3 A complex geometry solution for Weyl metrisability

In this section we will show that the Weyl metrisability problem for an oriented projective surface is globally equivalent to finding a section of the bundle of conformal inner products with holomorphic image.

3.1 The bundle of conformal inner products

Recall that a conformal inner product on a real vector space $V$ is an equivalence class $[b]$ of inner products on $V$ , where two inner products are called equivalent if one is a positive multiple of the other.

Let $N$ be a manifold of even dimension $2 n$ and let $F (N) \to N$ be the right principal $GL (2 n, R)$ -bundle of $1$ -frames over $N$ . We embed $GL (n, C)$ as a closed subgroup of $GL (2 n, R)$ . Let $F (N) / GL (n, C) \to N$ be the bundle whose fibre at $p \in N$ consists of the complex structures on $T_{p} N$ . It was observed in [7 , 24] that the choice of an affine connection $\nabla$ on $N$ induces an almost complex structure $J$ on $F (N) / GL (n, C)$ . If $\nabla$ is torsion-free, then $J$ is integrable if and only if the Weyl projective curvature tensor of $\nabla$ vanishes. In fact, $J$ only depends on the projective equivalence class of $\nabla$ . In the case where $N$ is oriented, this almost complex structure $J$ restricts to become an almost complex structure on the subbundle $F^{+} (N) / GL (n, C)$ where $F^{+} (N) \to N$ denotes bundle of positively oriented frames.

For the case of an oriented surface $M$ , the fibre of the bundle $ρ : C (M) = F^{+} (M) / GL (1, C) \to M$ at $p \in M$ may be identified with the space of conformal inner products on $T_{p} M$ . Consequently, a conformal structure on $M$ may also be thought of as a section of the bundle of conformal inner products $ρ : C (M) \to M$ . Note that in two dimensions the Weyl projective curvature tensors vanishes identically for every projective structure $[\nabla]$ . It follows that the almost complex structure $J$ is always integrable.

3.2 The complex surface $C (M)$ and Cartan’s connection

In this subsection we will characterise the complex structure on $C (M)$ in terms of the Cartan geometry $(π : B \to M, θ)$ associated to $[\nabla]$ . To this end let $C \subset H$ be the closed Lie subgroup consisting of elements $h_{a, b}$ with $a \in GL (1, C)$ where we identify $GL (1, C)$ with the non-zero $2$ -by- $2$ matrices of the form $(\begin{array}{rr} x & - y \\ y & x \end{array}) .$ Consider the smooth map $ν : B \to C (M), j_{0}^{2} φ \mapsto [(φ_{*} g_{E})_{φ (0)}] .$

Lemma 3.1

The map $ν : B \to C (M)$ makes $B$ into a right principal $C$ -bundle over $C (M)$ .

Proof. Since $C$ is a closed Lie subgroup of $H$ , it is sufficient to show that $ν$ is a smooth surjection whose fibres are the $C$ -orbits. Clearly $ν$ is smooth and surjective. Suppose $ν (j_{0}^{2} φ) = ν (j_{0}^{2} \tilde{φ})$ for some elements $j_{0}^{2} φ, j_{0}^{2} \tilde{φ} \in B$ . Then these two elements are in the same fibre of $π : B \to M$ , hence there exists $h_{a, b} \in H$ such that $j_{0}^{2} \tilde{φ} = j_{0}^{2} φ \cdot h_{a, b}$ and $\begin{matrix} (3.1) & c {(φ_{*} g_{E})}_{φ (0)} = {({(φ \circ f_{a, b})}_{*} g_{E})}_{φ (0)} \end{matrix}$ which is equivalent to $c (g_{E})_{0} = ({\tilde{a}}_{*} g_{E})_{0},$ where $\tilde{a} \in {GL}^{+} (2, R)$ is the linear map $x \mapsto (det a) a \cdot x$ and $c \in R^{+}$ . This is equivalent to $\tilde{a}$ being in $GL (1, C)$ or $h_{a, b} \in C$ . In other words, the $ν$ fibres are the $C$ -orbits.

