2 An EDS solution for local Weyl metrisability

In this section we will use the theory of exterior differential systems (eds) to show that locally every smooth affine torsion-free connection on a surface is projectively equivalent to a Weyl connection. The notation and terminology for eds are chosen to be consistent with [2].

2.1 Cartan’s projective connection

Weyl showed [27] that two affine connections with the same torsion \(\bar{\nabla}\) and \(\nabla\) on a smooth manifold \(N\) are projectively equivalent if and only if there exists a (unique) \(1\)-form \(\varepsilon \in \mathcal{A}^1(N)\) such that \[\tag{2.1} \bar{\nabla}_{X}Y-\nabla_{X}Y=\varepsilon(X)\,Y+\varepsilon(Y)\,X\] for every pair of vector fields \(X,Y\) on \(N\). In more geometric terms, (2.1) means that the parallel transports of projectively equivalent connections along any curve agree, when thought of as maps between projective space, thus justifying the name projective structure.

As an application of his method of equivalence, Cartan has shown how to associate a parabolic Cartan geometry to a manifold equipped with a projective structure \([\nabla]\). We will only state Cartan’s result for oriented projective surfaces, i.e. two-dimensional, connected, oriented, \(C^{\infty}\)-manifolds equipped with a \(C^{\infty}\) projective structure. For the general case the reader can consult Cartan’s original paper [6] or [15] for a more modern exposition (see [5] for background on parabolic Cartan geometries).

Let \(H \subset \mathrm{SL}(3,\mathbb{R})\) be the Lie group of matrices of the form \[H=\left\{\left(\begin{array}{cc} \det(a)^{-1} & b \\ 0 & a\end{array}\right)\;\Bigg\vert\; a\in \mathrm{GL}^+(2,\mathbb{R}), \; b^t \in \mathbb{R}^2\right\}.\] The elements of \(H\) will be denoted by \(h_{a,b}\).

Recall that a \(2\)-frame at \(p \in M\) is a \(2\)-jet \(j^2_0\varphi\) of a local diffeomorphism \(\varphi : U_{0} \to M\) which is defined in a neighbourhood of \(0 \in \mathbb{R}^2\) and satisfies \(\varphi(0)=p\). The fibre of \(\pi : B \to M\) at \(p \in M\) precisely consists of those \(2\)-frames \(j^2_0\varphi\) at \(p\) for which \(\varphi\) is orientation preserving at \(0\) and for which \(\varphi^{-1}\circ \gamma\) has vanishing curvature at \(0\) for every \([\nabla]\)-geodesic \(\gamma\) through \(p\). The Lie group \(\tilde{H}\) of \(2\)-jets of orientation preserving linear fractional transformations \(f_{a,b}\) \[x \mapsto \frac{a \cdot x}{1+b\cdot x}, \quad b^t \in \mathbb{R}^2, \;a \in \mathrm{GL}^+(2,\mathbb{R})\] acts smoothly from the right on \(B\) by \(j^2_0\varphi \cdot j^2_0f_{a,b}=j^2_0\left(\varphi \circ f_{a,b}\right)\). Note that \(H\) and \(\tilde{H}\) are isomorphic via the map \(h_{a,b} \mapsto j^2_0f_{\tilde{a},\tilde{b}}\) where \(\tilde{a}=\det(a)a\) and \(\tilde{b}=\det(a)b\). Henceforth we will use this identification whenever needed.

2.2 Coordinate sections of Cartan’s bundle

Let \(x=(x^1,x^2) : U \to \mathbb{R}^2\) be local orientation preserving coordinates on \(M\) and \(\Gamma^i_{kl} : U \to \mathbb{R}\) denote the Christoffel symbols of a representative of \([\nabla]\) with respect to the coordinates \(x\). Then the functions \[\Pi^i_{kl}=\Gamma^i_{kl}-\frac{1}{3}\left(\delta^i_k \sum_{j}\Gamma^j_{jl}+\delta^i_l\sum_{j}\Gamma^j_{jk}\right)\] are projective invariants in the sense that they do not depend on the representative chosen to compute them, but only on \(x\). Locally \([\nabla]\) can be recovered from the projective invariants \(\Pi^i_{kl}\) by defining them to be the Christoffel symbols with respect to \(x\) of an affine torsion-free connection \(\nabla^{\prime}\), which is well defined on \(U\) and projectively equivalent to \([\nabla]\). Consequently, two affine torsion-free connections on \(M\) are projectively equivalent, if and only if their Christoffel symbols give rise to the same functions \(\Pi^i_{kl}\). Associated to the coordinates \(x\) is a coordinate section \(\sigma_x : U \to B\) which assigns to every point \(p \in U\) the \(2\)-frame \(j^2_0\varphi \in B\) at \(p\) defined by \[\tag{2.3} \varphi(0)=p, \quad \partial_k(x\circ \varphi)^i(0)=\delta^i_k, \quad \partial_k\partial_l(x\circ \varphi)^i(0)=-\Pi^i_{kl}(p).\] This section indeed does take values in \(B\) as can be shown with a simple computation. The section \(\sigma_x: U \to B\) satisfies \[\tag{2.4} \sigma_x^*\theta^0_0=0, \quad \sigma_x^*\theta^1_0=\mathrm{d}x^1, \quad \sigma_x^*\theta^2_0=\mathrm{d}x^2,\] thus the structure equations (2.2) yield \[\begin{aligned} 0&=\sigma_x^*\mathrm{d}\theta^0_0=-\sigma_x^*\theta^0_1\wedge \mathrm{d}x^1-\sigma_x^*\theta^0_2\wedge \mathrm{d}x^2,\\ 0&=\sigma_x^*\mathrm{d}\theta^1_0=-\sigma_x^*\theta^1_1\wedge \mathrm{d}x^1-\sigma_x^*\theta^1_2\wedge \mathrm{d}x^2,\\ 0&=\sigma_x^*\mathrm{d}\theta^2_0=-\sigma_x^*\theta^2_1\wedge \mathrm{d}x^1-\sigma_x^*\theta^2_2\wedge \mathrm{d}x^2. \end{aligned}\] Since \(\sigma_x^*(\theta^0_0)=-\sigma_x^*(\theta^1_1+\theta^2_2)=0\) holds, Cartan’s lemma implies that there exist functions \(\kappa_0,\kappa_1,\kappa_2,\kappa_3\) and \(\zeta_1,\zeta_2,\zeta_3\) on \(U\) such that \(\sigma_x^*\theta=\eta_x\) where \[\tag{2.5} \eta_x=\left(\begin{array}{crr} 0& \zeta_1 \mathrm{d}x^1+\zeta_2 \mathrm{d}x^2 & \zeta_2 \mathrm{d}x^1+\zeta_3 \mathrm{d}x^2\\ \mathrm{d}x^1 & -\kappa_1 \mathrm{d}x^1-\kappa_2 \mathrm{d}x^2 & -\kappa_2 \mathrm{d}x^1-\kappa_3 \mathrm{d}x^2 \\ \mathrm{d}x^2 & \kappa_0 \mathrm{d}x^1+\kappa_1 \mathrm{d}x^2 & \kappa_1 \mathrm{d}x^1+\kappa_2 \mathrm{d}x^2\end{array}\right).\] The \(\theta\)-structure equations then imply that the functions \(\zeta_i\) and \(\kappa_j\) satisfy the relations \[\tag{2.6} \begin{aligned} \zeta_1&=\frac{\partial \kappa_1}{\partial x^1}-\frac{\partial \kappa_0}{\partial x^2}+2\kappa_1^2-2\kappa_0\kappa_2,\\ \zeta_2&=\frac{\partial \kappa_2}{\partial x^1}-\frac{\partial \kappa_1}{\partial x^2}+\kappa_1\kappa_2-\kappa_0\kappa_3,\\ \zeta_3&=\frac{\partial \kappa_3}{\partial x^1}-\frac{\partial \kappa_2}{\partial x^2}+2\kappa_2^2-2\kappa_1\kappa_3. \end{aligned}\] In terms of the \(\Pi^i_{kl}\), the functions \(\kappa_i : U \to \mathbb{R}\) can be expressed as \[\kappa_0=\Pi^2_{11}, \quad \kappa_1=\Pi^2_{12},\quad \kappa_2=\Pi^2_{22}, \quad \kappa_3=-\Pi^1_{22}.\] The coefficients \(\Pi^i_{kl}\) of Cartan’s projective connection were discovered independently of Cartan’s work from the invariant theoretic view point by Thomas [26]. They generalise to the \(n\)-dimensional case by replacing \(3\) with \(n+1\).

Note that the coordinate section \(\sigma_x : U \to B\) is an integral manifold of the eds with independence condition \((\mathcal{I},\Omega)\) where \[\mathcal{I}=\left\langle \theta^0_0,\mathrm{d}\theta^1_0,\mathrm{d}\theta^2_0\right\rangle\] and \(\Omega=\theta^1_0\wedge\theta^2_0\). Conversely, if \(s : U \to B\) is an integral manifold of the eds \((\mathcal{I},\Omega)\), then \(\pi \circ s : U \to M\) is a local diffeomorphism, thus locally, without losing generality, we can assume that \(U \subset M\) and \(s\) is a section of \(\pi : B \to M\). Assume \(U\) is simply connected, then \[s^*\mathrm{d}\theta^1_0=s^*\mathrm{d}\theta^2_0=0\] implies that there exist functions \((x^1,x^2) : U \to \mathbb{R}^2\) such that \[s^*\theta^1_0=\mathrm{d}x^1, \quad s^*\theta^2_0=\mathrm{d}x^2.\] Now \(s^*(\theta^1_0\wedge\theta^2_0)=\mathrm{d}x^1\wedge \mathrm{d}x^2 > 0\) implies that \(x=(x^1,x^2) : U \to \mathbb{R}^2\) is a local orientation preserving coordinate system on \(M\) satisfying \(\sigma_x=s\).

2.3 Coordinate sections and Weyl metrisability

Note that an affine torsion-free connection \(\nabla\) preserves a conformal structure \([g]\), if and only if for some (and hence any) Riemannian metric \(g \in [g]\), there exists a \(1\)-form \(\beta\), so that \[\tag{2.7} \nabla g=2 \beta \otimes g.\] The affine torsion-free connections preserving \([g]\) are called Weyl connections for \([g]\). Given a Riemannian metric \(g\) and a \(1\)-form \(\beta\), the affine torsion-free connection \(\mathrm{D}^{g,\beta}\) defined by \[(X,Y) \mapsto \mathrm{D}^g_{X}Y+g(X,Y)\beta^{\sharp}-\beta(X)Y-\beta(Y)X,\] is the unique Weyl connection for \([g]\) solving (2.7). Here \(\mathrm{D}^g\) denotes the Levi-Civita connection of \(g\) and \(\beta^{\sharp}\) the \(g\)-dual vector field to \(\beta\).

On the total space \(B\) of the Cartan geometry of an oriented projective surface \((M,[\nabla])\) consider the eds with independence condition \((\mathcal{I},\Omega)\) defined by \[\tag{2.8} \mathcal{I}=\left\langle \theta^0_0, \mathrm{d}\theta^1_0, \mathrm{d}\theta^2_0,\theta^1_0\wedge(3\,\theta^2_1+\theta^1_2),\theta^2_0\wedge(\theta^2_1+3\,\theta^1_2) \right \rangle, \quad \Omega =\theta^1_0\wedge\theta^2_0.\] This eds is of interest due to the following:

Lemma 2.2 translates the Weyl metrisability problem into finding integral manifolds of the eds \((\mathcal{I},\Omega)\). For the application of the theory of exterior differential systems it is more convenient to work with a linear Pfaffian system. Let \(A=B\times \mathbb{R}^2\) and denote by \(a_i : A \to \mathbb{R}\) the projection onto the \(i\)-th coordinate of \(\mathbb{R}^2\) and by \(\tau : A \to B\) the canonical projection. On \(A\) define \[\omega^1=\tau^*\theta^1_0, \quad \omega^2=\tau^*\theta^2_0,\] and \[\begin{aligned} \vartheta^1&=\tau^*\theta^1_1+a_1\omega^1+a_2\omega^2,\\ \vartheta^2&=\tau^*\theta^2_2-a_1\omega^1-a_2\omega^2,\\ \vartheta^3&=\tau^*\theta^1_2+a_2\omega^1+3a_1\omega^2,\\ \vartheta^4&=\tau^*\theta^2_1-3a_2\omega^1-a_1\omega^2. \end{aligned}\] Then it is straightforward to check that the integral manifolds of the eds \[\mathcal{I}^{\prime}=\left\langle \vartheta^1,\vartheta^2,\vartheta^3,\vartheta^4\right\rangle, \quad \Omega^{\prime}=\omega^1\wedge\omega^2,\] are in one-to-one correspondence with the integral manifolds of \((\mathcal{I},\Omega)\). We follow the strategy explained in [2] to prove: