2 Projective structures and conformal connections

In this section we assemble the essential facts about projective structures on surfaces and conformal connections that will be used during the proof of the main result. Here and throughout the article – unless stated otherwise – all manifolds are assumed to be connected and smoothness, i.e. infinite differentiability, is assumed. Also, we let \(\mathbb{R}^n\) denote the space of column vectors of height \(n\) with real entries and \(\mathbb{R}_n\) the space of row vectors of length \(n\) with real entries so that matrix multiplication \(\mathbb{R}_n \times \mathbb{R}^n \to \mathbb{R}\) is a non-degenerate pairing identifying \(\mathbb{R}_n\) with the dual vector space of \(\mathbb{R}^n\). Finally, we adhere to the convention of summing over repeated indices.

2.1 Projective structures

Recall that the space \(\mathfrak{A}(\Sigma)\) of affine torsion-free connections on a surface \(\Sigma\) is an affine space modelled on the space of sections of the real vector bundle \(V=S^2(T^*\Sigma)\otimes T\Sigma\).¹ We have a canonical trace mapping \(\mathrm{tr} : \Gamma(V) \to \Omega^1(\Sigma)\) as well as an inclusion \[\iota : \Omega^1(\Sigma) \to \Gamma(V), \quad \alpha \mapsto \alpha \otimes \mathrm{Id}+\mathrm{Id}\otimes \alpha,\] where we define \[\left(\alpha\otimes \mathrm{Id}\right)(v)w=\alpha(v)w \quad \text{and} \quad\left(\mathrm{Id}\otimes \alpha\right)(v)w=\alpha(w)v,\] for all \(v,w \in T\Sigma\). Consequently, the bundle \(V\) decomposes as \(V=V_0\oplus T^*\Sigma\) where \(V_0\) denotes the trace-free part of \(V\). The projection \(\Gamma(V) \to \Gamma(V_0)\) is given by \[\phi \mapsto \phi_0=\phi-\frac{1}{3}\iota\left(\operatorname{tr}\phi\right).\] Weyl [14] observed that two affine torsion-free connections \(\nabla\) and \(\nabla^{\prime}\) on \(\Sigma\) are projectively equivalent if and only if their difference is pure trace \[(\nabla-\nabla^{\prime})_0=0. \tag{2.1}\] We will denote the space of projective structures on \(\Sigma\) by \(\mathfrak{P}(\Sigma)\). From (2.1) we see that \(\mathfrak{P}(\Sigma)\) is an affine space modelled on the space of smooth sections of \(V_0\simeq S^3(T^*\Sigma)\otimes \Lambda^2(T\Sigma)\).

Cartan [4] (see [8] for a modern exposition) associates to an oriented projective surface \((\Sigma,\mathfrak{p})\) a Cartan geometry of type \((\mathrm{SL}(3,\mathbb{R}),G)\), which consists of a principal right \(G\)-bundle \(\pi : B \to \Sigma\) together with a Cartan connection \(\theta \in \Omega^1(B,\mathfrak{sl}(3,\mathbb{R}))\). The group \(G\simeq \mathbb{R}^2\rtimes \mathrm{GL}^+(2,\mathbb{R})\subset \mathrm{SL}(3,\mathbb{R})\) consists of matrices of the form \[b\rtimes a=\left(\begin{array}{cc} (\det a)^{-1} & b\\ 0 & a\end{array}\right),\] where \(a \in \mathrm{GL}^+(2,\mathbb{R})\) and \(b^t \in \mathbb{R}^2\). The Cartan connection \(\theta\) is an \(\mathfrak{sl}(3,\mathbb{R})\)-valued \(1\)-form on \(B\) which is equivariant with respect to the \(G\)-right action, maps every fundamental vector field \(X_v\) on \(B\) to its generator \(v \in \mathfrak{g}\), and restricts to be an isomorphism on each tangent space of \(B\). Furthermore, the Cartan geometry \((\pi : B \to \Sigma, \theta)\) has the following properties:

• Write \(\theta=(\theta^{\mu}_{\nu})_{\mu,\nu=0..2}\). Let \(X\) be a vector field on \(B\) satisfying \(\theta^i_0(X)=c^i\), \(\theta^i_j(X)=0\) and \(\theta^0_j(X)=0\) for real constants \((c^1,c^2)\neq (0,0)\), where \(i,j=1,2\). Then every integral curve of \(X\) projects to \(\Sigma\) to yield a geodesic of \(\mathfrak{p}\) and conversely every geodesic of \(\mathfrak{p}\) arises in this way;

• an orientation compatible volume form on \(\Sigma\) pulls-back to \(B\) to become a positive multiple of \(\theta^1_0\wedge\theta^2_0\);

• there exist real-valued functions \(W_1,W_2\) on \(B\) such that \[\tag{2.2} \mathrm{d}\theta+\theta\wedge\theta=\left(\begin{array}{ccc} 0 & W_1 \theta^1_0\wedge\theta^2_0 & W_2 \theta^1_0\wedge\theta^2_0\\ 0 & 0 & 0\\ 0 & 0 &0 \end{array}\right).\]

The fibre of \(B\) at a point \(p \in \Sigma\) consists of the \(2\)-jets of orientation preserving local diffeomorphisms \(\varphi\) with source \(0\in \mathbb{R}^2\) and target \(p\), so that \(\varphi^{-1}\) maps the geodesics of \(\mathfrak{p}\) passing through \(p\) to curves in \(\mathbb{R}^2\) having vanishing curvature at \(0\). The structure group \(G\) consists of the \(2\)-jets of orientation preserving fractional-linear transformations with source and target \(0 \in \mathbb{R}^2\). Explicitly, the identification between the matrix Lie group \(G\) and the Lie group of such \(2\)-jets is given by \(b\rtimes a \mapsto j^2_0f_{a,b}\) where \[\tag{2.3} f_{a,b} : x \mapsto \frac{(\det a) a\cdot x}{1+(\det a)b\cdot x}\] and \(\cdot\) denotes usual matrix multiplication. The group \(G\) acts on \(B\) from the right by pre-composition, that is, \[\tag{2.4} j^2_0\varphi\cdot j^2_0f_{a,b}=j^2_0(\varphi \circ f_{a,b}).\]

2.2 Conformal connections

Recall that an affine torsion-free connection \(\nabla\) on \(\Sigma\) is called a Weyl connection or conformal connection if \(\nabla\) preserves a conformal structure \([g]\) on \(\Sigma\). A torsion-free connection \(\nabla\) is \([g]\)-conformal if for some – and hence any – Riemannian metric \(g\) defining \([g]\) there exists a \(1\)-form \(\beta\in \Omega^1(\Sigma)\) such that \[\tag{2.7} \nabla g = 2 \beta \otimes g.\] Conversely, given a pair \((g,\beta)\) on \(\Sigma\), it follows from Koszul’s identity that there exists a unique affine torsion-free connection \(\nabla\) which satisfies (2.7), namely \[\tag{2.8} {}^{(g,\beta)}\nabla={}^g\nabla+g\otimes \beta^{\sharp}-\iota(\beta),\] where \({}^g\nabla\) denotes the Levi-Civita connection of \(g\) and \(\beta^{\sharp}\) the \(g\)-dual vector field to \(\beta\). For a smooth real-valued function \(u\) on \(\Sigma\) we have \[\tag{2.9} {}^{\exp(2u)g}\nabla={}^g\nabla-g\otimes{}^g\nabla u+\iota(\mathrm{d}u),\] and hence \[\tag{2.10} {}^{(\exp(2u)g,\beta+\mathrm{d}u)}\nabla={}^{(g,\beta)}\nabla.\] Fixing a Riemannian metric \(g\) defining \([g]\) identifies the space of \([g]\)-conformal connections with the space of \(1\)-forms on \(\Sigma\). It follows that the space of \([g]\)-conformal connections is an affine space modelled on \(\Omega^1(\Sigma)\). A conformal structure \([g]\) together with a choice of a particular \([g]\)-conformal connection \(\nabla\) is called a Weyl structure. We will denote the space of Weyl structures on \(\Sigma\) by \(\mathfrak{W}(\Sigma)\). Furthermore, a Weyl structure \(([g],\nabla)\) is called exact if \(\nabla\) is the Levi-Civita connection of a Riemannian metric \(g\) defining \([g]\). From (2.9) we see that the space of exact Weyl structures on \(\Sigma\) is in one-to-one correspondence with the space of Riemannian metrics on \(\Sigma\) modulo constant rescaling.

Let now \(\Sigma\) be oriented (pass to the orientable double cover in case \(\Sigma\) is not orientable) and fix a Riemannian metric \(g\) and a \(1\)-form \(\beta\) on \(\Sigma\). On the bundle \(\upsilon : F^+\to \Sigma\) of positively oriented coframes of \(\Sigma\) there exist unique real-valued functions \(g_{ij}=g_{ji}\) and \(b_i\) such that \[\tag{2.11} \upsilon^*g=g_{ij}\eta^i\otimes \eta^j \quad \text{and} \quad \upsilon^*\beta=b_i\eta^i.\] The \(\mathbb{R}_2\)-valued function \(b=(b_i)\) satisfies the equivariance property \[\tag{2.12} (R_a)^*b=ba.\] The Levi-Civita connection form of \(g\) is the unique \(\mathfrak{gl}(2,\mathbb{R})\)-valued connection \(1\)-form \(\varphi=(\varphi^i_j)\) satisfying \[\mathrm{d}\eta^i=-\varphi^i_j\wedge\eta^j\quad \text{and}\quad \mathrm{d}g_{ij}=g_{kj}\varphi^k_i+g_{ik}\varphi^k_j.\] The exterior derivative of \(\varphi\) can be expressed as \[\mathrm{d}\varphi^i_j=-\varphi^i_k\wedge\varphi^k_j+g_{jk}K\eta^i\wedge\eta^k,\] where the real-valued function \(K\) is constant on the \(\upsilon\)-fibres and hence can be regarded as a function on \(\Sigma\) which is the Gauss-curvature of \(g\). Infinitesimally, (2.12) translates to the existence of real-valued functions \(b_{ij}\) on \(F^+\) satisfying \[\mathrm{d}b_i=b_j\varphi^j_i+b_{ij}\eta^j.\] From (2.8) we see that the connection form \(\zeta=(\zeta^i_j)\) of the \([g]\)-conformal connection \({}^{(g,\beta)}\nabla\) can be written as \[\zeta^i_j=\varphi^i_j+\left(b_kg^{ki}g_{jl}-\delta^i_jb_l-\delta^i_lb_j\right)\eta^l,\] where the functions \(g^{ij}=g^{ji}\) satisfy \(g^{ik}g_{kj}=\delta^i_j\). It follows with the equivariance property of \(\eta\) and (2.11) that the equations \(g_{11}\equiv g_{22}\equiv 1\) and \(g_{12}\equiv 0\) define a reduction \(\lambda : F^+_g \to \Sigma\) of \(\upsilon : F^+ \to \Sigma\) with structure group \(\mathrm{SO}(2)\) which consists of the positively oriented coframes that are also \(g\)-orthonormal. On \(F^+_g\) we obtain \[\begin{aligned} 0&=\mathrm{d}g_{11}=2(g_{11}\varphi^1_1+g_{12}\varphi^2_1)=2\varphi^1_1,\\ 0&=\mathrm{d}g_{22}=2(g_{21}\varphi^1_2+g_{22}\varphi^2_2)=2\varphi^2_2,\\ 0&=\mathrm{d}g_{12}=g_{11}\varphi^1_2+g_{12}\varphi^2_2+g_{12}\varphi^1_1+g_{22}\varphi^2_1=\varphi^1_2+\varphi^2_1. \end{aligned}\] Therefore, writing \(\varphi:=\varphi^2_1\) we have the following structure equations on \(F^+_g\) \[\begin{aligned} \mathrm{d}\eta^1&=-\eta^2\wedge\varphi,\\ \mathrm{d}\eta^2&=\phantom{-}\eta^1\wedge\varphi,\\ \mathrm{d}\varphi&=-K\eta^1\wedge\eta^2,\\ \mathrm{d}b_i&=b_{ij}\eta^j+\varepsilon_{ij}b^j\varphi. \end{aligned}\] Furthermore, the connection form \(\zeta\) pulls-back to \(F^+_g\) to become³ \[\zeta=\begin{pmatrix} -b_1\eta^1-b_2\eta^2 & b_1\eta^2-b_2\eta^1-\varphi\\ -b_1\eta^2+b_2\eta^1+\varphi & -b_1\eta^1-b_2\eta^2\end{pmatrix}=\begin{pmatrix} -\beta & \star \beta -\varphi \\ \varphi -\star \beta & -\beta\end{pmatrix},\] where \(\star\) denotes the Hodge-star with respect to the orientation and metric \(g\). A simple calculation now shows that the components of the Schouten tensor are \[S=\begin{pmatrix} K+b_{11}+b_{22} & -\frac{1}{3}(b_{12}+b_{21})\\ \frac{1}{3}(b_{12}-b_{21}) & K+b_{11}+b_{22}\end{pmatrix}.\] Note that the diagonal entry of \(S\) is \(K-\delta \beta\) where \(\delta\) denotes the co-differential with respect to the orientation and metric \(g\).

If we now apply the formula (2.6) for the Cartan connection of the projective structure defined by \({}^{(g,\beta)}\nabla\) – whilst setting \(\xi \equiv 0\) – we obtain \[\tag{2.13} \phi=\left(\begin{array}{ccc} \frac{2}{3}\beta & (\delta \beta-K)\eta^1+\frac{1}{3}(\star \mathrm{d}\beta)\eta^2 & -\frac{1}{3}(\star \mathrm{d}\beta)\eta^1+(\delta\beta-K)\eta^2\\ \eta^1 & -\frac{1}{3}\beta & \star \beta-\varphi\\ \eta^2 & \varphi-\star\beta &-\frac{1}{3}\beta\end{array}\right).\] Denoting by \((\pi : B\to \Sigma, \theta)\) the Cartan geometry associated to the projective structure defined by \({}^{(g,\beta)}\nabla\), it follows from the uniqueness part of Cartan’s construction that there exists an \(\mathrm{SO}(2)\)-bundle embedding \(\psi : F^+_g \to B\) so that \(\psi^*\theta=\phi\).

For the sake of completeness we also record \[\mathrm{d}\phi+\phi\wedge\phi=\left(\begin{array}{ccc} 0 & \hat W_1 \phi^1_0\wedge\phi^2_0 &\hat W_2 \phi^1_0\wedge\phi^2_0\\ 0 & 0 & 0\\ 0 & 0 &0 \end{array}\right),\] where \[\hat W_1\phi^1_0+\hat W_2\phi^2_0=-\star \mathrm{d}(K-\delta\beta)+\frac{1}{3}\mathrm{d}\star \mathrm{d}\beta-2(K-\delta\beta)\star\beta-\frac{2}{3}\beta\star\mathrm{d}\beta.\]