We will now use the forms $α_{1} = θ_{0}^{1} + i θ_{0}^{2}$ and $α_{2} = (θ_{2}^{1} + θ_{1}^{2}) + i (θ_{2}^{2} - θ_{1}^{1})$ to define an almost complex structure on $C (M)$ . Note that the forms $α_{1}, α_{2}$ are $ν$ -semibasic, i.e. $α_{i} (X)$ vanishes for every vector field $X \in X (B)$ which is tangent to the fibres of $ν$ .

Proposition 3.2

There exists a unique complex structure $J$ on $C (M)$ such that a complex valued $1$ -form $μ \in A^{1} (C (M), C)$ is of type $(1, 0)$ if and only if $ν^{*} μ$ is a linear combination of ${α_{1}, α_{2}}$ with coefficients in $C^{\infty} (B, C)$ .

Proof. Let $T_{j}^{i}$ denote the vector fields dual to the coframing $θ_{j}^{i}$ . For $ξ \in T C (M)$ define $\begin{matrix} J (ξ) = ν^{'} (- θ_{0}^{2} (\tilde{ξ}) T_{0}^{1} + θ_{0}^{1} (\tilde{ξ}) T_{0}^{2} - \frac{1}{2} (θ_{2}^{2} - θ_{1}^{1}) (\tilde{ξ}) (T_{2}^{1} + T_{1}^{2}) + \\ + \frac{1}{2} (θ_{2}^{1} + θ_{1}^{2}) (\tilde{ξ}) (T_{2}^{2} - T_{1}^{1})), \end{matrix}$ where $\tilde{ξ} \in T B$ satisfies $ν^{'} (\tilde{ξ}) = ξ$ . Any other vector in $T B$ which is mapped to $ξ$ under $ν^{'}$ is of the form $(R_{c})^{'} (\tilde{ξ}) + χ$ for some $c \in C$ and $χ \in \ker ν^{'}$ . Using the identities $(R_{c})^{*} θ = c^{- 1} θ c, ν \circ R_{c} = ν, c \in C$ and the fact that $α_{1}, α_{2}$ are $ν$ -semibasic, it follows from straightforward computations that $J$ is a well defined almost complex structure on $C (M)$ which has all the desired properties. Moreover the structure equations (2.2) imply $\begin{aligned} d α_{1} = & (- 3 θ_{2}^{2} + i (2 θ_{2}^{1} + θ_{1}^{2})) \land α_{1} + (- θ_{0}^{2} + 2 i θ_{0}^{1}) \land α_{2}, \\ d α_{2} = & (θ_{2}^{0} - i θ_{1}^{0}) \land α_{1} + i (θ_{2}^{1} - θ_{1}^{2}) \land α_{2}, \end{aligned}$ and hence, by Newlander-Nirenberg [21], $J$ is integrable. Clearly such a complex structure is unique.

In fact it is not hard to show that every $ρ$ -fibre admits the structure of a Riemann surface biholomorphic to the unit disk $D^{2}$ such that the canonical inclusion into $C (M)$ is a holomorphic embedding.

3.3 The compatibility problem and holomorphic curves

In this subsection we will relate holomorphic curves in $C (M)$ to the Weyl metrisability problem. We will use the following lemma whose prove is elementary and thus omitted.

Lemma 3.3

Let $(X, J)$ be a complex surface, $μ_{1}, μ_{2} \in A^{1} (X, C)$ a basis for the $(1, 0)$ -forms of $J$ and $f : Σ \to X$ a $2$ -submanifold with $f^{*} (Re (μ_{1}) \land Im (μ_{1})) \neq 0.$ Then $f : Σ \to X$ is a holomorphic curve if and only if $f^{*} (μ_{1} \land μ_{2}) = 0$ . Moreover through every point $p \in X$ passes such a holomorphic curve.

