3 Twisted Weyl connections
We will see that timelike/spacelike minimal Lagrangian connections are twisted Weyl connections. In this section we study some properties of this class of connections that we will need later during the classification of spacelike minimal Lagrangian connections.
Let \([g]\) be a conformal structure on the smooth oriented surface \(\Sigma\). By a \([g]\)-Weyl connection on \(\Sigma\) we mean a torsion-free connection on \(T\Sigma\) preserving the conformal structure \([g]\). It follows from Koszul’s identity that a \([g]\)-Weyl connection can be written in the following form \[{}^{(g,\beta)}\nabla={}^g\nabla+g\otimes \beta^{\sharp}-\beta\otimes\mathrm{Id}-\mathrm{Id}\otimes \beta,\] where \(g \in [g]\), \(\beta \in \Omega^1(\Sigma)\) is a \(1\)-form and \(\beta^{\sharp}\) denotes the \(g\)-dual vector field to \(\beta\). We will use the notation \({}^{[g]}\nabla\) to denote a general \([g]\)-Weyl connection.
A twisted Weyl connection \(\nabla\) on \((\Sigma,[g])\) is a connection on the tangent bundle of \(\Sigma\) which can be written as \(\nabla={}^{[g]}\nabla+\alpha\) for some \([g]\)-Weyl connection \({}^{[g]}\nabla\) and some \(1\)-form \(\alpha\) with values in \(\mathrm{End}(T\Sigma)\) satisfying the following properties:
\(\alpha(X)\) is trace-free and \([g]\)-symmetric for all \(X \in \Gamma(T\Sigma)\);
\(\alpha(X)Y=\alpha(Y)X\) for all \(X,Y \in \Gamma(T\Sigma)\).
Note that if \(\alpha\) satisfies the above properties, then \({}^{[g]}\nabla+\alpha\) is torsion-free. Moreover, the covariant \(3\)-tensor obtained by lowering the upper index of \(\alpha\) with a metric \(g \in [g]\) gives a section of \(\Gamma(S^3_0(T^*\Sigma))\). Conversely, every \(\mathrm{End}(T\Sigma)\)-valued \(1\)-form on \(\Sigma\) satisfying the above properties arises in this way. In other words, fixing a Riemannian metric \(g \in [g]\) allows to identify the twist term \(\alpha\) with a cubic differential.
Fixing a metric \(g\in [g]\), the connection form \(\theta=(\theta^i_j)\) of a twisted Weyl connection is given by \[\theta^i_j=\varphi^i_j+\left(b_kg^{ki}g_{jl}-\delta^i_jb_l-\delta^i_lb_j+a^i_{jl}\right)\omega^l,\] where the map \((g_{ij}) : F \to S^2(\mathbb{R}_2)\) represents the metric \(g\), the map \((b_i) : F \to \mathbb{R}_2\) represents the \(1\)-form \(\beta\) and the map \((a^i_{jk}) : F \to \mathbb{R}^2\otimes S^2(\mathbb{R}_2)\) represents the \(1\)-form \(\alpha\). Moreover, \((\varphi^i_j)\) denote the Levi-Civita connection forms of \(g\). Reducing to the bundle \(F_g\) of \(g\)-orthonormal orientation preserving coframes, the connection form becomes \[\theta=\begin{pmatrix} -\beta & \star_g\beta-\varphi \\ \varphi-\star_g\beta & -\beta\end{pmatrix}+\begin{pmatrix} a^1_{11}\omega_1+a^1_{12}\omega_2 & a^1_{12}\omega^1+a^1_{22}\omega_2 \\ a^2_{11}\omega_1+a^2_{12}\omega_2 & a^2_{12}\omega_1+a^2_{22}\omega_2\end{pmatrix},\] where we use the identity \(\upsilon^*\left(\star_g\beta\right)=-b_2\omega_1+b_1\omega_2\). By definition, on \(F_g\) the functions \(a^i_{jk}\) satisfy the identities \[a^{i}_{jk}=a^i_{kj}, \qquad a^{k}_{kj}=0, \qquad \delta_{ki}a^{k}_{jl}=\delta_{kj}a^k_{il}.\] Thus, writing \(c_1=a^1_{11}\) and \(c_2=a^2_{22}\), we obtain \[\theta=\begin{pmatrix} -\beta & \star_g \beta-\varphi \\ \varphi-\star_g\beta & -\beta\end{pmatrix}+\begin{pmatrix} c_1\omega_1-c_2\omega_2 & -c_2\omega_1-c_1\omega_2\\ -c_2\omega_1-c_1\omega_2 & -c_1\omega_1+c_2 \omega_2\end{pmatrix}.\] In order to compute the curvature form of \(\theta\) we first recall that we write \(\upsilon^*\beta=b_i\omega_i\) and since \(b_i\omega_i\) is \(\mathrm{SO}(2)\)-invariant, it follows that there exist unique real-valued functions \(b_{ij}\) on \(F_g\) such that \[\begin{aligned} \mathrm{d}b_1&=b_{11}\omega_1+b_{12}\omega_2+b_2\varphi,\\ \mathrm{d}b_2&=b_{21}\omega_1+b_{22}\omega_2-b_1\varphi.\end{aligned}\] Recall also that the area form of \(g\) satisfies \(\upsilon^*dA_g=\omega_1\wedge\omega_2\) and since \(\star_g 1=dA_g\), we get \[\upsilon^*\delta_g \beta=-(b_{11}+b_{22}),\] as well as \[\upsilon^*\left(\mathrm{d}\!\star_g\!\beta\right)=(b_{11}+b_{22})\omega_1\wedge\omega_2.\] Since \(c_1+\mathrm{i}c_2\) represents a cubic differential on \(\Sigma\), there exist unique real-valued functions \(c_{ij}\) on \(F_g\) such that \[\begin{aligned} \mathrm{d}c_1&=c_{11}\omega_1+c_{12}\omega_2-3c_2\varphi,\\ \mathrm{d}c_2&=c_{21}\omega_1+c_{22}\omega_2+3c_1\varphi.\end{aligned}\] Consequently, a straightforward calculation shows that the curvature form \(\Theta=\mathrm{d}\theta+\theta\wedge\theta\) satisfies \[\tag{3.1} \begin{aligned} \Theta=\begin{pmatrix} -\mathrm{d}\beta & K_gdA_g+\mathrm{d}\star_g\beta-\frac{1}{2}|\alpha|^2_g\omega_1\wedge\omega_2\\ -K_gd A_g- \mathrm{d}\star_g\beta+\frac{1}{2}|\alpha|^2_g\omega_1\wedge\omega_2& -\mathrm{d}\beta \end{pmatrix}\\ + \begin{pmatrix} 2(b_1c_2+b_2c_1)-(c_{12}+c_{21}) & 2(b_1c_1-b_2c_2)+\left(c_{22}-c_{11}\right)\\ 2(b_1c_1-b_2c_2)+\left(c_{22}-c_{11}\right) & 2(-b_1c_2-b_2c_1)+(c_{12}+c_{21})\end{pmatrix} \omega_1\wedge\omega_2, \end{aligned}\] where we use the identity \(\upsilon^*|\alpha|^2_g=4\left((c_1)^2+(c_2)^2\right)\).
3.1 A characterisation of twisted Weyl connections
We obtain a natural differential operator \(\mathrm{D}_{[g]}\) acting on the space \(\mathfrak{A}(\Sigma)\) of torsion-free connections on \(T\Sigma\) \[\mathrm{D}_{[g]} : \mathfrak{A}(\Sigma) \to \Omega^2(\Sigma), \quad \nabla \mapsto \operatorname{tr}_g \mathrm{Ric}(\nabla)dA_g.\] Note that this operator does indeed only depend on the conformal equivalence class of \(g\). A twisted Weyl connection \(\nabla\) on \((\Sigma,[g])\) can be characterised by minimising the integral of \(\mathrm{D}_{[g]}\) among its projective equivalence class \(\mathfrak{p}(\nabla)\).
Suppose \(\nabla^{\prime}={}^{[g]}\nabla+\alpha\) is a twisted Weyl connection on the compact Riemann surface \((\Sigma,[g])\). Then \[\inf_{\nabla \in \mathfrak{p}(\nabla^{\prime})}\int_{\Sigma}\mathrm{D}_{[g]}(\nabla)=4\pi\chi(\Sigma)-\Vert\alpha\Vert^2_g\] and \(4\pi\chi(\Sigma)-\Vert\alpha\Vert^2_g\) is attained precisely on \(\nabla^{\prime}\).
Note that \[\Vert \alpha\Vert^2_{g}=\int_{\Sigma}|\alpha|^2_g\,dA_g\] does only depend on the conformal equivalence class of \(g\).
Proof of Proposition 3.2. Write \(\nabla^{\prime}={}^{(g,\beta)}\nabla+\alpha\) for some Riemannian metric \(g \in [g]\), some \(1\)-form \(\beta\) and some \(\mathrm{End}(T\Sigma)\)-valued \(1\)-form \(\alpha\) on \(\Sigma\) satisfying the properties of Definition 3.1. From (2.4) and the definition of the Schouten tensor it follows that \[\upsilon^*\left(\operatorname{tr}_g\mathrm{Ric}(\nabla^{\prime})dA_g\right)=\Theta^1_2-\Theta^2_1,\] where \(\Theta=(\Theta^i_j)\) denotes the curvature form of \(\nabla^{\prime}\) pulled-back to \(F_g\). Thus, equation (2.4) gives \[\tag{3.2} \operatorname{tr}_g\mathrm{Ric}(\nabla^{\prime})dA_g=2K_g+2\mathrm{d}\star_g\beta-|\alpha|^2_g dA_g\] and hence \[\tag{3.3} \int_\Sigma\operatorname{tr}_g\mathrm{Ric}(\nabla^{\prime})dA_g=4\pi\chi(\Sigma)-\Vert\alpha\Vert^2_g\] by the Stokes and the Gauss–Bonnet theorem.
It is a classical result due to Weyl [37] that two torsion-free connections \(\nabla^{1},\nabla^2\) on \(T\Sigma\) are projectively equivalent if and only if there exists a \(1\)-form \(\gamma\) on \(\Sigma\) such that \(\nabla^{1}-\nabla^2=\gamma\otimes \mathrm{Id}+\mathrm{Id}\otimes \gamma\). It follows that the connections in the projective equivalence class of \(\nabla^{\prime}\) can be written as \[\nabla=\nabla^{\prime}+\gamma\otimes \mathrm{Id}+\mathrm{Id}\otimes \gamma\] with \(\gamma \in \Omega^1(\Sigma)\). A simple computation gives \[\tag{3.4} \mathrm{Ric}(\nabla)=\mathrm{Ric}(\nabla^{\prime})+\gamma^2-\mathrm{Sym}\,\nabla^{\prime}\gamma+3\,\mathrm{d}\gamma,\] where \(\mathrm{Sym} : \Gamma(T^*\Sigma\otimes T^*\Sigma) \to \Gamma(S^2(T^*\Sigma))\) denotes the natural projection. We compute \[\begin{aligned} \operatorname{tr}_g \mathrm{Sym}\nabla^{\prime}\gamma\, dA_g&=\operatorname{tr}_g \mathrm{Sym}\left({}^g\nabla+g \otimes \beta^{\sharp}-\beta\otimes\mathrm{Id}-\mathrm{Id}\otimes \beta+\alpha\right)\gamma\,dA_g\\ &=\mathrm{d}\star_g\gamma+\left(2\gamma(\beta^{\sharp})-\gamma(\beta^{\sharp})-\gamma(\beta^{\sharp})\right)dA_g\\ &=\mathrm{d}\star_g\gamma,\end{aligned}\] where we used that \(\alpha(X)\) is trace-free and \([g]\)-symmetric for all \(X \in\Gamma(T\Sigma)\). Since the last summand of the right hand side of (3.4) is anti-symmetric, we obtain \[\begin{aligned} \int_\Sigma\operatorname{tr}_g \mathrm{Ric}(\nabla)dA_g&=\int_\Sigma\operatorname{tr}_g\mathrm{Ric}(\nabla)+\int_{\Sigma}\operatorname{tr}_g\gamma^2dA_g-\int_{\Sigma}\operatorname{tr}_g\mathrm{Sym}\nabla^{\prime}\gamma\, dA_g\\ &=4\pi\chi(\Sigma)-\Vert\alpha\Vert^2_g+\Vert \gamma\Vert^2_g-\int_{\Sigma}\mathrm{d}\star_g\gamma,\end{aligned}\] thus the claim follows from the Stokes theorem.
In [31] the following result is shown, albeit phrased in different language:
Let \((\Sigma,[g])\) be a Riemann surface. Then every torsion-free connection on \(T\Sigma\) is projectively equivalent to a unique twisted \([g]\)-Weyl connection.
Let \(\mathfrak{P}(\Sigma)\) denote the space of projective structures on \(\Sigma\). Using Proposition 3.2 and Proposition 3.4 we immediately obtain:
Let \((\Sigma,[g])\) be a compact Riemann surface. Then \[\sup_{\mathfrak{p} \in \mathfrak{P}(\Sigma)}\inf_{\nabla \in \mathfrak{p}} \int_{\Sigma}\operatorname{tr}_g \mathrm{Ric}(\nabla)dA_g=4\pi\chi(\Sigma).\]
A twisted Weyl connection \(\nabla\) on \((\Sigma,[g])\) defines an AH structure \((\mathfrak{p}(\nabla),[g])\) in the sense of [16, 17, 18], where \(\mathfrak{p}(\nabla)\) denotes the projective equivalence class arising from \(\nabla\). Moreover, the twisted Weyl connection agrees with the aligned representative of the associated AH structure \((\mathfrak{p}(\nabla),[g])\). In particular, the equations (3.2) and (3.3) have counterparts in the equations (5.8) and (7.12) of [17]. Also, the Proposition 3.4 corresponds to the existence of a unique aligned representative for an AH structure from [17].