% Mobius structures and two dimensional Einstein-Weyl geometry.
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% David M. J. Calderbank
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% Mathematics Institute, University of Warwick. Coventry CV4 7AL.
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% Short title: M\"obius structures and Einstein-Weyl geometry.
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\begin{document}
\title{M\"obius structures\\ and two dimensional Einstein-Weyl geometry}
\author{By \textit{David M. J. Calderbank} at Warwick}
\date{}
\maketitle

\begin{abstract} Two geometric structures on a conformal 2-manifold are
defined: M\"obius structures, which provide a non-integrable version of
the classical notion of a complex projective structure, and Einstein-Weyl
structures, which have been extensively studied in higher dimensions.
These structures are related and a classification of Einstein-Weyl
structures on compact Riemann surfaces is given.
\end{abstract}
%
Conformal $2$-manifolds possess a fascinatingly rich and elegant theory which
can be viewed in many ways: it is the theory of Riemann surfaces in complex
analysis, or of complex curves in algebraic geometry.  In this paper, a purely
differential geometric point of view will be taken, the aim being to introduce
two geometric structures that a conformal $2$-manifold might be equipped with,
and to study the relationship between them. These structures are closely
related to the projective and affine structures of Riemann surface theory.

The first structure can be viewed as a nonintegrable or nonholomorphic version
of a complex projective structure, and will be called a \emphdef{M\"obius
structure}. An integrable or flat M\"obius structure on a conformal
$2$-manifold induces a complex projective structure: the manifold possesses
an atlas whose transition functions are complex M\"obius
transformations. However, contrary to common usage~\cite{KP}, the M\"obius
structures discussed herein are not necessarily integrable: they possess a
curvature, analogous to the Cotton-York tensor of a conformal $3$-manifold,
whose vanishing is equivalent to integrability. M\"obius structures are also
different from real projective structures, in much the same way as conformal
and real projective structures differ in higher dimensions. (In one dimension,
M\"obius and real projective structures do coincide and are always
integrable.)

The other topic of interest here is Einstein-Weyl
geometry~\cite{Cartan,Hitchin3,Weyl}. This is the
geometry of a conformal manifold equipped with a compatible (or conformal)
torsion free connection, such that the symmetric tracefree part of the Ricci
tensor of this connection vanishes. These manifolds generalise Einstein
manifolds in a natural way, and have been investigated in some detail recently
(see~\cite{CP,Gauduchon4,Hitchin3} and references therein).  In~\cite{PT},
Pedersen and Tod posed the problem of classifying compact two dimensional
Einstein-Weyl manifolds---the possible geometries of compact \emph{three}
dimensional Einstein-Weyl manifolds have been classified (locally) by
Tod~\cite{Tod1}. However, the definition just given of an Einstein-Weyl
manifold is vacuous in the two dimensional case and Pedersen and Tod did not
offer an alternative definition. One of the main goals of this paper is to
give explicitly such a definition and present a classification of the compact
orientable examples.

In the first section the conformal and complex analytic descriptions of
surfaces are summarised and compared. In particular the Cauchy-Riemann
equations on various complex line bundles are identified as conformally
invariant differential operators. The aim here is both to illustrate
the special features of two dimensional conformal geometry, and to relate
it to the higher dimensional case. In section~\ref{ms}, M\"obius structures
are defined, together with the corresponding notion of Schwarzian derivative,
which generalises a definition of Osgood and Stowe~\cite{OS}.
The relationship with the usual Schwarzian derivative of complex analysis
is discussed and it is shown that a flat M\"obius structure is a complex
projective structure. In section~\ref{ew}, Einstein-Weyl structures on
surfaces are introduced, and compared to the higher dimensional geometries.
The definition is equivalent to a contracted Bianchi identity for the
compatible torsion free connection. More precisely, a conformal
surface with compatible covariant derivative $D$ is Einstein-Weyl iff
$$D\scal^D-2\divg^D F^D=0,$$
where $\divg^D=\trace D$, $\scal^D$ is the scalar curvature
of $D$, and $F^D$ is the Faraday 2-form of $D$, which is the curvature
of $D$ on an associated real line bundle.

After presenting this definition, I then show that the standard
theory of the Gauduchon gauge~\cite{Gauduchon1,Gauduchon4,Tod1}
continues to work in two dimensions, an observation which leads to the
following result.

\begin{thrm} Let $\Mn$ be a compact orientable surface which is
Einstein-Weyl. Then if the genus of $\Mn$ is greater than one, the
Einstein-Weyl structure is defined by a compatible metric of constant
curvature.
\end{thrm}

Since metrics of constant curvature on compact surfaces are well understood,
it remains to classify the Einstein-Weyl structures on the sphere and the
torus. This classification is carried out in the final section. The solutions
come in one parameter families with the constant curvature metrics as limits.

It is a pleasure to thank Henrik Pedersen for stimulating discussions
and for helpful comments on a preliminary version of this paper.

\section{Natural bundles and operators on a conformal surface}

The elegance of analysis on a conformal $2$-manifold $M$ is largely due to the
fact that $M$ is equivalently (if oriented) a complex $1$-manifold. This will
be viewed, half-jokingly, as a twistor correspondence for two dimensional
conformal geometry: the twistor space of an oriented conformal $2$-manifold is
the manifold itself, thought of as a holomorphic curve. In the unoriented
situation, the twistor space could be taken to be the oriented double cover.

In this section some of the bundles and differential operators arising
naturally on a conformal surface will be discussed, together with the
corresponding (i.e., the same!) objects on the twistor space. Firstly,
however, the notion of a density will be reviewed, since this will be crucial
in the sequel.

\begin{defn} Let $V$ be a real $n$-dimensional vector space and $w$ any real
number. Then a \emphdef{density of weight $w$} or \emphdef{$w$-density} on
$V$ is a map $\rho\colon(\Lambda^nV)\setdif0\to\R$ such that
$\rho(\lambda\omega)=|\lambda|^{-w/n}\rho(\omega)$ for all $\lambda\in\R^*$
and $\omega\in(\Lambda^nV)\setdif0$. The space of densities of weight
$w$ is denoted $L^w=L^w(V)$.
\end{defn}
$L^w$ naturally carries the representation $\lambda.\rho=|\lambda|^w\rho$
of the centre of $\GL(V)$ or equivalently the representation
$A.\rho=|\det A|^{w/n}\rho$ of $\GL(V)$. Note also:
\begin{itemize}
\item $L^w$ is an oriented one dimensional linear space
with dual space $L^{-w}$, and $L^0$ is canonically isomorphic to $\R$.
\item The absolute value defines a map from $\Lambda^nV\dual$ to $L^{-n}$. 
If $V$ is oriented then the $(-n)$-densities can be identified with the
volume forms.
\item The densities of $L^{-1}\tens V$ are canonically isomorphic to $\R$.
\end{itemize}

Suppose $\Mn$ is any manifold. Then the \emphdef{density line bundle}
$L^w=L^w_{T\Mn}$ of $\Mn$ is defined to be the bundle whose fibre at $x\in\Mn$
is $L^w(T_x\Mn)$. Equivalently it is the associated bundle
$\GL(\Mn)\times_{\GL(n)} L^w(n)$ where $\GL(\Mn)$ is the frame bundle of $\Mn$
and $L^w(n)$ is the space of $w$-densities of $\R^n$. The density bundles are
oriented (hence trivialisable) real line bundles, but there is no preferred
trivialisation.

Sections of $L=L^1$ may be thought of as scalar fields with dimensions of
length. This geometric dimensional analysis may also be applied to tensors:
\begin{defn} The tensor bundle $L^w\tens(T\Mn)^j\tens(T\dual\Mn)^k$
(and any subbundle, quotient bundle, element or section) will be said to
have \emphdef{weight} $w+j-k$, or \emphdef{dimensions} of [length]$^{w+j-k}$.
\end{defn}

The notion of density also allows one to define a conformal structure,
not as an equivalence class of inner products, but a genuine inner product:

\begin{defn}
A \emphdef{conformal structure} on a manifold $\Mn$ is an $L^2$ valued inner
product on $T\Mn$. More precisely it is a section $\conf\in\Cinf(\Mn,L^2\tens
S^2T\dual\Mn)$ which is everywhere positive definite. It will always be
assumed that $\conf$ is \emphdef{normalised} in the sense that
$|\det\conf|=1$.

If $\conf$ is viewed as a metric on the \emphdef{weightless tangent bundle}
$L^{-1}\tens T\Mn$, then the normalisation condition means that the
trivialisation of the densities of $L^{-1}\tens T\Mn$ determined by the metric
equals the canonical one. The conformal inner product of tangent vector
fields $X,Y$ will be denoted $\cip{X,Y}$, and is a section of $L^2$.
\end{defn}

Now let $\Mn$ be an oriented conformal $2$-manifold.  The isomorphism
$\SO(2)\cong\Un(1)$ means that $\Mn$ has a preferred almost complex structure
$J$, namely the section of $\SO(T\Mn)$ given by a positive rotation of $\pi/2$
in each tangent plane. Equivalently, the unique positive normalised weightless
$2$-form on $\Mn$ is $\omega(X,Y)=\cip{JX,Y}$, and $\conf+i\omega$ is a
Hermitian form on $T\Mn$ with values in $L^2\tens\C$.

As is well known, any conformal $2$-manifold is conformal flat (the existence
of isothermal coordinates), so the almost complex structure $J$ is integrable
and $T\Mn$ is a holomorphic line bundle.

\begin{notn} When tensoring a vector bundle with some $L^w$ (over $\R$),
the tensor product sign will often be omitted.  The tensor product of complex
line bundles $\cL_1,\cL_2$ over $\C$ will be denoted $\cL_1\ctens\cL_2$, and
the conjugate line bundle to $\cL$ will be denoted $\cconj\cL$ as usual. Note
that $L^2=L\tens L$ (over $\R$) whereas for complex line bundles $\cL$,
$\cpow[2]\cL$ will denote $\cL\ctens\cL$.
\end{notn}

Sections of the complex line bundle $\cpow[-p]{T\Mn}\ctens\cconj[-q]{T\Mn}=
\cpow[p]{T\dual\Mn}\ctens\cconj[q]{T\dual\Mn}$ are
often called \emphdef{$(p,q)$-differentials} (for $p,q\in\Z$).
If $q=0$ there is a $\overline\partial$ operator
$\cpow[p]{T\dual\Mn}\to\cpow[p]{T\dual\Mn}\ctens\cconj{T\dual\Mn}$ whose
kernel consists of the holomorphic sections.

The Hermitian form $\conf+i\omega$ is easily seen to be an isomorphism
$TM\ctens\cconj{TM}\to L^2\tens\C$ (and so $T\Mn\cong L^2\cconj{T\dual\Mn}$).
More generally there are the following correspondences between complex and
conformal geometric objects:
\begin{alignat*}{2}
\cpow[0]{T\dual\Mn}\ctens\cconj[0]{T\dual\Mn}&=\C
 & \R\dsum L^{-2}\Lambda^2T\Mn
 &\cong\R\dsum L^2\Lambda^2T\dual\Mn\cong\lie{co}(T\Mn)\\
\cpow{T\dual\Mn}\ctens\cconj{T\dual\Mn}&\quad
 \text{Area differentials}      & L^{-2}\dsum\Lambda^2T\dual\Mn&\\
\cpow{T\Mn}\ctens\cconj{T\dual\Mn}&\quad
 \text{Beltrami differentials}  & \Sym_0T\Mn&\cong L^{-2}S^2_0T\Mn\\
\cpow{T\dual\Mn}&\quad
 \text{Abelian differentials}   & T\dual\Mn&\\
\cpow[2]{T\dual\Mn}&\quad
 \text{Quadratic differentials} & S^2_0T\dual\Mn&\\
\cpow[p]{T\dual\Mn}&=\cpow[-p]{T\Mn}
 & S^p_0T\dual\Mn&=S^{-p}_0T\Mn
\end{alignat*}
Here $S^p_0$ denotes the subbundle of the $p$th symmetric power consisting of
tensors which are tracefree with respect to the conformal structure, and
$\Sym_0T\Mn$ denotes the bundle of symmetric tracefree endomorphism of $T\Mn$.

The holomorphic $(p,0)$-differentials correspond to solutions of conformally
invariant first order differential equations on $\Mn$. For the Abelian
differentials, the $\overline\partial$ operator is given by the $d+\delta$
operator, taking values in the area differentials.

More generally, to understand natural differential operators on a conformal
manifold $\Mn$ of any dimension, it is convenient to introduce the class of
torsion free covariant derivatives which are compatible with the conformal
structure. These are characterised by the following result.
\begin{fundthm}\tcite{Weyl}\label{FTCG}
On a conformal manifold $\Mn$ there is an affine bijection
between covariant derivatives on $L^1$ and torsion free connections on $T\Mn$
preserving the conformal structure. More explicitly, the covariant derivative
on $T\Mn$ is determined from the one on $L^1$ by the Koszul formula
\begin{align*}
2\cip{D_XY,Z}&=D_X\,\cip{Y,Z}+D_Y\,\cip{X,Z}-D_Z\,\cip{X,Y}\\
&+\cip{[X,Y],Z}-\cip{[X,Z],Y}-\cip{[Y,Z],X},
\end{align*}
where $X,Y,Z$ are vector fields and the conformal inner products
are sections of $L^2$.
\end{fundthm}

\begin{defn}
A covariant derivative on $L^1$ is called a \emphdef{Weyl derivative}, and will
be identified with the induced connection on $T\Mn$. A Weyl derivative
admitting a global nonzero parallel section is said to be \emphdef{exact}.
A compatible metric $g$ induces a global section of $L^1$ and hence an
exact Weyl derivative $D^g$.
\end{defn}

There is no preferred choice of Weyl derivative and so it is important to know
how different choices are related.

Suppose that $D$ and $\tilde D=D+\gamma$ are two Weyl derivatives on $L^1$,
where $\gamma$ is a $1$-form and suppose $\tilde D=D+\Gamma$ on $L^{w-1}T\Mn$,
for some $\lie{co}(T\Mn)$-valued $1$-form $\Gamma$. Then $\Gamma$ is given in
terms of $\gamma$ by the linearised Koszul formula:
$$\Gamma_X=w\gamma(X)\iden+\gamma\skwend X,\quad \text{where}\quad
(\gamma\skwend X)(Y)=\gamma(Y)X-\cip{X,Y}\gamma.$$ Here free use is being
made of the natural isomorphism $\sharp\colon T\dual\Mn\to L^{-2}T\Mn$ given
by the conformal structure.

One may then compute the curvatures of $D$ and $\tilde D$ on
$L^{w-1}T\Mn$. They are easily seen to be related by the formula:
\begin{align*}
R^{\tilde D,w}_{X,Y} = R^{D,w}_{X,Y} &+ w\,d\gamma(X,Y)\iden\\
&+\bigl(D_X\gamma-\gamma(X)\gamma+\tfrac12\cip{\gamma,\gamma}X\bigr)
\skwend Y\\
&-\bigl(D_Y\gamma-\gamma(Y)\gamma+\tfrac12\cip{\gamma,\gamma}Y\bigr)
\skwend X.
\end{align*}

The $d\gamma$ term comes from the curvature of the Weyl derivative on $L^1$.
This is a real $2$-form $F^D$, which (motivated by the links with classical
electromagnetism) will be called the \emphdef{Faraday curvature}.

Because Weyl derivatives have Faraday curvature, taking the trace of $R^{D,w}$
does not necessarily produce a symmetric tensor: the skew part is a multiple
of the Faraday curvature which depends on $w$. Consequently it turns out to
be more natural to work with the \emphdef{normalised Ricci tensor} of $D$
(also called the \emphdef{Rho tensor}):
$$r^D=r^D_0+\tfrac1{2n(n-1)}\scal^D\iden-\tfrac12 F^D.$$

Here $\scal^D$ is the scalar curvature of $D$ (which is a section of $L^{-2}$)
and $r^D_0$ is the normalised symmetric tracefree Ricci tensor
$r^D_0=\frac1{n-2}\sym_0\Ric^D$.

The symmetric tracefree part of the Ricci tensor vanishes automatically in two
dimensions and so, although $r^D_0$ at present \emph{makes no sense} in
dimension two, there is some hope that, once defined more carefully, it won't
be infinite. This will be carried out in the next section.

The reason for the factor $1/(n-2)$ in higher dimensions is that
$$R^{D,w}_{X,Y}= W_{X,Y}+wF^D(X,Y)\iden
-r^D(X)\skwend Y+r^D(Y)\skwend X,$$
where $W$ is the \emphdef{Weyl curvature} of the conformal structure.

The dependence of the normalised Ricci tensor on $D$ is given as follows.

\begin{prop}\label{RicChange}
If $D$ and $\tilde D=D+\gamma$ are Weyl derivatives on $(\Mn^n,\conf)$ then:
\begin{align*}
r^{\tilde D}_0&=r^D_0-\sym_0 D\gamma+(\gamma\tens\gamma
-\tfrac1n\cip{\gamma,\gamma}\iden)\\
\scal^{\tilde D}&=\scal^D-2(n-1)\trace D\gamma-(n-1)(n-2)\cip{\gamma,\gamma}\\
r^{\tilde D}&=r^D-(D\gamma-\gamma\tens\gamma+\tfrac12\cip{\gamma,\gamma}\iden).
\end{align*}
\end{prop}

Some examples of differential operators defined using a Weyl derivative $D$
will now be given, and the above formulae can be used to understand the
dependence of the operator on the choice of $D$.

Firstly, consider sections $X$ of the bundle $L^{w-1}T\Mn$
(for some $w\in\R$). Then $X\mapsto\sym_0 DX$ defines a first order
differential operator on this bundle. (As before $\sym_0$ denotes the
symmetric tracefree part.) If the Weyl derivative $D$ is
replaced by $D+\gamma$, then this operator changes by the zero order
term $(w-1)\sym_0(\gamma\tens X)$. The operator is therefore
conformally invariant for $w=1$: it is the \emphdef{conformal Killing
operator} $X\mapsto\cL_X\conf$ and its solutions are the conformal vector
fields. In two dimensions there is a ``twistor correspondence'': a
vector field is conformal iff it is holomorphic. A similar story applies
to many other first order differential operators~\cite{Fegan}.

Secondly, consider sections $\lambda$ of $L^w$. Comparing the dependence
of $\trace_\conf D^2\lambda$ on $D$ with that of the scalar curvature
$\scal^D$ leads to the conclusion that for $w=(2-n)/2$ the operator
$\trace D^2-\frac{n-2}{4(n-1)}\scal^D$ is independent of the choice
of $D$. This is the \emphdef{conformal Laplacian}. In two dimensions,
the scalar curvature term is not needed, since $\trace D^2$ is invariant
on ordinary functions. A function is harmonic iff its differential
is holomorphic.


\section{M\"obius structures}\label{ms}

The geometry of Weyl derivatives on conformal $2$-manifolds is impoverished by
the absense of a symmetric tracefree Ricci tensor analogous to $r^D_0$.  The
aim of this section is to repair this by introducing M\"obius structures,
which provide a notion of conformal structure in two dimensions more closely
analogous to the higher dimensional case.

Firstly, by way of motivation, the one dimensional case will be considered.  A
$1$-manifold automatically carries a conformal structure in the conventional
sense, but it is more natural to define a conformal (or M\"obius) $1$-manifold
as a $1$-manifold equipped with a (real) projective structure.  One definition
of a real projective structure on an $n$-manifold is a
$\SL(n+1,\R)$-connection on $J^1L^{n/(n+1)}$ such that the formal $1$-jets of
parallel sections are holonomic. This connection is equivalently given by a
Hessian: a second order linear differential operator from $L^{n/(n+1)}$ to
$S^2T\dual\Mn\tens L^{n/(n+1)}$ whose symbol is the identity.

On a $1$-manifold this Hessian becomes a Laplacian from $L^{1/2}$ to
$L^{-3/2}$. Higher dimensional conformal manifolds automatically
possess a Laplacian from $L^{(2-n)/2}$ to $L^{(-2-n)/2}$, namely the
conformal Laplacian mentioned at the end of the previous section. The fact
that there is no such operator in one dimension is related to the absence
of scalar curvature. The solution is to introduce a choice of Laplacian
as extra structure.

In two dimensions, however, the Laplacian \emph{does} make sense, but there is
another second order operator, the \emphdef{conformal tracefree Hessian},
which exists in higher dimensional conformal geometry, but has no two
dimensional analogue.

To find this operator, consider the tracefree Hessian $\sym_0 D^2$
defined by a Weyl derivative $D$. As an operator on $L^w$ this
depends upon $D$. More precisely, if $\tilde D=D+\gamma$ on $L^1$ and $\mu$ is
a section of $L^w$, then
\begin{multline*}
\sym_0\tilde D^2\mu=\sym_0 D^2\mu\\
+ 2(w-1)\sym_0\gamma\tens D\mu
+w(\sym_0 D\gamma)\mu+ w(w-2)\gamma\tens_0\gamma\,\mu.
\end{multline*}

For $w=1$ the first order term in this expression vanishes, and the zero order
term is \emph{identical} to the change in the normalised tracefree Ricci
tensor $r^D_0$ given in~\ref{RicChange}.  The following well known fact is
therefore obtained~\cite{BEG,GNotes}: the operator $\sym_0 D^2+r^D_0$ (acting
on sections of $L^1$ over a conformal manifold of dimension $n>2$) is
independent of the choice of the Weyl derivative $D$.

\begin{defn} A \emphdef{M\"obius structure} on a conformal manifold
$\Mn$ is a (smooth) second order linear differential operator $\MS$ from
$L^1$ to $S^2_0T\dual\Mn\tens L^1$ such that for some Weyl derivative $D$,
the operator $\MS-\sym_0 D^2$ is zero order.
\end{defn}
The calculations made above show that this condition on $\MS$ is independent
of $D$, that $\sym_0 D^2$ is a M\"obius structure for any Weyl derivative
$D$, and also give:
\begin{prop} On a conformal manifold of dimension $n>2$ there is a
canonical M\"obius structure $\MS=\sym_0 D^2+r^D_0$.
\end{prop}

On a two dimensional M\"obius manifold, the difference $\MS-\sym_0 D^2$
will be called the normalised tracefree Ricci tensor $r^D_0$ of $D$
with respect to $\MS$.

In higher dimensions, M\"obius structures other than the canonical one are
sometimes of interest. For example, in~\cite{OS}, Osgood and Stowe implicitly
study the M\"obius structure $\sym_0 (D^g)^2$ where $g$ is a Riemannian metric
on $\Mn$. This is the canonical structure of the induced conformal structure
iff $g$ is Einstein. The generalisation to Einstein-Weyl geometry will be
explained in the next section.

Firstly, though, these remarks require further justification, and the
definition of a M\"obius structure given here needs to be related to more
classical notions. An initial step in this direction is the following
easy fact.

\begin{prop} M\"obius structures (compatible with a given conformal structure
on $\Mn$) form an affine space modelled on the smooth sections of
$S^2_0T\dual\Mn$. In two dimensions, this is the space of smooth quadratic
differentials.
\end{prop}

Now define the \emphdef{Schwarzian derivative} of a conformal vector field $X$
or a conformal diffeomorphism $\theta$ to be the Lie derivative $\cL_X\MS$ or
the difference $\theta^*\MS-\MS$. Since the Lie derivative or pullback of a
density is first order in $X$ or $\theta$ and $\MS$ is a second order
operator, it follows that the Schwarzian derivative is third order in $X$ or
$\theta$. Also, since the conformal structure is preserved, the Schwarzian
derivative is a quadratic differential. A conformal vector field or
diffeomorphism with vanishing Schwarzian will be called \emphdef{M\"obius}.

In~\cite{OS}, Osgood and Stowe define a Schwarzian derivative of a conformal
diffeomorphism with respect to a metric $g$. It is obtained by applying the
operator $f\mapsto (D^g df-df\tens df)_0$ to the conformal scale factor of the
diffeomorphism (calculated with respect to $g$). It is elementary to check
that this definition coincides with the definition given here when
$\MS=\sym_0(D^g)^2$. The connection with Einstein geometry is very clear
in~\cite{OS}: Osgood and Stowe show, among other things, how the Schwarzian
derivative provides a beautiful setting for the work of Brinkmann on conformal
mappings of Einstein manifolds (see also W. K\"uhnel's article in~\cite{KP}
for an exposition of this).

M\"obius structures can be used to reinterpret and generalise these results,
but here the focus will be on the two dimensional case, since in higher
dimensions it seems natural to work with the canonical M\"obius structure,
making the choice of M\"obius structure less interesting.

In fact there is a canonical M\"obius structure also in two dimensions, but
the construction is deep and global in contrast to the simple local formula in
higher dimensions. To see this, it is first necessary to look at the basic
example of a M\"obius structure on a domain in the Riemann sphere.

The complex affine structure on the conformal plane identifies it, up to a
choice of origin and real axis, with $\C$. Using this it is straightforward to
write down a M\"obius structure as follows.

If $\lambda=f\mu$ is a section of $L^1$, where $f$ is a function and $\mu$
is a constant section, then:
$$\MS\lambda=\frac{d^2f}{dz^2}\,\mu\,dz^2.$$
Here $d/dz$ denotes the complex linear part of the derivative (which will also
be denoted by a prime) and $dz^2$ is a constant quadratic differential.

The claim is that this M\"obius structure is invariant under M\"obius
transformations: more precisely, the Schwarzian derivative defined above
reduces to the usual Schwarzian derivative of holomorphic functions. To see
this, suppose $\theta$ is a holomorphic diffeomorphism between open
subsets of $\C$. Then
$\theta_*\lambda=(|\theta'|\lambda)\circ\theta^{-1}$ and so:
$$(\theta^*\MS)\lambda=\theta^*\bigl(\MS(\theta_*\lambda)\bigr)
=|\theta'|^{-1}(\theta')^2\bigl(
(|\theta'|f)\circ\theta^{-1}\bigr)''\circ\theta\,\mu\,dz^2.$$
Since $\theta$ is holomorphic, $(\overline{\theta'})'=0$ and so
(after applying the chain rule) it is only necessary to compute
$(\theta')^{-1/2}\theta'
\bigl((\theta')^{-1}\bigl((\theta')^{1/2}f\bigr)'\bigr)'$.
A calculation then gives
$$(\theta^*\MS)\lambda=\MS\lambda+\tfrac12S(\theta)\lambda,$$
where $S(\theta)=\bigl(\frac{\theta''}{\theta'}\bigr)' -
\frac12\bigl(\frac{\theta''}{\theta'}\bigr)^2$ is the usual Schwarzian.
This proves the claim.

It now follows that any complex projective curve has a canonical M\"obius
structure. A M\"obius structure which arises in this way will be called
\emphdef{integrable}. Now the universal cover of any conformal surface is
conformally diffeomorphic to the sphere, the plane or the disc (The
Uniformisation Theorem) and the conformal transformations of these spaces are
all M\"obius. It also follows that any conformal surface admits a preferred
integrable M\"obius structure, although this structure depends on the global
properties of the surface.

Not all M\"obius structures are integrable, however. They possess a curvature
analogous to the Cotton-York tensor of a conformal $3$-manifold.

In general the Cotton-York tensor of a Weyl derivative $D$ on a conformal
manifold is defined by the formula $C^D=d^Dr^D$, where $r^D$ is the normalised
Ricci tensor viewed as a covector valued $1$-form. More precisely:
$$C^D_{X,Y} = (D_Xr^D)(Y,.) - (D_Yr^D)(X,.)$$

A painful computation of $d^{D+\gamma}r^{D+\gamma}$ leads to the well known
fact that $C^{D+\gamma}_{X,Y}=C^D_{X,Y}-W_{X,Y}\gamma$. Note that the same
calculation applies in the two dimensional case, provided $r^D_0$ is
interpreted as the normalised tracefree Ricci tensor with respect to a
fixed M\"obius structure. Since there is no Weyl curvature in dimensions
two and three, the Cotton-York tensor is an invariant of the M\"obius or
conformal structure in these dimensions.

In dimension three or more the second Bianchi identity implies
that $C^D$ is tracefree: $\sum C^D_{X,e_i}e_i=0$, where $e_i$ is
a weightless orthonormal basis. More explicitly, this
contracted Bianchi identity is given by the following.
\begin{prop}\label{ConfBianchi} On any Weyl manifold of dimension $n>2$,
$$\divg^D\left(r^D_0-\tfrac1{2n}\scal^D\iden+\tfrac12F^D\right)=0,$$ where
$\divg^D=\trace_\conf\circ D$ and in particular,
$\divg^DF^D=\sum_i(D_{e_i}F^D)(e_i,.)$.
\end{prop}

In two dimensions there is no such Bianchi identity and the Cotton-York tensor
$C^D$, far from being tracefree, is necessarily tracelike. Hence the
Cotton-York tensor of a M\"obius $2$-manifold may equivalently be viewed
as a $1$-form of weight $-3$.

\begin{defn} Let $\Mn$ be a conformal surface with M\"obius structure $\MS$.
Then the \emphdef{(Cotton-York) curvature} of $\Mn$ is the section $C^\MS$ of
$L^{-2}T\dual\Mn\cong\cconj{T\dual\Mn}\ctens T\dual\Mn^2$ defined by
$C^\MS=\divg^D\left(-r^D_0+\tfrac14\scal^D\iden-\tfrac12F^D\right)$, where $D$
is any Weyl derivative, and $r^D_0=\MS-\sym_0 D^2$. The M\"obius structure is
said to be \emphdef{flat} iff $C^\MS=0$.
\end{defn}
The tensor inside the divergence is $r^D(JX,JY)$.

If a M\"obius structure $\MS$ is replaced by $\MS+\Phi$ where $\Phi$ is a
quadratic differential, then for any fixed $D$,
$r^D_0(\MS+\Phi)=r^D_0(\MS)+\Phi$ and so $C^{\MS+\Phi}=C^\MS-\divg\Phi$.  Note
that the divergence is independent of $D$ on quadratic differentials and may
be identified with the $\overline\partial$ operator. Therefore \emph{flat}
M\"obius structures form an affine space modelled on \emph{holomorphic}
quadratic differentials.

The operator $\MS$ is equivalently given by a subbundle $\ME^2$ of $J^2L$,
which are the $2$-jets of formal solutions. If a M\"obius structure is flat
then this differential equation $\MS$ is completely integrable.  The
proof of this is a straightforward exercise in the formal theory of
differential equations~\cite{Spencer} and is sketched in the appendix. The
integrability in this case means that $\ME^2$ carries a flat connection whose
parallel sections correspond to local solutions of $\MS$. This is completely
analogous to the integrability, in higher dimensions, of conformal structures
with vanishing Cotton-York or Weyl curvature in higher dimensions---for
similar proofs in the higher dimensional case,
see~\cite{BEG,GNotes}. Alternatively, M\"obius structures could be defined
using Cartan connections, reducing integrability questions to the method of
equivalence. However, I feel that a description of a geometric structure in
terms of a Cartan connection should be a theorem rather than a definition, and
the construction of a connection on $\ME^2$ essentially establishes this
theorem.

It remains to check that an integrable M\"obius structure induces a
complex projective structure. This is a consequence of the following
observations.
\begin{itemize}
\item The local solutions of $\MS$ parallelise $\ME^2$. 
\item $\ME^2$ possesses a natural Lorentzian quadratic
form on each fibre which is preserved by parallel transport.
\item The symbol of $\ME^2$ defines a null line in each fibre and this section
of the bundle of null lines is not parallel, but in fact a local
diffeomorphism into the space of parallel null lines.
\end{itemize}
The Lorentzian structure is induced by the scalar curvature: for any positive
section $\mu$ of $L^1$, the scalar curvature of the corresponding metric
defines a function $\scal^\mu\mu^2$ on $\Mn$. At each point, this function
only depends on the $2$-jet of $\mu$ and so makes sense as a quadratic form on
$J^2L$. The other properties are more or less straightforward: for further
details again see~\cite{BEG,GNotes}. The space of null lines in a Lorentzian
vector space is the sphere with its natural M\"obius structure and
these local diffeomorphisms provide the required complex projective charts.

\section{Einstein-Weyl geometry in two dimensions}\label{ew}

\begin{defn} Let $(\Mn,\conf,D)$ be a Weyl manifold of dimension greater
than two. Then $\Mn$ is said to be \emphdef{Einstein-Weyl} iff $r^D_0=0$,
i.e., the symmetric tracefree part of the Ricci tensor vanishes.
\end{defn}

In other words, a Weyl derivative $D$ on a conformal manifold
is Einstein-Weyl iff $\sym_0 D^2$ is equal to the canonical
M\"obius structure. In two dimensions there is no canonical M\"obius
structure, but since the conformal structure is integrable, it is
natural to require instead that $\sym_0 D^2$ is an \emph{integrable}
M\"obius structure.

\begin{defn}\label{EW2def} Let $(\Mn,\conf,D)$ be a Weyl manifold
of dimension two. Then $\Mn$ is said to be
\emphdef{Einstein-Weyl} iff the M\"obius structure $\sym_0 D^2$ is flat.
\end{defn}

A $2$-manifold is usually said to be Einstein iff it has constant scalar
curvature, since this follows from the contracted Bianchi identity in
higher dimensions. The above defintion generalises this to the
Einstein-Weyl case.

\begin{prop}\label{EWBianchi} Suppose $\Mn$ is Einstein-Weyl. Then
$D\scal^D-n\divg^D F^D=0$ (here the trace is with the first entry of $F^D$).
\end{prop}
For $n>2$ this is immediate from the contracted Bianchi
identity~\ref{ConfBianchi} (see~\cite{PT,Gauduchon4}), whereas in two
dimensions, it is equivalent to the vanishing of the Cotton-York tensor of the
M\"obius structure $\sym_0 D^2$.

\begin{cor} A Weyl manifold $(\Mn,\conf,D)$ of dimension two
is Einstein-Weyl if and only if $D\scal^D-2\divg^D F^D=0$. Equivalently, the
section $\frac12\scal^D-F^D$ of $L^{-2}(\R\dsum L^{-2}\Lambda^2T\Mn)\cong
L^{-2}\tens\C$ is holomorphic with respect to $D$.
\end{cor}

Much of the theory of Einstein-Weyl manifolds, as described for instance
in~\cite{CP,Gauduchon4}, applies to the two dimensional case without
substantial change. In some cases the two dimensional proof is simpler,
an important such example being the existence of a Gauduchon gauge.

\begin{defn} Let $(\Mn,\conf,D)$ be a Weyl manifold. Then a compatible metric
$g$ is called a \emphdef{Gauduchon metric} or \emphdef{gauge} iff
$D=D^g+\omega^g$ with $\trace_\conf D^g\omega^g=0$.
\end{defn}

In two dimensions the divergence on $1$-forms is conformally invariant
and a Gauduchon gauge is a coclosed representative
for the space of $1$-forms $\omega$ with $D-\omega$ exact.

\begin{thm} A compact two dimensional Weyl manifold
admits a Gauduchon gauge, unique up to homothety.
\proofof{thm} This is an immediate consequence of the Hodge decomposition
for $1$-forms in two dimensions.
\end{proof}

A remarkable feature of the Gauduchon gauge on a compact Einstein-Weyl
manifold is that the Gauduchon $1$-form is dual to a Killing field
of the Gauduchon metric~\cite{Tod1}. This is closely related to the existence
of a \emphdef{Gauduchon constant}~\cite{Gauduchon4} generalising the
constant scalar curvature on an Einstein manifold. The two dimensional
formulation is as follows.

\begin{thm}\label{GKill} Let $\Mn$ be a \emph{compact} Weyl
$2$-manifold with $D=D^g+\omega^g$ in the Gauduchon gauge. Then
$\scal^D=\scal^g$ and if $D$ is Einstein-Weyl
then $\sharp_g\omega^g$ is a Killing field.

Conversely if $\Mn$ is any Weyl manifold with $D=D^g+\omega^g$ such that
$\sharp_g\omega^g$ is a Killing field, then $D$ is Einstein-Weyl iff the
section $\Gk=\scal^g-4|\omega^g|^2$ of $L^{-2}$ is constant (with respect
to $D^g$).
\proofof{thm} The Cotton-York tensor is an invariant of $\MS=\sym_0 D^2$ and
so computing it in terms of $D$ and $g$ gives:
$$\tfrac14D\scal^D-\tfrac12\divg^DF^D
=C^\MS=\divg^g\left(-r^g_0+\tfrac14\scal^g\iden\right),$$
where $r^g_0=\sym_0 D^2-\sym_0 (D^g)^2$ (acting on $L^1$). This implies
\begin{equation*}\tag{*}\begin{split}
\tfrac14D\scal^D-\tfrac12\divg^DF^D
=&2\cip{\sym_0D^g\omega^g,\omega^g}
+\tfrac14D^g\bigl(\scal^g-4|\omega^g|^2\bigr)\\
&+2(\divg^g\omega^g)\omega^g-\divg^g(\sym_0D^g\omega^g).
\end{split}\end{equation*}
If $D$ is Einstein-Weyl and $g$ is a Gauduchon metric then the left hand
side is zero and $\divg^g\omega^g=0$. Contracting the remaining terms
with $\omega^g$ leads to the equation:
$$\divg^g\bigl(\cip{\sym_0D^g\omega^g,\omega^g}-\tfrac14
\scal^g\omega^g\bigr) =2|\sym_0D^g\omega^g|^2.$$ Integration then
shows that $\sym_0D^g\omega^g=0$ and so $\sharp_g\omega^g$ is
Killing. Conversely, substituting $\sym D^g\omega^g=0$ into (*) shows that the
Einstein-Weyl condition in these circumstances is equivalent to the constancy
of $\Gk$.
\end{proof}
\begin{cor} Let $\Mn$ be a compact orientable conformal surface and suppose
$D$ is Einstein-Weyl. Then either $D$ is defined by a compatible metric
of constant curvature or the genus of $\Mn$ is less than two.
\proofof{cor} If $D$ is not given by a compatible metric then the
Gauduchon $1$-form is dual to a nonzero conformal vector field.
Such vector fields are precisely the holomorphic ones, and only
exist globally on the Riemann sphere and each torus.
\end{proof}

Each conformal structure on $S^1\times S^1$ has a family of flat
Weyl structures (which are therefore Einstein-Weyl) parameterised
by conformal vector fields. The only conformal vector fields on
a torus are the translations with respect to the flat metric $g$ (unique
up to homothety). Any such Killing field $K$ gives rise to a
$D^g$-parallel 1-form $\omega=g(K,.)$ and $D=D^g+\omega^g$ is flat.
Each such $D$ induces a flat M\"obius structure, equal to that
induced by $D^g-\omega^g$, and so the flat Weyl structures (which may
also be viewed as affine structures) form a branched $2$-fold covering
of the flat M\"obius structures (the projective structures).

In the next section, some less classical examples will be found.


\section{Classification of compact Einstein-Weyl geometries}

Let $\Mn$ be a compact Riemann surface with a Weyl derivative $D$.
Then $D=D^g+\omega^g$ where $D^g$ is the Weyl derivative of a compatible
metric $g$, and $\omega^g$ is a $1$-form with $\divg\omega^g=0$,
a condition which uniquely determines $\omega^g$ from $\Mn,D$
(the metric $g$ being unique up to homothety).
The Einstein-Weyl equation for this structure is equivalent to the following
two conditions:
\begin{itemize}
\item $\sharp_g\omega^g$ is a Killing field of $g$
\item $D^g(\scal^g-4|\omega^g|^2)=0$.
\end{itemize}
The Killing field is identically zero precisely when $D=D^g$ and
$g$ is a metric of constant curvature.

The aim of this section is to find all Einstein-Weyl structures
$D=D^g+\omega^g$ with $g$ admitting a \emph{nontrivial} Killing field
$K$ such that $\sharp_g\omega^g=AK$ for some constant $A$.
In this situation it is no longer necessary to assume $\Mn$ is
compact, and constant curvature metrics with symmetry arise when $A=0$.

The Einstein-Weyl equations may be integrated locally as follows. Suppose
that $K$ is nonvanishing on an open subset $U$ of $\Mn$ diffeomorphic to
a domain in $\C$. Since $K$ is holomorphic, isothermal
coordinates $x+it$ may be chosen locally such that
$K=\partial/\partial t$. The Gauduchon metric is then of the form
$g=v^2(dx^2+dt^2)$, and $\omega^g=Av^2\,dt^2$ where $v$ is an
unknown function of $x$. If $v=e^f$ then
$$\scal^D=\scal^g=-2\divg(df)=-2f''e^{-2f}\mu_g^{-2}$$
where $\mu_g$ is the trivialisation of $L^1$ determined by $g$.
The Einstein-Weyl equation now reduces to an ordinary differential
equation for $f$, namely
$$f''e^{-2f}+2A^2e^{2f}=B,$$
which readily integrates to give
$$v^{-2}dv^2=(-A^2v^4+Bv^2+C)dx^2.$$
Therefore the Einstein-Weyl structure is locally of the form
\begin{align*}
g&=P(v)^{-1}dv^2 + v^2 dt^2\\
\omega&=A v^2\,dt,\\
\tag*{\text{where}} P(v) &=-A^2v^4 + Bv^2 +C,
\end{align*}
and $A,B,C$ are arbitrary constants. Integrating the equation
$v'(x)^2=P(v)v^2$ gives the solution in the original
isothermal coordinates, but it is perhaps simpler to introduce a
new coordinate $r$ by $v'(r)^2=P(v)$. The metric is now
$$g=dr^2 +v(r)^2dt^2$$
and $v(r)$ is an elliptic function since $P$ is a quartic polynomial.
In order to have a real solution $v(r)$ it is necessary that
$P(v)$ is somewhere nonnegative. These solutions are given by the
following Jacobian elliptic functions and their limits.
\begin{equation*}
v(r)=\begin{cases}
\lambda\, \mathrm{cn}(\mu r+\alpha,k)\quad\text{or}\quad
\lambda\, \mathrm{sd}(\mu r+\alpha,k) &\text{if $C>0\qquad\qquad (1)$}\\
\lambda\, \mathrm{dn}(\mu r+\alpha,k)\quad\text{or}\quad
\lambda\, \mathrm{nd}(\mu r+\alpha,k) &\text{if $C<0\qquad\qquad (2)$},
\end{cases}
\end{equation*}
where $\alpha$ is a constant of integration and $\lambda,\mu,k$ are
constants depending on $A,B,C$. The Einstein-Weyl structure remains
unchanged if $v$ is rescaled and $r$ is affinely reparameterised,
provided $t$ is rescaled appropriately, and so the solutions in fact
only depend (locally) on one parameter.

The two forms of the solution given in each case above are equivalent by
period translation, but behave differently in the limit $k\to1$ when the
(real) period becomes infinite: cn and dn both reduce to sech, whereas sd and
nd reduce to sinh and cosh. The sech solution corresponds to $C=0$ and gives a
nonexact Einstein-Weyl structure on $\R^2$ or $S^1\times\R$, whereas the
hyperbolic solutions arise when $A=0$ and the Einstein-Weyl structure is the
hyperbolic metric. In the limit $k\to0$, cn$\to\cos$, sd$\to\sin$, while
dn,nd$\to1$. These limits therefore give the spherical metric and the flat
Weyl structure.

It remains to consider when these solutions are defined on a compact surface
(necessarily of genus less than two, since only these admit metrics with
Killing fields). If $w$ ranges over a half-period of cn or sd, (1) gives a one
parameter family of global solutions on $S^2$, whereas dn and nd are periodic
and nonvanishing, and so the solutions in (2) are defined on each torus
$S^1\times S^1$. The constant solutions correspond to the flat Weyl structures
discussed at the end of the previous section.  The non-constant solutions are
more complicated, but are interesting from the point of view of
Einstein-Weyl geometry, since in four or more dimensions compact Einstein-Weyl
manifolds with nontrivial Faraday curvature necessarily have finite
fundamental group.

\section{Appendix: Integrability of M\"obius structures}

In this appendix, I sketch the proof that the subbundle $\ME^2$ of $J^2L$,
corresponding to a M\"obius structure $\MS$, carries a natural connection
whose curvature is given by the Cotton-York tensor of $\MS$.
The proof is not
central to the main thrust of the paper, and will perhaps seem rather obscure
to readers not familiar with the formal theory of differential equations.
The calculations are purely formal, because M\"obius geometry is a
geometry of ``finite type'' much like conformal geometry in higher
dimensions (see~\cite{Guillemin})---indeed, to emphasise the common
framework, I will only specialise to the two dimensional case at the
end of the proof. It is possible to convert the following into detailed
and explicit calculations bypassing this conceptual framework,
but I will not attempt that here.

The \emphdef{symbol} $\cg^2$ of $\ME^2$ is the line bundle of tracelike
tensors in $S^2T\dual\tens L$ and $\ME^2$ is an extension of $J^1L$ by
$\cg^2$. The induced first order equation on $L$ is trivial and so
one sets $\cg^1=T\dual\tens L$.

The prolongation of $\ME^2$ is the bundle $\ME^3=J^1\ME^2\intersect
J^3L\subseteq J^1J^2L$, which may be viewed as a third order differential
equation on $L$, or a first order differential equation on the bundle
$\ME^2$. It is the latter point of view which will be pursued. The
corresponding differential operator is the restriction of the \emphdef{jet
derivative} to $\ME^2\subseteq J^2L$, i.e., the first order operator from
$J^2L$ to $T\dual\wedge J^2L:=(T\dual\tens J^2L)\big/(S^3T\dual\tens L)$
characterising the holonomic sections of $J^2L$.

Since the prolongation $\cg^3=(T\dual\tens\cg^2)\intersect(S^2T^*\tens\cg^1)$
of the symbol is zero, the natural map from $T\dual\tens\ME^2$ to
$T\dual\wedge J^2L$ is injective. The first obstruction to the integrability
of $\ME^2$ is the failure of the jet derivative to have its image in this
bundle. This obstruction therefore takes its values in $(T\dual\wedge
J^2L)\big/(T\dual\tens\ME^2)$, which may be identified with the symbol
cohomology group $(T\dual\wedge S^2T\dual\tens
L)\big/(T\dual\wedge\cg^2)$. If the first obstruction vanishes then $\ME^3$
defines a connection on $\ME^2$ and since the higher symbol cohomology groups
vanish, there are no further obstructions and the connection is
flat~\cite{Goldschmidt}. The local parallel sections are holonomic sections
of $\ME^2$ and hence $\ME^2$ is integrable. It therefore remains to show that
the first integrability obstruction is given by the Cotton-York curvature.

The computation of the integrability obstruction is most easily carried out by
introducing an exact Weyl derivative $D$ on $L$. Using the induced covariant
derivative on $T\dual$ (which is not flat in general), $D$ defines an
isomorphism $$J^2L\cong (S^2T\dual\tens L)\dsum (T\dual\tens L)\dsum L$$ and
covariant derivatives on the summands. A covariant derivative on $J^2L$ may
then be defined by the formulae:
$$D(\phi,\alpha,\lambda) =
(D\phi-R^D\cdot\alpha,D\alpha-\phi,D\lambda-\alpha),$$ where
$3(R^D\cdot\alpha)_X(Y,Z)=(R^D_{X,Y}\alpha)(Z)+(R^D_{X,Z}\alpha)(Y)$.  The
reason for this definition is that if $(\phi,\alpha,\lambda)$ is holonomic
then is covariant derivative is $(D^3\lambda-R^D\cdot D\lambda,0,0)$ which is
a section of $S^3T\dual\tens L$, hence in the kernel of the symbol of the jet
derivative. It follows that the jet derivative in general is given by the
symbol applied to $D(\phi,\alpha,\lambda)$.

Now suppose $(\phi,\alpha,\lambda)$ is a section of $\ME^2$. Applying the jet
derivative and removing the tracelike part gives
$$\bigl(-D(r^D_0\lambda)-R^D.\alpha\pmod{S^3T\dual\tens L},D\alpha-\phi,
D\lambda-\alpha\bigl).$$
since the tracefree part of $\phi$ is equal to $-r^D_0\tens\lambda$ which is
the tracefree part of $-r^D\tens\lambda$ since $F^D=0$. Modulo the section
$((D\lambda-\alpha)\tens(-r^D_0),D\alpha-\phi,D\lambda-\alpha)$ of
$T\dual\tens\ME^2$ this is equivalent to
$$\bigl(-(Dr^D_0)\lambda-\alpha\tens r^D_0-R^D.\alpha\pmod{S^3T\dual\tens L},
0,0\bigr).$$ This integrability obstruction depends only on the 1-jet
$(\alpha,\lambda)$ and so is given by a first order operator on $L$.  It is
now straightforward to see that the symbol of this operator is the Weyl
curvature, and that in two and three dimensions it reduces to the Cotton-York
curvature tensor.


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\end{thebibliography}

\noindent Mathematics Institute, University of Warwick, Coventry CV4 7AL.
England.
\medskip

\noindent email: \texttt{davidmjc@maths.warwick.ac.uk}

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