Remark 3.4

Here a $2$ -submanifold $f : Σ \to X$ is called a holomorphic curve if $(J \circ f^{'}) (T_{p} Σ) = f^{'} (T_{p} Σ)$ for every $p \in Σ$ .

Let $[g]$ be a conformal structure on $(M, [\nabla])$ and $x : U \to R^{2}$ local orientation preserving coordinates which are isothermal for $[g]$ . Then it is easy to check that the coordinate section $σ_{x} : U \to B$ satisfies $ν \circ σ_{x} = [g] |_{U}$ .

We will now relate conformal structures $[g] : M \to C (M)$ , which are holomorphic curves in $C (M)$ to the eds with independence condition $(I, Ω)$ on $B$ given by $I = ⟨ θ_{0}^{0}, d θ_{0}^{1}, d θ_{0}^{2}, Re (α_{1} \land α_{2}), Im (α_{1} \land α_{2}) ⟩, Ω = Re (α_{1}) \land Im (α_{1}) .$ Note that we may write $(θ_{0}^{1} \land (3 θ_{1}^{2} + θ_{2}^{1})) + i (θ_{0}^{2} \land (θ_{1}^{2} + 3 θ_{2}^{1})) = α_{1} \land α_{2} + 3 i {\bar{α}}_{1} \land θ_{0}^{0} + 2 i d {\bar{α}}_{1}$ where ${\bar{α}}_{1} = θ_{0}^{1} - i θ_{0}^{2}$ . It follows that the eds $(I, Ω)$ equals the eds (2.8).

Lemma 3.5

Let $[g]$ be a conformal structure on $(M, [\nabla])$ and $x = (x^{1}, x^{2}) : U \to R^{2}$ local orientation preserving $[g]$ -isothermal coordinates. Then the coordinate section $σ_{x} : U \to B$ is an integral manifold of $(I, Ω)$ if and only if $[g] |_{U} : U \to C (M)$ is a holomorphic curve.

Proof. Let $s : ρ^{- 1} (U) \to B$ be a local section of the bundle $ν : B \to C (M)$ and let $μ_{1} = s^{*} α_{1}, μ_{2} = s^{*} α_{2}$ be a local basis for the $(1, 0)$ -forms on $ρ^{- 1} (U)$ . Note that such sections exist, since the principal bundle $H \to H / C$ is trivial. Now $ν^{*} μ_{k} = (s \circ ν)^{*} α_{k} = R_{t}^{*} α_{k},$ for some smooth function $t : π^{- 1} (U) \to C$ . Write the elements of $C \subset H$ in the form $c_{z, w} = (\begin{array}{ccr} | z |^{- 2} & Re (w) & Im (w) \\ 0 & Re (z) & - Im (z) \\ 0 & Im (z) & Re (z) \end{array}),$ for some complex numbers $z \neq 0$ and $w$ . Since $θ$ is a Cartan connection, we have $R_{h}^{*} θ = h^{- 1} θ h,$ for every $h \in H$ which yields together with a short computation $\begin{matrix} (3.2) & R_{c_{z, w}}^{*} (\begin{matrix} α_{1} \\ α_{2} \end{matrix}) = (\begin{array}{cc} \bar{z} / | z |^{4} & 0 \\ i \bar{w} / \bar{z} & \bar{z} / z \end{array}) (\begin{matrix} α_{1} \\ α_{2} \end{matrix}) . \end{matrix}$ Using (3.2) it follows $ν^{*} μ_{1} = λ_{1} α_{1}, ν^{*} μ_{2} = λ_{2} α_{1} + λ_{3} α_{2},$ for some smooth functions $λ_{k} : π^{- 1} (U) \to C$ with $λ_{1} λ_{3} \neq 0$ . Then (2.5) and the identity $[g] |_{U} = ν \circ σ_{x}$ yield $\begin{aligned} σ_{x}^{*} (Re (α_{1}) \land Im (α_{1})) = d x^{1} \land d x^{2} & \neq 0, \\ σ_{x}^{*} d α_{1} = d (d x^{1} + i d x^{2}) & = 0, \\ σ_{x}^{*} θ_{0}^{0} & = 0, \end{aligned}$ and $\begin{matrix} (3.3) & ([g] |_{U})^{*} μ_{1} = (ν \circ σ_{x})^{*} μ_{1} = σ_{x}^{*} (λ_{1} α_{1}) = (λ_{1} \circ σ_{x}) (d x^{1} + i d x^{2}) \neq 0, \end{matrix}$ which shows that $([g] |_{U})^{*} (Re (μ_{1}) \land Im (μ_{1})) \neq 0$ . Therefore according to Lemma 3.3, $[g] |_{U} : U \to C (M)$ is a holomorphic curve if and only if $([g] |_{U})^{*} (μ_{1} \land μ_{2}) = (ν \circ σ_{x})^{*} (μ_{1} \land μ_{2}) = (λ_{1} λ_{3} \circ σ_{x}) σ_{x}^{*} (α_{1} \land α_{2}) = 0,$ which finishes the proof.

The eds $(I, Ω)$ precisely governs the Weyl metrisability problem for an oriented projective surface.

Proposition 3.6

Let $[g]$ be a conformal structure on $(M, [\nabla])$ . Then the following two statements are equivalent:

There exists a Weyl connection for $[g]$ on $M$ which is projectively equivalent to $\nabla$ .
The coordinate section $σ_{x} : U \to B$ associated to any local orientation preserving $[g]$ -isothermal coordinate chart $x = (x^{1}, x^{2}) : U \to R^{2}$ is an integral manifold of $(I, Ω)$ .

Proof. (i) $\Rightarrow$ (ii): This direction is an immediate consequence of Lemma 2.2.
(ii) $\Rightarrow$ (i): Let $x : U \to R^{2}$ , be local orientation preserving isothermal coordinates for $[g]$ and $σ_{x} : U \to B$ the corresponding coordinate section which is an integral manifold of $(I, Ω)$ . Fix a $[g] -$ representative $g$ , then $g |_{U} = e^{2 f} x^{*} g_{E}$ for some smooth $f : U \to R$ . Since $σ_{x}$ is an integral manifold, the projective invariants $κ_{i} : U \to R$ with respect to $x$ satisfy $3 κ_{1} = κ_{3}$ and $3 κ_{2} = κ_{0}$ . On $U$ define the $1$ -form $β = - κ_{3} d x^{1} + κ_{0} d x^{2} + d f,$ then the Weyl connection $D^{g, β}$ on $U$ associated to the pair $(g |_{U}, β)$ is projectively equivalent to $\nabla$ . Let $\tilde{x} : \tilde{U} \to R^{2}$ be another local orientation preserving isothermal coordinate chart for $[g]$ overlapping with $U$ . Writing $g |_{\tilde{U}} = e^{2 \tilde{f}} {\tilde{x}}^{*} g_{E}$ for some smooth $\tilde{f} : \tilde{U} \to R$ and ${\tilde{κ}}_{i} : \tilde{U} \to R$ for the projective invariants of $[\nabla]$ with respect to $\tilde{x}$ , then again the Weyl connection $D^{g, \tilde{β}}$ on $\tilde{U}$ associated to the pair $(g |_{\tilde{U}}, \tilde{β})$ with $\tilde{β} = - {\tilde{κ}}_{3} d {\tilde{x}}^{1} + {\tilde{κ}}_{0} d {\tilde{x}}^{2} + d \tilde{f}$ is projectively equivalent to $\nabla$ . On $U \cap \tilde{U}$ we have $[D^{g} + g \otimes β^{♯}] = [\nabla] = [D^{g} + g \otimes {\tilde{β}}^{♯}],$ using Weyl’s result (2.1), this is equivalent to the existence of a $1$ -form $ε$ on $U \cap \tilde{U}$ such that $D_{X}^{g} Y + g (X, Y) β^{#} = D_{X}^{g} Y + g (X, Y) {\tilde{β}}^{#} + ε (X) Y + ε (Y) X$ for every pair of vector fields $X, Y$ on $U \cap \tilde{U}$ . In particular the choice of a basis of $g$ -orthonormal vector fields implies $ε = 0$ . Thus $β = \tilde{β}$ on $U \cap \tilde{U}$ and therefore, using a coordinate cover, $β$ extends to a well defined global $1$ -form which proves the existence of a smooth Weyl connection on $M$ which is projectively equivalent to $\nabla$ .

Summarising the results found so far we have the main

Theorem 3.7

A conformal structure $[g]$ on an oriented projective surface $(M, [\nabla])$ is preserved by a $[\nabla]$ -representative if and only if $[g] : M \to C (M)$ is a holomorphic curve.

Proof. This follows immediately from Lemma 3.5 and Proposition 3.6.

Remark 3.8

It is easy to check that for a given projective structure $[\nabla]$ on $M$ every holomorphic curve $[g] : M \to C (M)$ determines a unique Weyl connection which is projectively equivalent to $\nabla$ . Theorem 3.7 therefore gives a one-to-one correspondence between the Weyl connections on an oriented surface whose unparametrised geodesics are prescribed by a projective structure $[\nabla]$ and sections of the fibre bundle $ρ : C (M) \to M$ which are holomorphic curves. Note also that a conformal structure $[g]$ on an oriented surface determines a unique complex structure whose holomorphic coordinates are given by orientation preserving isothermal coordinates for $[g]$ . It follows that a conformal structure $[g]$ on an oriented projective surface $(M, [\nabla])$ is preserved by a $[\nabla]$ -representative if and only if $[g] : M \to C (M)$ is holomorphic with respect to the complex structure on $C (M)$ and the complex structure on $M$ induced by $[g]$ and the orientation.

Corollary 3.9

An affine torsion-free connection $\nabla$ on a surface $M$ is locally projectively equivalent to a Weyl connection.

Proof. Since the statement is local we may assume that $M$ is oriented. Let $(π : B \to M, θ)$ be the Cartan geometry associated to $[\nabla]$ . For a given point $p \in M$ , choose $q \in C (M)$ with $ρ (q) = p$ and a coordinate neighbourhood $U_{p}$ . Let $μ_{1}, μ_{2}$ be a basis for the $(1, 0)$ -forms on $ρ^{- 1} (U_{p})$ as constructed in Lemma 3.5. Using Lemma 3.3 there exists a complex $2$ -submanifold $f : Σ \to ρ^{- 1} (U_{p})$ passing through $q$ for which $f^{*} (Re (μ_{1}) \land Im (μ_{1})) \neq 0$ . Since the $π : B \to M$ pullback of a volume form on $M$ is a nowhere vanishing multiple of $θ_{0}^{1} \land θ_{0}^{2}$ , the $ρ$ pullback of a volume form on $U_{p}$ is a nowhere vanishing multiple of $Re (μ_{1}) \land Im (μ_{1})$ and hence $ρ \circ f : Σ \to U_{p}$ is a local diffeomorphism. Composing $f$ with the locally available inverse of this local diffeomorphism one gets a local section of the bundle of conformal inner products which is defined in a neighbourhood of $p$ and which is a holomorphic curve. Using Theorem 3.7 it follows that $\nabla$ is locally projectively equivalent to a Weyl connection.

3 A complex geometry solution for Weyl metrisability

3.1 The bundle of conformal inner products

3.2 The complex surface C(M) and Cartan’s connection

3.3 The compatibility problem and holomorphic curves

3.2 The complex surface $C (M)$ and Cartan’s connection