4 The flat case
In this section we use Theorem 3.7 and results from algebraic geometry to globally identify the set of Weyl connections on the \(2\)-sphere whose geodesics are the great circles.
4.1 Cartan’s connection in the flat case
Consider the projective \(2\)-sphere \(\mathbb{S}^2=\left(\mathbb{R}^3\setminus\{0\}\right)/\mathbb{R}^{+}\) equipped with its “outwards” orientation and projective structure \([\nabla_0]\) whose geodesics are the “great circles” \(\mathbb{S}^1 \subset\mathbb{S}^2\), i.e. subspaces of the form \(E\cap \mathbb{S}^2\) for some \(2\)-plane \(E\subset \mathbb{R}^3\). Let \((\pi : B\to \mathbb{S}^2,\theta)\) denote Cartan’s structure bundle of \((\mathbb{S}^2,[\nabla_0])\). Note that \(\mathrm{SL}(3,\mathbb{R})\) is a right principal \(H\)-bundle over \(\mathbb{S}^2\) with base-point projection \[\tilde{\pi} : \mathrm{SL}(3,\mathbb{R}) \to \mathbb{S}^2, \quad g=(g_0,g_1,g_2) \mapsto [g_0],\] where an element \(g \in \mathrm{SL}(3,\mathbb{R})\) is written as three column vectors \((g_0,g_1,g_2)\). Now let \(\psi : \mathrm{SL}(3,\mathbb{R}) \to B\) be the map which associates to \(g \in \mathrm{SL}(3,R)\) the \(2\)-frame generated by the map \(f_g\) where \[f_g : \mathbb{R}^2 \to \mathbb{S}^2,\quad x \mapsto \left[g\cdot \left(\begin{array}{c} x \\ 1 \end{array}\right)\right].\] It turns out that \(\psi\) is an \(H\)-bundle isomorphism which pulls-back \(\theta\) to the Maurer Cartan form \(\omega\) of \(\mathrm{SL}(3,\mathbb{R})\). In particular since \(\mathrm{d}\omega+\omega\wedge\omega=0\), the functions \(L_i : \mathrm{SL}(3,\mathbb{R}) \to \mathbb{R}\) vanish. Oriented projective surfaces for which the functions \(L_i\) vanish are called projectively flat or simply flat.
4.2 The bundle of conformal inner products in the flat case
In the canonical flat case \((\mathbb{S}^2,[\nabla_0])\) the bundle of conformal inner products \(\rho : \mathcal{C}(\mathbb{S}^2) \to\mathbb{S}^2\) can be identified explicitly. Let \([b] \in \mathcal{C}(\mathbb{S}^2)\) be a conformal inner product at \([x]=\rho([b])\). Identify \(T_{[x]}\mathbb{S}^2\) with \(x^{\perp} \subset\mathbb{R}^3\) where \(\perp\) denotes the orthogonal complement of \(x\) in \(\mathbb{R}^3\) with respect to the Euclidean standard metric. Let \((v_1,v_2) \in \mathbb{R}^3\times \mathbb{R}^3\) be a positively oriented conformal basis for \([b]\). The pair \((v_1,v_2)\) is unique up to a transformation of the form \[\left(r\left(v_1\cos\varphi -v_2\sin\varphi\right),r\left(v_1\sin\varphi+v_2\cos\varphi\right)\right)\] for some \(r \in \mathbb{R}^+\) and \(\varphi \in [0,2\pi]\). Thus we may uniquely identify \([b]\) with an element \(re^{\mathrm{i}\varphi}\left(v_1+\mathrm{i}v_2\right)\) in \(\mathbb{CP}^2\). Note that the image of the standard embedding \(\iota : \mathbb{RP}^2 \to \mathbb{CP}^2\) precisely consists of those elements \([z] \in \mathbb{CP}^2\) for which \(\operatorname{\mathrm{Re}}(z)\) and \(\operatorname{\mathrm{Im}}(z)\) are linearly dependent. It is easy to verify that the just described map is a diffeomorphism \(\mathcal{C}(\mathbb{S}^2) \to \mathbb{CP}^2\setminus\mathbb{RP}^2\) which will be denoted by \(\psi\). Thus \(\rho_0=\rho \circ \psi^{-1} :\mathbb{CP}^2\setminus\mathbb{RP}^2\to \mathbb{S}^2\) makes \(\mathbb{CP}^2\setminus\mathbb{RP}^2\) into a \(D^2\)-bundle over \(\mathbb{S}^2\) whose projection map is explicitly given by \[\rho_0 : \mathbb{CP}^2\setminus\mathbb{RP}^2\to \mathbb{S}^2,\quad [z] \mapsto [\operatorname{\mathrm{Re}}(z)\wedge \operatorname{\mathrm{Im}}(z)].\]
For \((M,[\nabla])=(\mathbb{S}^2,[\nabla_0])\) there exists a biholomorphic fibre bundle isomorphism \(\varphi : \mathcal{C}(M) \to \mathbb{CP}^2\setminus\mathbb{RP}^2\) covering the identity on \(\mathbb{S}^2\).
Proof. Suppose there exists a smooth surjection \(\lambda : \mathrm{SL}(3,\mathbb{R}) \to \mathbb{CP}^2\setminus\mathbb{RP}^2\) whose fibres are the \(C\)-orbits and which pulls back the \((1,\!0)\)-forms of \(\mathbb{CP}^2\setminus\mathbb{RP}^2\) to linear combinations of \(\alpha_1\) and \(\alpha_2\). Then it is easy to check that the map \(\varphi=\lambda \circ \nu^{-1} : \mathcal{C}(\mathbb{S}^2) \to \mathbb{CP}^2\setminus\mathbb{RP}^2\) is well defined and has all the desired properties. Consider the smooth map \(\tilde{\lambda} : B=\mathrm{SL}(3,\mathbb{R}) \to \mathbb{C}^3\) given by \[\left(g_0 \; g_1 \; g_2\right) \mapsto g_0\wedge \left(g_1+\mathrm{i}g_2\right).\] We have \[\tag{4.1} \tilde{\lambda} \circ R_{h_{a,b}} = \det a^{-1} \left(1,\mathrm{i}\right)\cdot a^t \cdot \left(\begin{array}{c}\operatorname{\mathrm{Re}}(\tilde{\lambda}) \\ \operatorname{\mathrm{Im}}(\tilde{\lambda})\end{array}\right)\] and \[\tag{4.2} \mathrm{d}\tilde{\lambda}=\mathrm{i}g_1 \wedge g_2\left(\theta^1_0+\mathrm{i}\theta^2_0\right)+\operatorname{\mathrm{Im}}(\tilde{\lambda})\theta^2_1-\operatorname{\mathrm{Re}}(\tilde{\lambda})\theta^2_2+\mathrm{i}\operatorname{\mathrm{Re}}(\tilde{\lambda})\theta^1_2-\mathrm{i}\operatorname{\mathrm{Im}}(\tilde{\lambda})\theta^1_1.\] Denote by \(q : \mathbb{C}^3\setminus\left\{0\right\} \to \mathbb{CP}^2\) the quotient projection, then (4.1) implies that \(\lambda=q \circ \tilde{\lambda}\) is a smooth surjection onto \(\mathbb{CP}^2\setminus\mathbb{RP}^2\) whose fibres are the \(C\)-orbits. Moreover it follows with (4.2) and straightforward computations that \(\lambda\) pulls back the \((1,\! 0)\)-forms of \(\mathbb{CP}^2\setminus\mathbb{RP}^2\) to linear combinations of \(\alpha_1, \alpha_2\).
4.3 Weyl connections on \(\mathbb{S}^2\) and smooth quadrics \(C \subset \mathbb{CP}^2\)
Theorem 3.7 and Proposition 4.1 now allow to prove:
The Weyl connections on the \(2\)-sphere whose unparametrised geodesics are the great circles are in one-to-one correspondence with the smooth quadrics (i.e. smooth algebraic curves of degree \(2\)) \(\mathcal{C} \subset \mathbb{CP}^2\) without real points.
The proof can be adapted from [3].1 The proof given here relies on Theorem 3.7 and Proposition 4.1. Another proof could be given by using results from [16, 17].
Proof of Corollary 4.2. Suppose \(\nabla\) is a Weyl connection for some conformal structure \([g]\) on \(\mathbb{S}^2\) whose geodesics are the great circles. Then by Theorem 3.7 and Proposition 4.1 \([g] : \mathbb{S}^2 \to \mathbb{CP}^2\setminus\mathbb{RP}^2\) is a holomorphic curve and hence by Chow’s Theorem \(\mathcal{C}=[g](\mathbb{S}^2)\subset \mathbb{CP}^2\setminus\mathbb{RP}^2\) is a smooth algebraic curve whose genus is \(0\), since its the image of the \(2\)-sphere under a section of a fibre bundle. Note that by standard results of algebraic geometry, the genus \(g\) and degree \(d\) of a smooth plane algebraic curve satisfy the relation \(g=(d-1)(d-2)/2\). It follows that \(\mathcal{C}\) is either a line or a quadric. Since every line in \(\mathbb{CP}^2\) has a real point, \(\mathcal{C}\) must be a quadric.
Conversely let \(\mathcal{C}\subset \mathbb{CP}^2\) be a smooth quadric without real points. In order to show that \(\mathcal{C}\) is the image of a smooth section of \(\rho : \mathbb{CP}^2\setminus\mathbb{RP}^2\to \mathbb{S}^2\), which is a holomorphic curve, it is sufficient to show that \(\rho_0\vert_\mathcal{C} : \mathcal{C} \to \mathbb{S}^2\) is a diffeomorphism. The fibre of \(\rho : \mathbb{CP}^2\setminus\mathbb{RP}^2\to \mathbb{S}^2\) at \(u=[(u_1,u_2,u_3)^t] \in \mathbb{S}^2\) is an open subset of the real line \(u_1z_1+u_2z_2+u_3z_3=0\). Smoothness of \(\mathcal{C}\) implies that \(\mathcal{C}\) cannot contain that line as a component and since a quadric in \(\mathbb{CP}^2\) without real points does not have any real tangent lines, it follows from Bezout’s theorem that \(\mathcal{C}\) intersects that line transversely in two distinct points. Clearly the line intersects \(\mathcal{C}\) in \(\mathbb{CP}^1\setminus \mathbb{RP}^1\) which has two connected components: \(\rho^{-1}(u)\cup \rho^{-1}(-u)\). The intersection of the quadric with the line consists of one point in each of these components. It follows that \(\rho\vert_\mathcal{C} : \mathcal{C} \to \mathbb{S}^2\) is a submersion and hence by the compactness of \(\mathcal{C}\) and \(\mathbb{S}^2\) a covering map which is at most \(2\)-to-\(1\). But since \(\mathcal{C}\) is diffeomorphic to \(\mathbb{S}^2\), \(\rho\) must restrict to \(\mathcal{C}\) to be a diffeomorphism onto \(\mathbb{S}^2\). Hence \((\rho\vert_\mathcal{C})^{-1} : \mathbb{S}^2 \to \mathbb{CP}^2\setminus\mathbb{RP}^2\) is a smooth section and holomorphic curve whose image is \(\mathcal{C}\). According to Theorem 3.7 this section determines a unique Weyl connection on \(\mathbb{S}^2\) whose geodesics are the great circles.