mathbb{R}_2\right)/\mathrm{GL}^+(2,\mathbb{R})\simeq T^*\Sigma\] denote the quotient projection. Writing \[\psi=\mathrm{d}\xi-\xi\theta-\xi\omega\xi-\omega^tS^t\] or in components \(\psi=(\psi_i)\) with \[\psi_i=\mathrm{d}\xi_i-\xi_j\theta^j_i-\xi_j\omega^j\xi_i-S_{ji}\omega^j,\] we consider the covariant \(2\)-tensor field \(T_{\nabla}=\psi\omega:=\psi_i\otimes \omega^i\). Note that the \(\pi\)-semibasic \(1\)-forms on \(F\times \mathbb{R}_2\) are given by the components of \(\omega\) and \(\mathrm{d}\xi-\xi\theta\), or, equivalently, by the components of \(\omega\) and \(\psi\). Indeed, the components of \(\omega\) are semibasic by the definition of \(\omega\). Moreover, if \(X_v\) for \(v \in \mathfrak{gl}{(2,\mathbb{R})}\) is a fundamental vector field for the coframe bundle \(F \to \Sigma\), that is, the vector field associated to the flow \(R_{\exp(tv)}\), then \[\begin{aligned} (\mathrm{d}\xi-\xi\theta)(X_v)&=-\xi\theta(X_v)+\lim_{t\to 0}\frac{1}{t}\left((R_{\exp(tv)})^*\xi-\xi\right)\\ &=-\xi v+\lim_{t\to 0}\frac{1}{t}\left(\xi\exp(tv)-\xi\right)=-\xi v+\xi v=0,\end{aligned}\] where we have used that the connection \(\theta\) maps a fundamental vector field \(X_v\) to its generator \(v \in \mathfrak{gl}(2,\mathbb{R})\). Since the fundamental vector fields span the vector fields tangent to fibres of \(\pi\), it follows that the components of \(\mathrm{d}\xi-\xi\theta\) are \(\pi\)-semibasic. Moreover, \[\tag{2.7} R_a^*\psi=\mathrm{d}\xi\, a-\xi a a^{-1}\theta a-\xi a a^{-1}\omega\xi a-\omega^t (a^{-1})^t a^tS^ta=\psi a,\] \[\tag{2.8} R_a^*\omega=a^{-1}\omega,\] for all \(a \in \mathrm{GL}^+(2,\mathbb{R})\), it follows that the \(\pi\)-semibasic tensor field \(T_{\nabla}\) is invariant under the \(\mathrm{GL}^+(2,\mathbb{R})\)-right action and hence there exists a unique symmetric covariant \(2\)-tensor field \(h_{\nabla}\) and a unique anti-symmetric covariant \(2\)-tensor field \(\Omega_{\nabla}\) on \(T^*\Sigma\) such that \[\pi^*\left(h_{\nabla}+\Omega_{\nabla}\right)=T_{\nabla}.\] Using the structure equation (2.1), we compute \[\begin{aligned} \pi^*\Omega_{\nabla}&=\mathrm{d}\xi_i\wedge \omega^i-\xi_j\theta^j_i\wedge\omega^i-\xi_i\xi_j\omega^j\wedge\omega^i-S_{ji}\omega^j\wedge\omega^i\\ &=\mathrm{d}\xi_i\wedge\omega^i+\xi_i\mathrm{d}\omega^i-S_{ij}\omega^i\wedge\omega^j\\ &=\mathrm{d}(\xi_i\omega^i)-S_{[ij]}\omega^i\wedge\omega^j.\end{aligned}\] The \(1\)-form \(\xi\omega=\xi_i\omega^i\) on \(F\times \mathbb{R}_2\) is \(\pi\)-semibasic and \(R_a\) invariant, hence the \(\pi\)-pullback of a unique \(1\)-form \(\tau\) on \(T^*\Sigma\) which is the tautological \(1\)-form of \(T^*\Sigma\). Recall that the canonical symplectic form on \(T^*\Sigma\) is \(\Omega_0=\mathrm{d}\tau\), hence \(\Omega_{\nabla}\) defines a symplectic structure on \(T^*\Sigma\) which is the canonical symplectic structure twisted with the (closed) \(2\)-form \(\frac{1}{3}\mathrm{Ric}^{-}(\nabla)\) \[\Omega_{\nabla}=\Omega_0+\Upsilon^*\left(\frac{1}{3}\mathrm{Ric}^{-}(\nabla)\right),\] where \(\Upsilon : T^*\Sigma \to \Sigma\) denotes the basepoint projection. In particular, denoting by \(o : \Sigma \to T^*\Sigma\) the zero \(\Upsilon\)-section, the definition of the Schouten tensor gives \[o^*\Omega_{\nabla}=\frac{1}{3}\mathrm{Ric}^{-}(\nabla).\] This shows:

Which motivates:

Also, we obtain for the symmetric part \[\pi^*h_{\nabla}=\psi\circ \omega:=\psi_1\circ \omega^1+\psi_2\circ \omega^2\] where \(\circ\) denotes the symmetric tensor product. Since the four \(1\)-forms \(\psi_1,\psi_2,\omega^1,\omega^2\) are linearly independent, it follows that \(h_{\nabla}\) is non-degenerate and hence defines a pseudo-Riemannian metric of split signature \((1,1,-1,-1)\) on \(T^*\Sigma\).

From the definition of the Schouten tensor and \(h_{\nabla}\) we immediately obtain \[\tag{2.9} o^*h_{\nabla}=-\mathrm{Ric}^+(\nabla).\] Following standard pseudo-Riemannian submanifold theory, we call a tangent vector \(v\) timelike if \(h_{\nabla}(v,v)<0\) and spacelike if \(h_{\nabla}(v,v)>0\). Thus (2.9) motivates: