Gauge theory on projective surfaces and anti-self-dual Einstein metrics in dimension four
Given a projective structure on a surface $\mf$, we show how to canonically construct a neutral signature Einstein metric with non-zero scalar curvature on the total space $M$ of a certain rank $2$ affine bundle $M \to \mf$. The resulting Einstein metric has anti-self-dual conformal curvature and admits a parallel field of anti-self-dual planes. We show that Einstein metrics arising from this construction admit smooth deformations to Ricci--flat metrics if the corresponding projective structures admit a representative connection with totally skew-symmetric Ricci tensor. The homogeneous Einstein metric corresponding to the flat projective structure on $\RP^2$ is the non-compact real form of the Fubini-Study metric on $M=\mathrm{SL(3, \R)/\mathrm{GL(2, \R)$. Our construction is a combination of the Cartan and tractor bundle approaches to projective differential geometry with a projectively invariant gauge theory of a connection and a pair of Higgs fields developed by Calderbank.
1509.04276.tex
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\begin{document} \date{September 15, 2015}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title[Projective surfaces and anti-self-dual Einstein metrics]{Gauge theory on projective surfaces and anti-self-dual Einstein metrics in dimension four}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\author[M.~Dunajski]{Maciej Dunajski}
\address{Department of Applied Mathematics and Theoretical Physics\\ 
University of Cambridge\\ Wilberforce Road, Cambridge CB3 0WA, UK.}
\email{m.dunajski@damtp.cam.ac.uk}
\author[T.~Mettler]{Thomas Mettler}
\address{Institut f\"ur Mathematik\\
Goethe-Universit\"at Frankfurt\\
Robert-Mayer-Str.~10, 60325 Frankfurt, Germany.
}
\email{mettler@math.uni-frankfurt.de}
\begin{abstract}
Given a projective structure on a surface $\mf$, 
we show how to canonically construct a neutral signature Einstein metric with non-zero scalar curvature on the total space $M$ of a certain rank $2$ affine bundle $M \to \mf$. The resulting Einstein metric has anti-self-dual conformal curvature and admits a parallel field of anti-self-dual planes. We show that Einstein metrics arising from this construction
admit  smooth deformations to Ricci--flat metrics if the corresponding projective structures admit a representative connection with totally skew-symmetric Ricci tensor. The homogeneous Einstein metric corresponding to the flat projective structure
on $\RP^2$ is the non-compact real form of the Fubini-Study metric on  $M=\mathrm{SL}(3, \R)/\mathrm{GL}(2, \R)$.

 Our construction is a combination of the Cartan and tractor bundle approaches to projective differential geometry with a
projectively invariant  gauge theory of a connection and a pair of Higgs fields developed by Calderbank.
\end{abstract}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}

The aim of this paper is to canonically associate a  neutral
 signature Einstein metric
to any two--dimensional  projective structure. Recall that a projective structure $[\nabla]$ on a surface $\mf$ is an equivalence class
of affine torsion--free connections on $T\mf$ which share the same unparametrised geodesics. We shall provide two 
different constructions of this metric: one as a $\mathrm{GL}(2, \R)$ quotient
of the total space of the Cartan bundle associated to $(\mf, [\nabla])$, 
and another one as a solution to Calderbank's 
projective pair equations \cite{Cal1} corresponding to the tractor connection on a rank-$3$ vector bundle $E$ over $\mf$. The Einstein metric is defined on the total space of a rank-$2$ affine bundle $M \to N$ which is the complement of an $\RP^1$ sub-bundle of the projectivised tractor bundle $\PP(E)\to N$ of $(N,[\nabla])$. If a local trivialisation is chosen the metric takes the form
\be
\label{main_metric}
g=\left(d\xi_i-\left(\Gamma_{ij}^k \xi_k-\Lambda \xi_i\xi_j- \Lambda^{-1}\Rho_{ji}\right)\d x^j\right)\odot \d x^i
\ee
where the indices $i,j,k$ take values from $1$ to $2$, the coordinates $x=(x^i)$ refer to a local chart $U$ in $\mf$, the coordinates
$\xi=(\xi_i)$ are defined on an affine chart of the fibres of $\PP(E)$, 
$\Gamma_{ik}^j$ are the Christoffel symbols of a $[\nabla]$-representative connection, $\Rho_{ij}$ is the Schouten tensor of $\Gamma^j_{ik}$, given in terms of 
the Ricci tensor $R_{ij}$ by $\Rho_{ij}=R_{(ij)}+\frac{1}{3}R_{[ij]}$ and $\Lambda=1$.
We shall show (Theorem \ref{theo_char}) that the formula 
(\ref{main_metric}) with $\Lambda=1$ defines a metric which does not depend on the choice of a connection in the projective class. 

All  metrics in the one--parameter family (\ref{main_metric}) with arbitrary non--zero $\Lambda$
have anti-self-dual (ASD) Weyl tensor, and are Einstein with non-zero scalar curvature equal to 
 $-24\Lambda$. Moreover (Theorem \ref{theo_asd})
all ASD Einstein metrics which admit a parallel ASD totally null distributions are locally of the form (\ref{main_metric}). In Theorem \ref{theo_kill} we shall establish a one-to-one correspondence between projective vector fields on $(\mf, [\nabla])$ and Killing vectors on $(M, g)$ which additionally preserve
a canonically defined symplectic structure on $M$ given by
\be
\label{s_form}
\Omega= d\xi_i\wedge d x^i+\frac{1}{\Lambda}\Rho_{ij} d x^i\wedge d x^j.
\ee
This symplectic structure is parallel w.r.t the Levi--Civita connection of $g$ only iff the projective Cotton tensor of $(\mf, [\nabla])$ vanishes. In Theorem 
\ref{maincorrespondence} we shall construct the pair $(g, \Omega)$ for the choice $\Lambda=1$ as a bi-La\-grang\-ian structure on
the quotient of the Cartan bundle of $(N, [\nabla])$ by $\mathrm{GL}(2, \R)$.

The flat projective structure on $\mf=\RP^2$ gives rise
to a non-flat homogeneous metric on $M=\mathrm{SL}(3, \R)/\mathrm{GL}(2, \R)$ which is a neutral signature analogue of the Fubini--Study metric. In this case
the Schouten tensor vanishes, and $g$ admits a flat limit
where $\Lambda=0$. Motivated by this example we shall find all projective structures where the limit $\Lambda\rightarrow 0$ exists. These projective structures are characterised by the   existence of a connection with totally skew-symmetric Ricci tensor, and the resulting metrics (\ref{main_metric}) with $\Lambda=0$ 
are Ricci--flat and hence hyper-symplectic. These metrics are given by formula (\ref{skew_metrics}).

This paper mainly concerns itself with the two-dimensional case, but there are obvious higher dimensional generalisations which we briefly discuss in an 
Appendix.

\subsection*{Acknowledgments} The authors wish to thank
Andreas \v{C}ap, Andrzej Derdzi\'nski and
Nigel Hitchin for helpful discussions regarding the contents of this paper. 
TM is grateful for travel support via the grant SNF 200020\_144438 of the Swiss National Science Foundation. 

\section{Projective differential geometry and anti-self-duality}
In this preliminary Section we shall summarise known facts about projective 
structures on a surface, and neutral signature conformal structures in four dimensions which underlie the results of the paper.

\subsection{Algebraic preliminaries}
As usual, we let $\mathbb{R}^n$ the 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. Matrix multiplication $\mathbb{R}_n \times \mathbb{R}^n \to \mathbb{R}$ is a non-degenerate pairing identifying $\R_n$ with the dual vector space of $\R^n$. 

Let $\mathbb{RP}^2=(\R^3\setminus\{0\})/\R^*$ denote space of lines in $\R^3$ through the origin, i.e., two-dimensional real projective space. For any nonzero $x \in \R^3$ let $[x]$ denote its corresponding point in $\mathbb{RP}^2$. Let $\mathbb{RP}_2=(\R_3\setminus\{0\})/\R^*$ denote the dual projective space and likewise for any nonzero $\xi \in \R_3$ we denote by $[\xi]$ its corresponding point in $\mathbb{RP}_2$.

The group $\mathrm{SL}(3,\R)$ acts from the left on $\R^3$ by matrix multiplication and this action descends to define a transitive left action on $\mathbb{RP}^2$. Likewise, $\mathrm{SL}(3,\R)$ acts on $\R_3$ from the left by the rule
$$
h \cdot \xi=\xi h^{-1}
$$
for $h \in \mathrm{SL}(3,\R)$ and this actions descends to define a transitive left action on $\mathbb{RP}_2$. The stabiliser subgroup of $[x_0] \in \mathbb{RP}^2$ where $x_0={}^t(1\;0\;0)$ will be denoted by $G$, so that $\mathbb{RP}^2\simeq \mathrm{SL}(3,\R)/G$. The elements of $G\subset \mathrm{SL}(3,\R)$ are matrices of the form
$$
b\rtimes a=\left(\begin{array}{cc} \det a^{-1} & b \\ 0 & a\end{array}\right),
$$
with $a \in \mathrm{GL}(2,\R)$ and $b \in \R_2$. Denoting by $\mathbb{RP}_1\subset \mathbb{RP}_2$ the projective line consisting of those elements $[\xi] \in \mathbb{RP}_2$ which satisfy $[\xi]\cdot [x_0]=0$, the group $G$ acts faithfully from the left by affine transformations on the affine $2$-space $\mathbb{A}_2=\mathbb{RP}_2\setminus \mathbb{RP}_1$. Indeed, if we represent an element in $\mathbb{A}_2$ by a vector $(1,\xi) \in \R_3$ with $\xi \in \R_2$, we obtain
$$
(1,\xi)\left(\begin{array}{cc} \det a^{-1} & b \\ 0 & a\end{array}\right)^{-1}=\left(\det a,-ba^{-1}\det a+\xi a^{-1}\right)
$$
so that the induced affine transformation is
$$
(b\rtimes a)\cdot \xi=\xi a^{-1}\det a^{-1}-ba^{-1}.
$$    
Consequently, we may naturally think of $G$ as the $2$-dimensional real affine group. 


\subsection{Projective differential geometry}
\label{projective_sec}
Let $\mf$ be a connected smooth surface. 
A projective structure $[\nabla]$ on $\mf$ is an equivalence class
of torsion--free connections on $T\mf$ which share the same unparametrised geodesics.
Two connections $\nabla$ and $\hat{\nabla}$ belong to the same projective structure if their geodesic sprays have the same projections to the projectivised tangent bundle $\PP(T\mf)$ or equivalently, 
if there exists a one--form $\Upsilon$ on $\mf$ such that
\be
\label{connection_change}
{\hat{\Gamma}^i}_{jk}={{\Gamma}^i}_{jk}+\delta^i_j\Upsilon_k+
\delta^i_k\Upsilon_j.
\ee
Given a connection $\nabla\in[\nabla]$, its curvature  is defined by
\[
[\nabla_i, \nabla_j]X^k={{R_{ij}}^k}{}_l X^l.
\]
It can be  uniquely decomposed as
\be
\label{formula_for_rho}
{{R_{ij}}^k}{}_l= 
\delta_i{}^k\Rho_{jl}-\delta_j{}^k\Rho_{il} -2\Rho_{[ij]}\delta_l{}^k,
\ee
where $\Rho_{ij}$ is the Schouten tensor.
In dimension higher than two there is an additional (Weyl) term in this 
decomposition which vanishes identically in the surface case.
If we change the connection in the
projective class using (\ref{connection_change}) then
\be
\label{schouten_change}
\hat{\Rho}_{ij}={\Rho}_{ij}-\nabla_i\Upsilon_j+\Upsilon_i\Upsilon_j, \quad
\hat{\Rho}_{[ij]}=\Rho_{[ij]}-\nabla_{[i}\Upsilon_{j]}.
\ee
For a non-zero real number $w$ let $\mathcal{E}(w):=(\Lambda^2(T^*\mf))^{-w/3}$ be the line bundle of projective densities of weight $w$.\footnote{More precisely, by this we mean the bundle associated to the coframe bundle of $N$ via the $\mathrm{GL}(2,\R)$ representation $a \mapsto |\!\det a|^{w/3}$.} The torsion-free connection $\nabla$ on $TN$ induces a connection on $\mathcal{E}(w)$, which -- by standard abuse of notation -- we denote by $\nabla$ as well. Under the projective change~\eqref{connection_change} we obtain 
\[
\hat{\nabla}_i s=\nabla_i s+w\;\Upsilon_i s,
\]
where $s$ is any section of $\mathcal{E}(w)$. 

If $N$ is orientable, we may restrict attention to connections in $[\nabla]$ which preserve some fixed volume form $\epsilon\in \Gamma(\Lambda^2(T^*\mf))$,  
so that $\nabla_i\epsilon_{jk}=0$. We shall refer to such 
connections as {\em special} \cite{EM}. Note that special connections always exist globally. For special connections
the Schouten tensor is symmetric, that is $\Rho_{[ij]}=0$.
The residual freedom in special connections within 
a given projective class is given
by (\ref{connection_change}) where $\Upsilon_i=\nabla_i f$ is a gradient of an arbitrary smooth function $f:\mf\rightarrow \R$. The special condition 
is preserved if  $\hat{\epsilon}_{ij}=e^{3f}\epsilon_{ij}$.

\subsection{The Cartan geometry of a projective surface}\label{cartgeomsec}

In~\cite{MR1504846}, Cartan associates to a projective structure $[\nabla]$ on a smooth surface $\mf$ a Cartan geometry $(\pi : \projb \to \mf,\theta)$ of type $(\mathrm{SL}(3,\R),G)$ which consists of a principal right $G$-bundle $\pi : \projb \to \mf$ together with a Cartan connection $\theta\in \Omega^1(\projb,\mathfrak{sl}(3,\R))$ having the following properties:
\begin{itemize}
\item[(i)] $\theta(\bX_v)=v$ for all fundamental vector fields $\bX_v$ on $\projb$;
\item[(ii)] $\theta_u : T_u\projb\to \mathfrak{sl}(3,\R)$ is an isomorphism for all $u \in \projb$;
\item[(iii)] $R_g^*\theta=\mathrm{Ad}(g^{-1})\theta=g^{-1}\theta g$ for all $ g\in G$;  
 \item[(iv)] Writing
$$
\theta=\left(\begin{array}{cc} -\tr \phi & \eta \\ \omega & \phi\end{array}\right)
$$
for an $\R^2$-valued $1$-form $\omega=(\omega^i)$, an $\R_2$-valued $1$-form $\eta=(\eta_i)$ and a $\mathfrak{gl}(2,\R)$-valued $1$-form $\phi=(\phi^i_j)$, the leaves of the foliation defined by the equations $\omega^2=\phi^2_1=0$ project to $\mf$ to be the geodesics of $[\nabla]$;   
\item[(v)] The curvature $2$-form $\Theta$ satisfies 
\begin{equation}\label{struceqproj}
\Theta=\d \theta+\theta\wedge\theta=\left(\begin{array}{cc} 0 & L(\omega\wedge\omega) \\ 0 & 0\end{array}\right),
\end{equation}
for a smooth curvature function $L : \projb \to \mathrm{Hom}\left(\R^2\wedge\R^2,\R_2\right)$.
\end{itemize} 
Note the Bianchi-identity
$$
\d \Theta=\Theta\wedge\theta-\theta\wedge\Theta,
$$
the algebraic part of which reads
\begin{equation}\label{algbianchi}
0=L(\omega\wedge\omega)\wedge\omega. 
\end{equation}
A projective structure $[\nabla]$ is called~\textit{flat} if locally $[\nabla]$ is defined by a flat connection. A consequence of Cartan's construction is that a projective structure is flat if and only if $L$ vanishes identically. 


\begin{rmk}
Cartan's bundle is unique in the following sense: If $(\hat{\pi} : \hat{B}_{[\nabla]} \to \mf,\hat{\theta})$ is another Cartan geometry of type $(\mathrm{SL}(3,\R),G)$ satisfying the properties (iii),(iv),(v), then there exists a $G$-bundle isomorphism $\psi : \projb \to \hat{B}_{[\nabla]}$ so that $\psi^*\hat{\theta}=\theta$.
\end{rmk}

Fixing a representative connection $\nabla$ in the projective equivalence class $[\nabla]$ allows to give an explicit description of the Cartan geometry $(\pi : \projb \to \mf, \theta)$. Let $\upsilon : F \to \mf$ denote the coframe bundle of $\mf$ whose fibre at at point $p \in \mf$ consists of the linear isomorphisms $u : T_p\mf \to \R^2$. The group $\mathrm{GL}(2,\R)$ acts transitively from the right on each $\upsilon$-fibre by the rule $R_a(u)=u \cdot a=a^{-1}\circ u$ for all $a \in \mathrm{GL}(2,\R)$. This action turns $\upsilon : F \to \mf$ into a principal right $\mathrm{GL}(2,\R)$-bundle. The bundle $F \to \mf$ is equipped with a tautological $\R^2$-valued $1$-form $\omega=(\omega^i)$ satisfying the equivariance property $(R_a)^*\omega=a^{-1}\omega$, where the $1$-form $\omega$ is defined by $\omega_u=u\circ \upsilon^{\prime}_{u}$.

Suppose $\varphi=(\varphi^i_j) \in \Omega^1(F,\mathfrak{gl}(2,\R))$ is the connection $1$-form of $\nabla \in [\nabla]$, then we have the structure equations
$$
\aligned
\d \omega^i&=-\varphi^i_j\wedge\omega^j,\\
\d\varphi^k_l+\varphi^k_j\wedge\varphi^j_l&=\frac{1}{2}\left(\delta_i{}^k\Rho_{jl}-\delta_j{}^k\Rho_{il} -2\Rho_{[ij]}\delta_l{}^k\right)\omega^i\wedge\omega^j,
\endaligned
$$
where -- by slight abuse of notation -- the $\R_2\otimes \R_2$-valued map $\Rho=(\Rho_{ij})$ represents the Schouten tensor of $\nabla$. We define a right $G$-action on $F\times \R_2$ by the rule
$$
(u,\xi)\cdot (b\rtimes a)=\left(\det a^{-1} a^{-1} \circ u,\xi a \det a + b\det a\right),
$$
for all $b\rtimes a \in G$. Denoting by $\pi : F\times \R_2 \to N$ the basepoint projection of the first factor, this action turns $\pi : F\times \R_2 \to \mf$ into a principal right $G$-bundle over $\mf$. On $F\times \R_2$ we define the $\mathfrak{sl}(3,\R)$-valued $1$-form
$$
\theta=\left(\begin{array}{cc} -\frac{1}{3}\tr \varphi+\xi \omega & -\d \xi+\xi\varphi-\Rho^t\omega-\xi\omega\xi\\  \omega & \varphi-\frac{1}{3}\mathrm{I}\tr \varphi-\omega\xi  \end{array}\right).
$$
Then $(\pi : F \times \R_2 \to \mf,\theta)$ is a Cartan geometry of type $(\mathrm{SL}(3,\R),G)$ satisfying the properties (iii) to (v) for the projective structure defined by $\nabla$. It follows from the uniqueness part of Cartan's construction that $(\pi : F \times \R_2\to\mf,\theta)$ is isomorphic to Cartan's bundle.

Note that we may think of the left action of $G\subset \mathrm{SL}(3,\R)$ on $\R_3$ by matrix multiplication as a (linear) $G$-representation and consequently, we obtain an associated rank-$3$ vector bundle $E$ for every projective surface $(\mf,[\nabla])$. The vector bundle $E$ is commonly referred to as the~\textit{tractor bundle} of $(\mf,[\nabla])$. Interest in $E$ stems from the fact that it comes canonically equipped with an $\mathrm{SL}(3,\R)$ connection which is flat if and only if $(\mf,[\nabla])$ is. We refer the reader to~\cite{BEG} for additional details.

\subsection{Anti-self-duality}
Let $M$ be an oriented four--dimensional manifold with a metric
$g$ of signature $(2, 2)$. The Hodge $\ast$ operator is an
involution on two-forms, and induces a decomposition
\be \label{splitting}
\Lambda^{2}(T^*M) = \Lambda_{+}^{2}(T^*M) \oplus \Lambda_{-}^{2}(T^*M)
\ee
of two-forms
into self-dual (SD)
and anti-self-dual (ASD)  components, which
only depends on the conformal class of $g$. 
The Riemann tensor of $g$ 
has the symmetry $R_{abcd}=R_{[ab][cd]}$ so can be thought of
as a map $\mathcal{R}: \Lambda^{2}(T^*M) \rightarrow \Lambda^{2}(T^*M)$
which admits a decomposition   under (\ref{splitting}):
\be \label{decomp}
{\mathcal R}=
\left(
\mbox{
\begin{tabular}{c|c}
&\\
$C_+-2\Lambda$&$\phi$\\ &\\
\cline{1-2}&\\
$\phi$ & $C_--2\Lambda$\\&\\
\end{tabular}
} \right) .
\ee
Here $C_{\pm}$ are the SD and ASD parts 
of the (conformal) Weyl tensor, $\phi$ is  the
trace-free Ricci curvature, and $-24\Lambda$ is the scalar curvature which acts
by scalar multiplication. 
The metric $g$ is ASD if $C_{+}=0$. It is ASD and Einstein if 
$C_{+}=0$ and $\phi=0$. Finally it is ASD Ricci--flat 
(or equivalently hyper-symplectic) if $C_{+}=\phi=\Lambda=0$. In this case
the Riemann tensor is also anti-self-dual.
\vskip3pt
Locally there exist real rank-two vector bundles $\spp, \spp'$  (spin-bundles) over $M$ equipped with parallel symplectic structures
$\ep, \ep'$ such that
\be
\label{can_bun_iso}
T M\cong {\spp}\otimes {\spp'}
\ee
is a  canonical bundle isomorphism, and
\[
g(v_1\otimes w_1,v_2\otimes w_2)
=\varepsilon(v_1,v_2)\varepsilon'(w_1, w_2)
\]
for $v_1, v_2\in \Gamma(\spp)$ and $w_1, w_2\in \Gamma(\spp')$.
A vector $V\in \Gamma(TM)$ is called null if $g(V, V)=0$. Any null vector is of the form
$V=\lambda \otimes \pi$ where $\lambda$, and $\pi$ are sections of
$\spp$ and $\spp'$ respectively.
An $\alpha$--plane (respectively a $\beta$--plane) 
is a two--dimensional plane in $T_pM$
spanned by null vectors of the above form with $\pi$ (respectively
$\lambda$) fixed, and
an $\alpha$--surface ($\beta$--surface) is a two--dimensional surface in $\zeta\subset M$ such 
that its tangent plane at every point is an $\alpha$--plane ($\beta$--plane). The 
seminal theorem  of Penrose \cite{penrose} states that   a 
maximal, three dimensional, family of $\alpha$--surfaces exists 
in $M$ iff $C_+=0$.

\section{From projective to bi-La\-grang\-ian structures}

In this section we show how to canonically construct a bi-La\-grang\-ian structure on the total space of a certain rank $2$ affine bundle over a projective surface $(N,[\nabla])$. 

For what follows it will be necessary to have an interpretation of vector fields on $\mf$ in terms of $G$-equivariant maps on $\projb$. To this end let 
$$
(b\rtimes a)^{-1}=\begin{pmatrix} \det a & -(\det a)b a^{-1} \\ 0 & a^{-1}\end{pmatrix}.
$$
Hence
$$
(R_{b\rtimes a})^*\theta=\begin{pmatrix} \det a & -(\det a)b a^{-1} \\ 0 & a^{-1}\end{pmatrix}\begin{pmatrix} -\tr \phi & \eta \\ \omega & \phi \end{pmatrix}\begin{pmatrix} \det a^{-1} & b \\ 0 & a\end{pmatrix}
$$
the right hand side of which becomes 
$$
\begin{pmatrix} -\det a \tr \phi-(\det a) ba^{-1}\omega & (\det a) \eta-(\det a)ba^{-1}\phi \\ a^{-1}\omega & a^{-1}\phi \end{pmatrix}\begin{pmatrix} \det a^{-1} & b \\ 0 & a\end{pmatrix}
$$
and thus
\begin{equation}\label{rightactioncartan}
(R_{b\rtimes a})^*\theta=\begin{pmatrix} -\tr \phi-b\cdot a^{-1}\omega & * \\ (\det a^{-1})a^{-1}\omega & a^{-1} \omega b+a^{-1}\phi a \end{pmatrix}
\end{equation}
with
\begin{equation}\label{rightactioncartan2}
*=-(\det a)b\tr\phi-(\det a)ba^{-1}\omega b+\eta (\det a)a-(\det a)ba^{-1}\phi a.
\end{equation}
Since
\begin{equation}\label{yetanothereq}
(R_{b\rtimes a})^*\omega=(\det a^{-1})a^{-1}\omega,
\end{equation}
it follows that the tangent bundle of $\mf$ is the bundle associated to the representation $\chi : G \to \mathrm{GL}(2,\R), 
$
where $\chi$ is defined by the rule
$$
\chi(b\rtimes a)x=(\det a)ax,
$$
for all $x \in \R^2$ and $b\rtimes a \in G$. Consequently, the vector fields on $\mf$ are represented by smooth $\R^2$-valued functions $X$ on $\projb$ that are equivariant with respect to $\chi$, that is,
\begin{equation}\label{vectfieldrightaction}
(R_{b\rtimes a})^*X=\chi((b\rtimes a)^{-1})X=(\det a^{-1})a^{-1}X
\end{equation}
for all $b\rtimes a \in G$.

Recall that the group $G$ also acts faithfully on $\R_2$ by affine transformations defined by the rule
$$
\left(b\rtimes a\right)\cdot \xi=\xi a^{-1}\det a^{-1}-ba^{-1}
$$
for $b\rtimes a \in G$. Therefore, the bundle associated to $\projb$ via this affine $G$-action is a rank-$2$ affine bundle $M\to \mf$ and from~\eqref{vectfieldrightaction} we see that its associated vector bundle is the cotangent bundle of $\mf$. We will refer to $M$ as the~\textit{canonical affine bundle} of $(\mf,[\nabla])$. Note that by construction $M \subset \mathbb{P}(E)$.

By definition, an element of $M$ is an equivalence class $[u,\xi]$ with $u\in \projb$ and $\xi \in \R_2$ subject to the equivalence relation
$$
(u_1,\xi_1)\sim (u_2,\xi_2) \quad \iff \quad u_2=u_1\cdot b\rtimes a\;\;\land \;\; \xi_2=(b\rtimes a)^{-1}\cdot \xi_1, \quad b\rtimes a \in G.  
$$  
Clearly, every element of $M$ has a representative $(u,0)$, unique up to a $\mathrm{GL}(2,\R)$ transformation, where here $\mathrm{GL}(2,\R)\subset G$ consists of those elements $b\rtimes a \in G$ satisfying $b=0$. For simplicity of notation we will henceforth write $a$ instead of $0\rtimes a$ for the elements of $\mathrm{GL}(2,\R)\subset G$. It follows that as a smooth manifold, $M$ is canonically diffeomorphic to the quotient $\projb/\mathrm{GL}(2,\R)$.
\begin{rmk}
It can be shown that the sections of $M \to N$ are in one-to-one correspondence with the $[\nabla]$-representative connections. The submanifold geometry in $M$ of representative connections will be studied in depth in a forthcoming article by the second author~\cite{tm15}.
\end{rmk}

\subsection{A bundle embedding}\label{subsec:bundleembedding}

It turns out that we can embed $\projb \to M$ as subbundle of the coframe bundle $F\to M$ of $M$. Here, we define a coframe at $p \in M$ to be a linear isomorphism $T_pM \to \R_2\oplus \R^2$ and we denote the tautological $\R_2\oplus\R^2$-valued $1$-form on $F$ by $\zeta$. 

By definition of $M$, a vector field $\bX$ on $M$ is represented by a unique $(\R_2\oplus \R^2)$-valued function $(X_+,X_-)$ on $\projb$ satisfying the equivariance condition
\begin{equation}\label{anothereq}
R_{a}^*X_+=X_+ a\det a, \quad R_{a}^*X_-=(\det a^{-1})a^{-1}X_-. 
\end{equation}
Therefore, we obtain a unique map $\psi : \projb \to F$ having the property that for every vector field $\bX$ on $M$ and for all $u \in \projb$  
$$
\psi(u)(\bX(\mu(u)))=(X_+(u),X_-(u)),
$$
where $(X_+,X_-)$ is the function on $\projb$ representing $\bX$. Clearly, $\psi$ is a smooth embedding.  Furthermore, from~\eqref{anothereq} we obtain
$$
\psi(u\cdot a)=\psi(u)\cdot\chi(a)
$$
where $\chi : G\ni \mathrm{GL}(2,\R) \to \mathrm{Aut}(\R_2\oplus\R^2)$ is the Lie group embedding defined by the rule
$$
\chi\left(a\right)(\xi,x)=\left(\xi a \det a,(\det a^{-1})a^{-1} x\right).
$$
Consequently, the pair $(\psi,\chi)$ embeds $\projb \to M$ as a subbundle of the coframe bundle of $M$ whose structure group is isomorphic to $\mathrm{GL}(2,\R)$. Furthermore, unraveling the definition of $\zeta$, it follows that we have
\begin{equation}\label{pullbackcanonical}
\psi^*\zeta=(\eta,\omega). 
\end{equation}
The induced geometric structure on $M$ defined by the reduction of the coframe bundle of $M$ is a bi-La\-grang\-ian structure, so we will study these structures next. 
 
\subsection{Bi-La\-grang\-ian structures}

A \textit{bi-La\-grang\-ian} structure on smooth $4$-mani\-fold $M$ (or more generally an even dimensional manifold) consists of a symplectic structure $\Omega$ together with a splitting of the tangent bundle of $M$ into a direct sum of $\Omega$-La\-grang\-ian subbundles $E_{\pm}$
$$
TM= E_+\oplus E_-. 
$$
A manifold equipped with a bi-La\-grang\-ian structure will be called a bi-La\-grang\-ian manifold. The endomorphism $I : TM \to TM$ defined by
$$
\bv=\bv_++\bv_-\mapsto \bv_+-\bv_-, \quad \bv \in TM, \bv_{\pm} \in E_{\pm}
$$
is $\Omega$-skew-symmetric, satisfies $I^2=\mathrm{Id}$ and its $\pm 1$-eigenbundle is $E_{\pm}$. Clearly, $I$ is the unique endomorphism of the tangent bundle having these properties and therefore, we may equivalently think of a bi-La\-grang\-ian structure as a pair $(\Omega,I)$ consisting of a symplectic structure $\Omega$ and a $\Omega$-skew-symmetric endomorphism $I : TM \to TM$ whose square is the identity.

Note also, that we may use the pair $(\Omega,I)$ to define a pseudo-Riemannian metric 
$$
g(\bv,\bw)=\Omega(\bv,I(\bw)), \quad \bv,\bw \in TM, 
$$
whose signature is $(2,2)$ and for which $I$ is skew-symmetric. Of course, a bi-La\-grang\-ian structure is also equivalently described in terms of the pair $(g,I)$ or the pair $(g,\Omega)$. 

\begin{rmk}
What we call a bi-La\-grang\-ian structure is also referred to as an~\textit{almost para-K\"ahler structure} and a~\textit{para-K\"ahler structure} provided $E_{\pm}$ are both Frobenius integrable. Note that in~\cite{bryantboch} the term bi-La\-grang\-ian structure is reserved for the case where both $E_{\pm}$ are Frobenius integrable. 
\end{rmk} 

A bi-La\-grang\-ian structure admits an interpretation as a reduction of the structure group of the coframe bundle of $M$. To this end consider the symmetric bilinear form of signature $(2,2)$ on $\R_2\oplus \R^2$
$$
\left\langle (\xi_1,x^1),(\xi_2,x^2)\right\rangle=-\frac{1}{2}\left(\xi_1x^2+\xi_2x^1\right)
$$
and the skew-symmetric non-degenerate bilinear form
$$
\rangle(\xi_1,x^1),(\xi_2,x^2)\langle=\frac{1}{2}\left(\xi_1x^2-\xi_2x^1\right).
$$
The two bilinear forms are related by the endomorphism $\iota$ sending $(\xi,x) \mapsto (\xi,-x)$. The endomorphism $\iota$ satisfies $\iota^2=\mathrm{Id}$ and its $1$-eigenspace is $\R_2\oplus \{0\}$ and its $-1$-eigenspace is $\{0\}\oplus \R^2$. By construction, both eigenspaces are null and La\-grang\-ian, that is, both bilinear forms vanish identically when restricted to the $\iota$-eigenspaces. The group $\mathrm{GL}(2,\R)$ acts from the left on $\R_2\oplus \R^2$ by
$$
a\cdot (\xi,x)=\left(\xi a^{-1},ax\right)
$$
and this action preserves both bilinear forms. We henceforth identify $\mathrm{GL}(2,\R)$ with its image subgroup in $\mathrm{Aut}( \R_2\oplus \R^2)$. In fact, $\mathrm{GL}(2,\R)$ is the largest subgroup of $\mathrm{Aut}(\R_2\oplus \R^2)$ preserving both bilinear forms.   

Given a bi-La\-grang\-ian structure $(\Omega,I)$ on $M$ we say that a coframe $u$ at $p \in M$ is~\textit{adapted} to $(\Omega,I)$ if for all $\bv, \bw\in T_pM$ 
$$
\Omega_p(\bv,\bw)=\rangle u(\bv),u(\bw)\langle \quad \text{and}\quad \left(u\circ I\right)(\bv)=\left(\iota\circ u\right)(\bv). 
$$
The set of all coframes of $M$ adapted to $(\Omega,I)$ defines a reduction $\lambda : \bilb \to M$ of the coframe bundle $F \to M$ of $M$ with structure group $\mathrm{GL}(2,\R)$. Conversely, every reduction of the coframe bundle of $M$ with structure group $\mathrm{GL}(2,\R)$ defines a unique pair $(\Omega,I)$, consisting of a non-degenerate $2$-form on $M$ and a $\Omega$-skew symmetric endomorphism $I : TM \to TM$ whose square is the identity. Note however that $\Omega$ need not be closed.   

The tautological $\R_2\oplus\R^2$-valued $1$-form $\zeta$ on $\bilb$ will be written as $\zeta=(\eta,\omega)$, so that $\eta=(\eta_i)$ is an $\R_2$-valued $1$-form on $\bilb$ and $\omega=(\omega^i)$ is an $\R^2$-valued $1$-form on $\bilb$. By construction, we have
$$
\lambda^*\Omega=-\eta\wedge\omega\equiv-\eta_i\wedge\omega^i.
$$
Furthermore, let $\hat{L}_{\pm}=\left(\lambda^{\prime}\right)^{-1}(E_{\pm})\subset T\bilb$, then the subbundle $E_+$ is defined by the equations $\eta=0$ and the subbundle $E_-$ is defined by the equations $\omega=0$.   


A linear connection on $F$ is said to be adapted to $(\Omega,I)$ if it pulls back to $\bilb$ to become a principal $\mathrm{GL}(2,\R)$-connection on $\bilb$. An adapted connection is given by a $\mathfrak{gl}(2,\R)$-valued equivariant $1$-form $\nu$ on $\bilb$ such that
$$
\aligned
\d\eta&=-\eta\wedge\nu+\frac{1}{2}T_+\left((\eta,\omega)\wedge(\eta,\omega)\right),\\
\d \omega&=-\nu\wedge\omega+\frac{1}{2}T_-\left((\eta,\omega)\wedge(\eta,\omega)\right),\\
\endaligned
$$
for some torsion map $T_+$ on $\bilb$ with values in $\mathrm{Hom}(\Lambda^2(\R_2\oplus \R^2),\R_2)$ and some torsion map $T_-$ on $\bilb$ with values in $\mathrm{Hom}(\Lambda^2(\R_2\oplus \R^2),\R^2)$, both of which are equivariant with respect to the $\mathrm{GL}(2,\R)$ right action. It is an easy exercise in linear algebra to check that for every bi-La\-grang\-ian structure there exists a unique adapted connection $\nu$ so that
\begin{equation}\label{firststruceqbil}
\aligned
\d\eta&=-\eta\wedge\nu+\frac{1}{2}T_+\left(\omega\wedge\omega\right),\\
\d \omega&=-\nu\wedge\omega+\frac{1}{2}T_-\left(\eta\wedge\eta\right),\\
\endaligned
\end{equation}
with $T_+$ taking values in $\mathrm{Hom}(\Lambda^2\R^2,\R_2)$ and $T_-$ taking values in $\mathrm{Hom}(\Lambda^2\R_2,\R^2)$. It follows that $E_{\pm}$ is integrable if and only if $T_{\pm}$ vanishes identically. Furthermore, the identity $\d(\eta\wedge\omega)=0$ implies
$$
T_+(\omega\wedge\omega)\wedge\omega=0\quad \text{and}\quad T_-(\eta\wedge\eta)\wedge\eta=0.
$$
 The linear connection $\nu$ on the bundle of adapted frames induces connections on the tensor bundles of $M$ in the usual way. By construction, the induced connection ${}^{\nu}\nabla$ on $TM$ is the unique (affine) connection  with torsion $\tau$ satisfying
$$
{}^{\nu}\nabla \Omega=0 \quad \text{and}\quad {}^{\nu}\nabla I=0 \quad\text{and}\quad \tau(X_+,X_-)=0, 
$$
for all $X_{\pm} \in \Gamma(E_{\pm})$. To the to best of our knowledge, the connection ${}^{\nu}\nabla$ was first studied by Libermann~\cite{libermann}, so we call $\nu$ the~\textit{Libermann connection}. Of course, if $\tau$ vanishes identically, then ${}^{\nu}\nabla$ is just the Levi-Civita connection of $g$. 

\subsection{From projective to bi-La\-grang\-ian structures}

Denoting by $\bilb$ the bundle of adapted coframes of a bi-La\-grang\-ian structure $(\Omega,I)$ and by $\projb$ the Cartan bundle of a projective structure $[\nabla]$, we obtain:
\begin{theo}\label{maincorrespondence}
Let $(\mf,[\nabla])$ be a projective surface with Cartan bundle $(\pi : \projb \to \mf,\theta)$. Then there exists a bi-La\-grang\-ian structure $(\Omega,I)$ on the quotient $M=\projb/\mathrm{GL}(2,\R)$ having the following property: There exists a $\mathrm{GL}(2,\R)$-bundle isomorphism $\psi : \projb \to \bilb$ so that
$$
\psi^*\left(\begin{array}{cc} -\frac{1}{3}\tr \nu & \eta \\ \omega & \nu-\frac{1}{3}\mathrm{Id}\tr \nu\end{array}\right)=\theta, 
$$
where $(\eta,\omega)$ denotes the tautological $1$-form on $\bilb$ and $\nu$ the Libermann connection. Moreover, the $E_-$-bundle of the bi-Lagrangian structure $(\Omega,I)$ is always integrable and the $E_+$-bundle is integrable if and only if $[\nabla]$ is flat.  
\end{theo}
\begin{proof}
We write
$$
\theta=\left(\begin{array}{cc} -\tr \phi & \hat{\eta} \\ \hat{\omega} & \phi\end{array}\right)
$$
for the Cartan connection on $\projb$. From \S\ref{subsec:bundleembedding} we know that we have an embedding $(\psi,\chi)$ of $\projb \to M$ as a $\mathrm{GL}(2,\R)$-subbundle $\lambda : \bilb \to M$ of the coframe bundle of $M$. Furthermore, $\psi$ satisfies 
$$
\psi^*(\eta,\omega)=(\hat{\eta},\hat{\omega}),
$$ 
where $(\eta,\omega)$ denotes the tautological $(\R_2\oplus\R^2)$-valued $1$-form on $\bilb$. Therefore, we obtain a unique non-degenerate $2$-form $\Omega$ on $M$ and a unique $\Omega$-skew-symmetric endomorphism $I : TM \to TM$ whose square is the identity. The $2$-form $\Omega$ pulled back to $\bilb$ becomes $-\eta\wedge \omega$. The structure equations~\eqref{struceqproj} imply that we have
\begin{equation}\label{struceqpart}
\aligned
\d \hat{\omega}&=-(\phi+\mathrm{I}\tr \phi)\wedge\hat{\omega},\\
\d \hat{\eta}&=-\hat{\eta}\wedge(\phi+\mathrm{I}\tr \phi)+L(\hat{\omega}\wedge\hat{\omega}). 
\endaligned
\end{equation}
In particular, we obtain
$$
\aligned
\d \left(\hat{\eta}\wedge\hat{\omega}\right)&=\left[-\hat{\eta}\wedge(\phi+\mathrm{I}\tr \phi)+L(\hat{\omega}\wedge\hat{\omega})\right]\wedge\hat{\omega}-\hat{\eta}\wedge\left[-(\phi+\mathrm{I}\tr \phi)\wedge\hat{\omega}\right]\\
&=L\left(\hat{\omega}\wedge\hat{\omega}\right)\wedge\hat{\omega}=0,
\endaligned
$$
where the last equality follows since $\mf$ is two-dimensional. This shows that $\Omega$ is symplectic, so that the pair $(\Omega,I)$ defines a bi-La\-grang\-ian structure on $M$. The equivariance properties of $\theta$ and~\eqref{struceqpart} imply that the $\psi$-pushforward of $\phi+\mathrm{I}\tr\phi$ is a principal right $\mathrm{GL}(2,\R)$-connection on $\bilb$ which satisfies~\eqref{firststruceqbil} with $T_-\equiv 0$ and $T_+=L\circ \psi^{-1}$. In particular, $E_-$ is always integrable and $E_+$ is integrable if and only if $L$ vanishes identically, that is, $[\nabla]$ is flat. Denoting by $\nu$ the Libermann connection of $(\Omega,I)$, we obtain from its uniqueness that
\begin{equation}\label{pullblibermann}
\psi^*\nu=\phi+\mathrm{I}\tr \phi, 
\end{equation}
which completes the proof. 
\end{proof}
Denoting by $\rho : M \to \mf$ the basepoint projection, we immediately obtain:
\begin{col}\label{vectisom}
For every open set $U\subset \mf$ the Lie algebra of projective vector fields $\mathcal{P}_{[\nabla]}(U)$ is isomorphic to the Lie algebra of bi-La\-grang\-ian vector fields $\mathcal{B}_{(\Omega,I)}(\rho^{-1}(U))$. 
\end{col}
\begin{rmk}
Here we call a vector field $X$ defined on some open subset $U\subset (N,[\nabla])$~\textit{projective}, if its flow maps unparametrised geodesics of $[\nabla]$ to unparametrised geodesics of $[\nabla]$. The set of such vector fields on $U$ is a Lie subalgebra of the Lie algebra of vector fields on $U$ which we will denote by $\mathcal{P}_{[\nabla]}(U)$. Likewise, we call a vector field $X$ defined on some open subset $V\subset (M,\Omega,I)$~\textit{bi-La\-grang\-ian} if its (local) flow preserves both $\Omega$ and $I$. The set of such vector fields on $V$ is a Lie subalgebra of the Lie algebra of vector fields on $V$ which we will denote by $\mathcal{B}_{(\Omega,I)}(V)$. 
\end{rmk}
\begin{proof}[Proof of Corollary~\ref{vectisom}]
By standard results about Cartan geometries (c.f.~\cite{parabook}), the projective vector fields on $U\subset (N,[\nabla])$ are in one-to-one correspondence with the vector fields on $\pi^{-1}(U)\subset \projb$ whose flow preserves the Cartan connection $\theta$. Theorem~\ref{maincorrespondence} implies that such a vector field corresponds to a vector field on $\psi(\pi^{-1}(U))\subset \bilb$ preserving both the tautological form $(\eta,\omega)$ and the Libermann connection. Again, by standard results about $G$-structures~\cite{parabook}, such vector fields are in one-to-one correspondence with vector fields on $\rho^{-1}(U)$ preserving both $\Omega$ and $I$.   
\end{proof}
\begin{rmk}
In addition to the proof of Corollary~\ref{vectisom} relying on standard results about $G$-structures and Cartan geometries, we will provide an explicit lift of a projective vector field on $N$ to a bi-La\-grang\-ian vector field on $M$ 
in Theorem \ref{theo_kill}. In addition, in \S\ref{killlift} a stronger result 
will be established: any Killing vector on $(M, g)$ is necessarily 
bi--La\-grang\-ian, and arises from some 
projective vector field on $N$.
\end{rmk}

\begin{rmk}
Note that if $\bX_{x}$ is a vector field on $\projb$ having the property that
$$
\omega(\bX_{x})=x, \quad \eta(\bX_{x})=0, \quad \phi(\bX_{x})=0, 
$$
for some non-zero $x \in \R^2$, then the the integral curve of $\bX_{x}$, when projected to $\mf$, becomes a geodesic of $[\nabla]$. Conversely every geodesic of $[\nabla]$ arises in this way ~(see for instance \cite{kobanag} for details). Likewise, a geodesic of the Libermann connection arises as the projection of an integral curve of a horizontal vector field on $\bilb$ which is constant on the canonical $1$-form. It follows that the geodesics on $(\mf,[\nabla])$ correspond to the geodesics of the Libermann connection on $(M,\Omega,I)$ that are everywhere tangent to $E_-$.  
\end{rmk}
\subsection{The choice of a representative connection}

Recall from \S\ref{cartgeomsec} that the choice of a representative connection $\nabla \in [\nabla]$ gives a $G$-bundle isomorphism $\projb\simeq F \times \R_2$. In particular, we obtain an affine bundle isomorphism $\psi : (F\times\R_2)/\mathrm{GL}(2,\R) \to M$. By construction, the quotient $(F\times\R_2)/\mathrm{GL}(2,\R)$ is just the cotangent bundle of $\mf$. Denoting by $\lambda : F\times \R_2 \to T^*\mf$ the basepoint projection, we obtain 
\begin{equation}\label{metrisymppreferred}
\aligned
(\psi\circ \lambda)^*g&=-\left(-\d \xi+\xi\varphi-\Rho^t\omega-\xi\omega\xi\right)\odot \omega,\\
(\psi\circ \lambda)^*\Omega&=\omega\wedge\left(-\d \xi+\xi\varphi-\Rho^t\omega-\xi\omega\xi\right),
\endaligned
\end{equation}
where the $\R_2\otimes \R_2$-valued map $\Rho=(\Rho_{ij})$ on $F$ represents the Schouten tensor and the $\varphi$ the connection form of $\nabla$. A choice of local coordinates $x=(x^i) : U \to \R^2$ on $\mf$ induces a trivialisation $\upsilon^{-1}(U)\simeq U \times \mathrm{GL}(2,\R)$ so that 
$$
\omega=a^{-1}\d x,\quad \varphi=a^{-1}\d a +a^{-1}\Gamma \d x \,a,
$$
where $\Gamma=(\Gamma^i_{jk})$ denote the Christoffel symbols of $\nabla$ with respect to $x$ and $a : U \times \mathrm{GL}(2,\R) \to \mathrm{GL}(2,\R)$ the projection onto the latter factor. Taking the quotient by $\mathrm{GL}(2,\R)$, we obtain from~\eqref{metrisymppreferred} the following coordinate expressions
\begin{equation}\label{coordexpmetsymp}
\aligned
g&=\left(\d \xi_i\odot\d x^i -\left(\xi_l\Gamma^l_{ij}-\Rho_{(ij)}-\xi_i\xi_j\right)\d x^i\odot \d x^{j}\right),\\
\Omega&=\d \xi_i \wedge \d x^i+\Rho_{[ij]}\d x^i\wedge \d x^j.
\endaligned
\end{equation}

\section{Gauge theory on surfaces with projective structure}
Let $(\mf, [\nabla])$ be a projective structure on a surface, and let
$E\rightarrow \mf$ be a vector bundle  with
$\mathfrak{g}$--connection $A$, where $\mathfrak{g}$ is some Lie algebra. 
Let $\phi$ be a one-form on $N$, called the Higgs pair, 
with values  in the adjoint 
bundle $\mbox{adj}(E)$. In an open set $U\subset \mf$ we shall write 
$\phi=\phi_i dx^i$. Let $\nabla\in [\nabla]$ be a chosen connection in 
the projective class. The Calderbank equations \cite{Cal1,Cal2} hold if
\be
\label{cal_2}
D_{(i}\phi_{j)}=0,
\ee
where 
\[
D_i\phi_j:=\partial_i\phi_j-\Gamma_{ij}^k\phi_k-[A_i, \phi_j].
\]
Equivalently,
 the Higgs pair 
is constant along the charged geodesic
spray on $T\mf$ i.e.
\be
\label{cal_1}
{\bf \Theta}^A(\phi):=\Big(\pi^i\frac{\partial}{\partial x^i}-\Gamma_{ij}^k\pi^i\pi^j\frac{\partial}{\partial \pi^k}\Big) (\phi)-[A, \phi]=0
\ee
where  $\pi^i$ are coordinates on the fibres of $T\mf$, and 
$\phi=\phi_i\pi^i$ and  $A=A_i\pi^i$ are $\mathfrak{g}$--valued linear
functions on $T\mf$.
The equations (\ref{cal_2}) do not depend on the choice of the connection 
$\nabla$ in 
the 
projective class if the Higgs pair $\phi$ has projective weight $2$.

In the \S\ref{main_tractor_section}  we shall show how the Calderbank equations with the gauge group
$\mathrm{SL}(3, \R)$ regarded as a subgroup of the group of diffeomorphisms of
$\RP^2$ leads to neutral signature anti--self--dual Einstein metrics 
in four dimensions, which are also bi--La\-grang\-ian and given by 
(\ref{coordexpmetsymp}).
We shall first list some other (implicit) occurrences
of these equations for other gauge groups.

\subsection{Null reductions of anti-self-dual Yang--Mills equations.} 
If the projective structure is flat, then (\ref{cal_2}) is 
the symmetry reduction of the anti-self-dual Yang--Mills
(ASDYM) equation on $\R^{2, 2}$ by two null translations and such 
that the $(2, 2)$ 
metric $g$ restricted to the two--dimensional space of orbits $\mf=\R^2$
is totally isotropic, and the bi-vector generated by the null translations is anti-self-dual.


To see it, consider a $\mathfrak{g}$--valued connection one--form $A$ on 
$\R^{2, 2}$, and set
$F=dA+A\wedge A$.
In local coordinates adapted to $\R^{2, 2}=T\mf$ with $x^i$ the coordinates 
on $\mf$,
the null isometries are $\partial/\partial \xi_i$, and
the metric is
\[
g=dx^1d\xi_1+dx^2d\xi_2.
\]
Choose an orientation on $\R^{2, 2}$ such that the two--form $dx^1\wedge dx^2$ is ASD.
Defining two Higgs fields $\phi_1=\partial/\partial \xi_2\hook A, \phi_2=\partial/\partial \xi_1\hook A$, 
the ASDYM equations $F=-*F$ yield \cite{MW}
\be
\label{null_ASD}
D_1 \phi_1=0, \quad D_2 \phi_2=0, \quad D_1 \phi_2+D_2 \phi_1=0,
\ee
where $D=d+A_1dx^1+A_2dx^2$ is a covariant derivative on $\mf$ induced by $A$.
In \cite{tafel} these equations have been solved completely for the gauge 
group $\mathrm{SL}(2)$.
\subsubsection{Prolongation of the Calderbank equations}
Instead of regarding both the connection and the Higgs pair as unknowns we
shall assume that the connection is given and consider  (\ref{null_ASD})
as a system of PDEs for the Higgs pair.
To determine all derivatives of the Higgs pair in (\ref{null_ASD})
we prolong the system once, and define a section $\mu$ of $\mbox{adj}(E)$ by
\[
D_i\phi_j=\frac{1}{2}\mu\epsilon_{ij},
\]
where $\epsilon=dx^1\wedge dx^2$ is the parallel volume form of $\nabla\in [\nabla]$.
Commuting the covariant derivatives now leads to a closed 
system and therefore a connection
\[
D_i\left(\begin{array}{c}
\phi_j\\ 
\mu
\end{array} \right)=\left(\begin{array}{c}
\frac{1}{2}\mu\epsilon_{ij}\\ 
2[\phi_i, {\mathcal F}]
\end{array} \right),
\]
where ${\mathcal F}=[D_1, D_2]$ is the $\mathfrak{g}$--valued curvature of the connection 
$A$. The system is now closed. Commuting the covariant derivatives on $\mu$ leads to an integrability condition
\[
[{\mathcal F}, \mu]-2D_1[\phi_2, {\mathcal F}]+2D_2[\phi_1, {\mathcal F}]=0.
\]
\subsection{Killing equations.} 
If the connection $A$ is flat, and $\mathfrak{g}=\R$ then
the Calderbank equations become the projectively invariant
Killing equations.
\subsection{Anti-self-dual conformal structures with null conformal Killing vectors.}
Let $\mathfrak{g}$ be a subalgebra of infinite dimensional Lie algebra of vector fields $\mathfrak{diff}(\Sigma)$ of vector fields on a surface $\Sigma$
consisting of those elements of $\mathfrak{diff}(\Sigma)$ which commute with a 
fixed vector field $K$ on $\Sigma$. Let $M\rightarrow\mf$ be a surface bundle over $\mf$, with two dimensional fibres $\Sigma$.
In this case the Calderbank equations
are solvable by quadrature and the two-dimensional distribution 
\be
\label{twistor_dist}
{\mathcal D}=\{
{\bf \Theta}^A:=\pi^i\frac{\p}{\partial x^i}-\Gamma_{ij}^k\pi^i\pi^j\frac{\p}{\partial \pi^k}
-A_i(x)\pi^i, \; \phi=\pi^i\phi_i\}
\ee
spanning an $\RP^1$ worth of null self--dual surfaces ($\alpha$--surfaces) through
each point of $M$ is the twistor distribution
for the most general ASD $(2, 2)$ conformal 
structure
which admits a null conformal Killing vector $K$ \cite{DW, Cal1, nakata}.
\subsection{Walker's Riemannian extensions}
The conformal structure resulting from the distribution
(\ref{twistor_dist}) is a generalisation of the Walker
lift \cite{walker,Gilkey}.  To recover the Walker lift 
\be
\label{walker}
g=d\xi_i\odot dx^i-\Gamma_{ij}^k\;\xi_k\; dx^i\odot dx^j,
\ee
take
the gauge algebra $\mathfrak{g}=\mathfrak{gl}(2,\R)$ which generates linear
transformations of $\Sigma=\R^2$. If the coordinates on $\Sigma$ 
are $(\xi_1, \xi_2)$,
the elements of $\mathfrak{gl}(2,\R)$ are vector fields of the 
form 
$
{{\bf t}_i}^j= \xi_i\frac{\p}{\partial \xi_j}.
$
Taking the 
connection $A$ and the  Higgs one--form given 
by
\[
A_i=-\Gamma_{ij}^k\;\xi_k\frac{\p}{\partial \xi_j}, \quad \phi_i=(b^k\xi_k)\epsilon_{ij}\frac{\p}{\partial \xi_j}
\]
where $b^k$ is a non--zero constant
leads to an integrable distribution (\ref{twistor_dist}), as
then
\begin{eqnarray*}
[{\bf \Theta}^A, \phi] &=&\pi^ib^j
\Big( {\Gamma_{jk}}^k\xi_i+ {\Gamma_{ij}}^k\xi_k\Big)
\epsilon_{lm}\pi^l\frac{\p}{\partial \xi_m}\\
&=&0 \quad(\mbox{mod}\; \phi).
\end{eqnarray*}
The resulting
metric (\ref{walker}) on $M=T\mf$ is then uniquely determined by the condition that  the integral two--surfaces of ${\mathcal D}$ in $T\mf\times\RP^1$ project down to self-dual totally null surfaces 
on $T\mf$. 
The generators of the gauge group
satisfy
$
[{\bf v}, K]={\bf v},
$
where $K=\xi_1\partial/\partial \xi_1+\xi_2\partial/\partial \xi_2$ is a 
conformal null Killing vector
of (\ref{walker}).

We shall end this subsection by clarifying  the connection between
changing the projective class of $(\mf, [\nabla]$), and conformal 
rescalings of the metric $g$ on $T^*\mf$. 
We shall 
restrict our discussion to {\em special connections} in 
$[\nabla]$ which preserve some volume.
Consider the effect of transformation (\ref{connection_change}) with $\Upsilon_i=\nabla_i f$, together with rescaling the fibers of $T\mf\rightarrow \mf$
\[
\xi_i\rightarrow \hat{\xi}_i=e^{2f}\xi_i
\]
on the Walker lift\footnote{In \cite{DT10} (see also 
\cite{BDE,hamerl,hamerl2} for other applications of this lift) 
it was proven that a `similar' metric
\be
\label{walker_dt}
g=d\xi_i\odot dx^i-\Pi_{ij}^k\;\xi_k\; dx^i\odot dx^j,
\ee
constructed out of the 
Thomas symbols $\Pi_{ij}^k=\Gamma_{ij}^k-\frac{1}{3}\Gamma_{il}^l\delta^k_j
-\frac{1}{3}\Gamma_{jl}^l\delta^k_i$
is anti--self--dual  and null--K\"ahler  (with ASD null--K\"ahler two--form) for any choice of $\Gamma_{ij}^k$.
The Walker lift (\ref{walker}) is conformally equivalent 
(up to a diffeomorphism) to the projective
Walker lift (\ref{walker_dt})
 only if
$\Gamma_{ij}^j=\nabla_i F$ for some function $F$ on $N$.
}  (\ref{walker}). A straightforward calculation yields
\[
\hat{g}=e^{2f} g.
\]
Thus conformal scales on $T\mf$ correspond to projective scales on $\mf$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Tractor connection and ASD Einstein metrics}
\label{main_tractor_section}
In this Section we shall consider the Calderbank equations, where
the gauge group is $\mathrm{SL}(3, \R)$, and $E$ is the tractor bundle
for the projective structure $[\nabla]$.

Let $\mathcal{E}(1)$ be the line bundle of projective densities of weight 1 
(see Section \ref{projective_sec}).
Consider a rank-three vector bundle $E={\mathcal E}(1)\oplus
(T^*\mf\otimes{\mathcal E}(1))$ over  $\mf$ with connection \cite{BEG}
\be
\label{tractor_con}
{\quad{\mathcal{D}}_i \left(\begin{array}{c}
\sigma\\ 
\mu_j
\end{array} \right)= 
\left(\begin{array}{c} \nabla_i\sigma-\mu_i \\ 
\nabla_i\mu_j+\Rho_{ij}\sigma
\end{array} \right),}
\ee
where $\Rho_{ij}$ is the (not necessarily symmetric) Schouten tensor of projective geometry. The splitting of the tractor bundle depends on a choice of a connection $\nabla$ in the projective class $[\nabla]$, and 
under (\ref{connection_change})
changes according to 
\be
\label{tractor_change}
\left(\begin{array}{c}
\hat{\sigma}\\ 
\hat{\mu}_j
\end{array} \right)=
\left(\begin{array}{c}
\sigma\\ 
\mu_j+\Upsilon_j\sigma
\end{array} \right).
\ee
Using the tractor indices $\alpha, \beta, \dots =0, 1, 2$ we can rewrite the connection (\ref{tractor_con}) in terms of its Christoffel symbols
$\gamma_{i\alpha}^\beta$  as
\[
\gamma_{i0}^0=0, \quad \gamma_{i0}^j=\delta_i^j,\quad
\gamma_{ij}^k=\Gamma_{ij}^k, \quad \gamma_{ij}^0=-\Rho_{ij}.
\]
The vector fields
\[
{{\bf t}_\alpha}^\beta=\psi_\alpha\frac{\p}{\partial \psi_\beta}
\]
generate the linear action of $\mathrm{GL}(3, \R)$ on the fibres of $E$.
These generators descent to eight vector fields
(which we shall also denote ${{\bf t}_\alpha}^\beta$) which generate
the action of $\mathrm{SL}(3, \R)$ on the fibres of the projective tractor
bundle $\PP(E)$ which is a quotient of $E$ by the Euler vector field
$\sum_{\alpha=0}^2 {{\bf t}_\alpha}^\alpha$. Setting $\xi_i=\psi_i/\psi_0$ yields
\[
{{\bf t}_i}^j=\xi_i\frac{\p}{\partial \xi_j}, \quad  
{{\bf t}_i}^0=-\xi_i\xi_j\frac{\p}{\partial \xi_j}, \quad
{{\bf t}_0}^i=\frac{\p}{\partial \xi_i}, \quad {{\bf t}_0}^0=-\xi_j\frac{\p}{\partial \xi_j}.
\]
Consider the Calderbank equations with the gauge group $\mathrm{SL}(3, \R)\subset\mbox{Diff}(\RP^2)$, where
the connection is given by a vector--valued one-form
\[
A=A_idx^i=-\gamma_{i\beta}^{\alpha} dx^i\otimes\;{{\bf t}_\alpha}^\beta
\]
so that
\[
A_i=(\Rho_{ij}+\xi_i\xi_j  -\Gamma_{ij}^k\;\xi_k)\frac{\p}{\partial \xi_j}.
\]
The Calderbank equations are solved by the Higgs pair
\[
\phi_i=\epsilon_{ij}\frac{\p}{\partial \xi_j}.
\] 
Let $M$ be a complement of a projective line in the total space of the bundle $\PP(E)$.
The corresponding contravariant metric on $M$ is constructed by
demanding that the leaves of the rank-2 distribution (\ref{twistor_dist})
${\mathcal D}\subset T(M\times\RP^1)$ project down to self-dual
two-surfaces on $M$. This gives
$
\epsilon^{ij}(\partial/\partial x^i -A_i)\odot \phi_j,
$
or, in the covariant form,
\be
\label{einstein_lift}
g=(d\xi_i-(\Gamma_{ij}^k \xi_k -\xi_i\xi_j-\Rho_{ji})dx^j)\odot dx^i,
\ee
so that we have recovered the bi-La\-grang\-ian structure
(\ref{coordexpmetsymp}).
\begin{theo}
\label{theo_char}
Formula (\ref{einstein_lift}) defines a metric which does not depend on a
choice of a connection in a projective class.
\label{theo_einstein}
\end{theo}
\noindent
\begin{proof}
If we change the connection in the
projective class using (\ref{connection_change}) then the Schouten tensor changes by
(\ref{schouten_change}). To establish the invariance of (\ref{einstein_lift}) we translate the fibre coordinates according to
\[
\hat{\xi}_i=\xi_i+\Upsilon_i
\]
in agreement with (\ref{tractor_change}).
Then
\begin{eqnarray*}
&&(d\hat{\xi}_i-({\hat{\Gamma}_{ij}}^k \hat{\xi}_k-\hat{\xi}_i\hat{\xi}_j-\hat{\Rho}_{ji})dx^j)\odot dx^i
= d\xi_i\odot dx^i +\\
&&\Big(\xi_{(j} \Upsilon_{i)} -  \Gamma_{ij}^k\xi_k
-\xi_i\Upsilon_j-\xi_j\Upsilon_i- \Gamma_{ij}^k\Upsilon_k-2\Upsilon_i\Upsilon_j
+\xi_i\xi_j+\xi_i\Upsilon_j+\xi_j\Upsilon_i+\Upsilon_i\Upsilon_j
\\
&&+\Rho_{ji}-\nabla_{(j}\Upsilon_{i)}+\Upsilon_i\Upsilon_j\Big)dx^i\odot dx^j\\
&&=
\left(d \xi_i-\left(\Gamma_{ij}^k \xi_k
-\xi_i\xi_j-{\Rho}_{ji}\right)dx^j\right)\odot dx^j.
\end{eqnarray*}
\end{proof}
The metric is anti-self-dual, and Einstein
with scalar curvature equal to $-24$. The anti-self-duality is a consequence of the fact
that the connection $A$ and the Higgs field $\phi_i\pi^i$ satisfy the 
Calderbank equations \cite{Cal1}. We shall establish the anti-self-duality 
for a larger class of metrics of the modified Walker 
form \cite{spanish,derdzinski2}, by verifying the integrability of the 
twistor distribution  (\ref{twistor_dist}). 
\begin{prop} The conformal class defined by the metric
\be
\label{el2}
g=\left(d \xi_i-\left({\Gamma}_{ij}^k \xi_k -\xi_i\xi_j-{M}_{ij}\right)dx^j\right)\odot dx^i
\ee
is anti-self-dual for any $\Gamma_{ij}^k=\Gamma_{ij}^k(x^l)$
and ${M}_{ij}={M}_{ij}(x^l)$. Moreover the metric (\ref{el2}) is Einstein
with non-zero scalar curvature if and only if  $M_{ij}=\Rho_{(ij)}$ is the
symmetrised projective Schouten tensor of $\Gamma_{ij}^k$.
\end{prop}
\noindent
\begin{proof}
We shall compute the Lie bracket
\begin{eqnarray*}
[{\bf \Theta}^A, \phi]&=&[\pi^i\Big(\frac{\p}{\partial x^i}+  
\Gamma_{ij}^k\;\xi_k\frac{\p}{\partial \xi_j}-\xi_i\xi_j\frac{\p}{\partial \xi_j}
-M_{ij}\frac{\p}{\partial \xi_j}\Big) -\Gamma_{ij}^k\pi^i\pi^j\frac{\p}{\p\pi^k}
, \pi^i\epsilon_{ij}\frac{\p}{\partial \xi_j}]\\
&=& -\pi^i\Gamma_{ij}^k \epsilon_{km}\Big(\pi^j\frac{\p}{\partial \xi_m}-\pi^m\frac{\p}{\partial \xi_j}\Big) +\xi_i\pi^i \pi^j\epsilon_{jk}\frac{\p}{\partial \xi_k}   \\
&=& (-\pi^j{\Gamma_{jk}}^k +\pi^j\xi_j) \phi
\in\mbox{span}\{{\bf \Theta}^A, \phi\}.
\end{eqnarray*}
Therefore ${\mathcal D}$ is Frobenius integrable, and there exists a 
one-parameter family of self-dual two-surfaces through each point of $M$,
and (by Penrose's Nonlinear Graviton Theorem \cite{penrose}) the conformal structure
of $g$ is ASD. The last part of the proposition is established by directly computing the Einstein tensor.
\end{proof}
\subsection{The characterisation of the projective-to-Einstein lift.}
A general, real analytic ASD Einstein metric in four dimensions depends on two arbitrary functions of three variables. If the projective structure on $\mf$ is 
real  analytic, then
metrics of the form (\ref{main_metric})
depend on two arbitrary functions of two variables: two out of six functions
$\Gamma_{ij}^k$ can be eliminated by a diffeomorphism of $\mf$, and
two more functions can be eliminated by making a projective change
of connection (\ref{connection_change})  which, by Theorem \ref{theo_einstein},
leaves the metric invariant.
In this Section we shall invariantly characterise the ASD Einstein metrics  arising from projective structures
by the ansatz (\ref{main_metric}).


Recall \cite{Walker_1, Gilkey, Chudecki} that
a distribution ${\mathcal D}\subset TM$ is called parallel if
${{}^g\nabla}_X Y\in\Gamma ({\mathcal D})$ if  
$X\in\Gamma ({\mathcal D})$, where ${{}^g\nabla}$ is the Levi--Civita connection 
of $g$. 
Thus, if ${\mathcal D}$ is parallel, then
it is necessarily Frobenius integrable
as $[X, Y]={{}^g\nabla}_X Y - {{}^g\nabla}_Y X  \in\Gamma ({\mathcal D})$
if  $X, Y\in\Gamma ({\mathcal D})$.
\begin{theo}
\label{theo_asd}
Let $(M, g)$ be an Einstein metric with scalar curvature scaled to $-24$,
such that Weyl tensor is ASD, and there exists a parallel 
ASD totally null distribution. Then $g$ is conformally flat, or it is locally 
of the 
form (\ref{einstein_lift}), with $M=T^*\mf$  for some projective structure
$[\nabla]$ on $\mf$.
\end{theo}
\noindent
\begin{proof}
We shall rely on  two isomorphisms: $TM=\spp\otimes \spp'$, and 
${\Lambda^2}_-=\spp\odot \spp$. Let the ASD totally null distribution
correspond to an ASD two-form $\Theta\in\Gamma({\Lambda^2}_-)$, or equivalently to a section $\iota\in\Gamma(\spp)$. The Frobenius integrability conditions imply the local existence of two functions
$\xi_1$ and $\xi_2$ on $M$ such that $\mbox{Ker}(\Theta)=\mbox{span}\{\partial/\partial \xi_1, \partial/\partial \xi_2\}$. We can rescale $\iota$ so that the corresponding two--form is 
closed, and proportional to $dx^1\wedge dx^2$ for some functions $(x^1, x^2)$
which are constant on each $\beta$--surface in the two parameter family.
The functions $(\xi_1, \xi_2)$ are then the coordinates on the $\beta$--surface.
The corresponding metric takes the form
\[
g=d\xi_i\odot dx^i+\Theta_{ij}(x,\xi)dx^i\odot dx^j
\]
for some symmetric two-by-two matrix $\Theta$. The anti--self--duality 
condition on the Weyl tensor forces the components of $\Theta$ to be at most 
cubic in $(\xi_1, \xi_2)$, with some additional algebraic relations between the components. Imposing the Einstein  
condition  (with non--zero Ricci scalar scaled to $-24$) 
gives $\Theta_{ij}=\xi_i\xi_j+\Rho_{ji}-\Gamma_{ij}^k\xi_k$, where
the functions $\Gamma_{ij}^k$ do not depend on $p$s and are otherwise arbitrary,
and the functions $\Rho_{ij}$ are determined by (\ref{formula_for_rho}).
\end{proof}
\begin{rmk}
Existence of a neutral metric with a two--plane distribution 
impose
topological restrictions on $M$. If $M$ is compact then \cite{atiyah, HH}
\[
\chi[M] \equiv 0 \ \mbox{mod} \ 2, \ \ \ \ \ \chi[M] \equiv \tau[M] \
\mbox{mod} \ 4,
\]
where $\tau[M]$ and $\chi[M]$ are the signature and Euler characteristic
respectively.
\end{rmk}
\begin{rmk}
Rescaling the fibre coordinates in (\ref{einstein_lift}) allows us to change 
the Ricci scalar from $-24$ to  $-24\Lambda$, where $\Lambda$ is 
an arbitrary non-zero constant. This gives the metric (\ref{main_metric})
\[
g=\left(d\xi_i-\left(\Gamma_{ij}^k \xi_k-\Lambda \xi_i\xi_j- \frac{1}{\Lambda}\Rho_{ji}\right)dx^j\right)\odot dx^i.
\]
The one--parameter family of metrics (\ref{einstein_lift}) does not admit the Ricci flat limit unless $\Rho_{(ij)}=0$. The  special case $\Rho_{ij}=0$
provides the homogeneous model of our construction which will be discussed in
\S\ref{sec_homo}.
\end{rmk}
\begin{rmk}
Performing the analogous rescaling on the bi--La\-grang\-ian structure
(\ref{coordexpmetsymp}) gives the  {\em charged} 
symplectic structure\footnote{This terminology is motivated by
the Hamiltonian description of a charged particle moving on a manifold, where the canonical symplectic structure on the cotangent bundle needs to be modified by a pull-back of a closed two-form (magnetic field) from the base manifold. In our case the two-form is the skew-symmetric part
of the Schouten tensor, and the inverse of the Ricci scalar plays 
a role of electric charge. This magnetic term can always be set to zero by an appropriate choice of a connection in a projective class - here we find it convenient not to make any choices at this stage.} $\Omega$ given by
(\ref{s_form}). This two-form is ASD with respect to our choice of orientation, and
the metric (\ref{main_metric}). We find that
\[
{}^g\nabla\Omega=4L
\]
where $L=\epsilon^{bc}\nabla_{b}P_{ca}dx^a\otimes (dx^1\wedge dx^2)$ is the Liouville tensor which vanishes iff $[\nabla]$ is projectively flat.
\end{rmk}
\section{Killing vector fields from projective vector fields}\label{killlift}
In this Section we shall establish a correspondence between symmetries of a projective structure and isometries of the corresponding Einstein metric.
As a spin--off from our analysis we shall show how to reduce a problem of solving a system of second order PDEs for projective symmetries, to 
a problem involving a system of first order PDEs for Killing vectors. 

We shall first consider a correspondence between
symmetries of affine connection and Killing vectors for a Walker metric 
(\ref{walker}). Let $\Gamma_{ij}^k$ be an affine connection on $\mf$. A vector field $K$ 
on $\mf$ 
is called affine
if
\be
{\mathcal L}_K \Gamma_{ij}^k\equiv \frac{\p^2 K^k}{\partial x^i \partial x^j}
+K^m\frac{\p}{\partial x^m} \Gamma_{ij}^k- \Gamma_{ij}^m\frac{\partial K^k}{\partial x^m}
+\Gamma_{im}^k\frac{\partial K^m}{\partial x^j}+\Gamma_{jm}^k\frac{\partial K^m}{\partial x^i}
= 0
\ee
or equivalently if
\[
\nabla_i\nabla_j K^k ={R_{ij}{}^k}{}_l K^l.
\]
Any vector field on $\mf$ corresponds to a linear function on $T^*\mf$,
which in 
local coordinates is given by $K^i\xi_i$. This function, together
with the canonical symplectic structure on $T^*\mf$ gives rise to the 
Hamiltonian vector field
\be
\label{complete_lift}
\widetilde{K}=K^i\frac{\p}{\partial x^i}-\xi_j\frac{\partial K^j}{\partial x^i}\frac{\p}{\partial \xi_i}.
\ee
This vector field is some-times referred to as the complete lift \cite{yano}.
\begin{prop}
\label{prop_kill_1}
Let $K$ be an affine vector field for a connection $\nabla$ on $U\subset N$.
 Then
its complete lift  (\ref{complete_lift}) is a Killing vector field
for the Walker lift (\ref{walker}).
\end{prop}
\noindent 
\begin{proof}
Consider the one--parameter group of transformations generated by the vector 
field (\ref{complete_lift})
\[
x^i\longrightarrow x^i+\epsilon\; K^i+O(\epsilon^2), \quad \xi_i\longrightarrow
\xi_i-\epsilon\; \xi_j\frac{\partial K^j}{\partial x^i}+O(\epsilon^2).
\]
This yields
\begin{eqnarray*}
g&&\longrightarrow g+\epsilon\{\xi_j K^idx^j d\xi_i-\xi_i K^j dx^id\xi_j
-(\xi_j\xi_i\xi_k K^j)dx^idx^k\\
&&-2\Gamma_{ik}^j\xi_j(\xi_m K^i)dx^kdx^m
+\Gamma_{ik}^j\xi_m\xi_j K^m dx^idx^k-K^m(\xi_m \Gamma_{ik}^j)\xi_j dx^idx^k\}
+O(\epsilon^2)
\\
&&=g-\epsilon\; \xi_k{\mathcal L}_{{K}} (\Gamma_{ij}^k) dx^i\odot dx^j+O(\epsilon^2).
\end{eqnarray*}
Therefore
\be
\label{formula}
{\mathcal L}_{\widetilde{K}} g=-
\xi_k{\mathcal L}_{{K}} (\Gamma_{ij}^k) dx^i\odot dx^j,
\ee
and the result follows.
\end{proof}
Recall that a vector field is projective if its flow  
maps unparametrised geodesics of an affine connection $\nabla$ to 
unparametrised geodesics. This happens if 
\be
\label{projective_field}
{\mathcal L}_K \Gamma_{ij}^k={\delta_i}^k \rho_j+ {\delta_j}^k \rho_i
\ee
for some one-form $\rho_i$.
\begin{prop}
Let $K$ be a projective vector field with $\rho_i=\nabla_i f$. Then
\be
\label{vect_theo}
K-\xi_j\frac{\partial K^j}{\partial x^i}\frac{\p}{\partial \xi_i}+f\xi_i\frac{\p}{\partial \xi_i}
\ee
is a conformal Killing vector field for the Walker metric
(\ref{walker}).
\end{prop}
\noindent
\begin{proof}
The proof is similar to that of Proposition (\ref{prop_kill_1}). The one-parameter
group of transformation generated by (\ref{vect_theo}) is
\[
x^i\longrightarrow x^i+\epsilon\; K^i+O(\epsilon^2), \quad \xi_i\longrightarrow
\xi_i-\epsilon\; \xi_j\frac{\partial K^j}{\partial x^i} -\epsilon f\xi_i +O(\epsilon^2),
\]
which gives
\[
g\longrightarrow 
g-\epsilon\; \xi_k{\mathcal L}_{{K}} (\Gamma_{ij}^k) dx^i\odot dx^j
-\xi_kdx^k\odot df+\epsilon f\;g
+O(\epsilon^2).
\]
This does not change the conformal class iff $K$ satisfies
(\ref{projective_field}) with $\rho_i=\nabla_i f$.
\end{proof}
Finally we give the main result of this Section, and establish a one--to--one correspondence between projective vector fields on $(\mf, [\nabla])$, and
Killing vector fields on the Einstein lift on $M$.
\begin{theo}
\label{theo_kill}
Let $K$ be a projective vector field on $(U, [\nabla])$, where $U\subset N$. 
Then
\be
\label{symp_kil}
{\mathcal K}:=K-\xi_j\frac{\partial K^j}{\partial x^i}\frac{\p}{\partial \xi_i}+
\frac{1}{\Lambda}\rho_i\frac{\p}{\partial \xi_i}
\ee
is a Killing vector for the Einstein Riemannian extension 
(\ref{main_metric}) which is symplectic with respect to the charged symplectic form (\ref{s_form}).
Conversely, any Killing vector field for (\ref{main_metric}) is a lift 
(\ref{symp_kil}) from $\mf$ of some projective vector field.
\end{theo}
\noindent
\begin{proof} The integrability conditions for (\ref{projective_field})
are \cite{yano2} (note however that that our sign conventions for the Schouten tensor
differ from that in \cite{yano2}, so the sign of the RHS of (\ref{rho_integrability})
is opposite to what is given in \cite{yano2})
\be
\label{rho_integrability}
{\mathcal L}_K\Rho_{ij}=-\nabla_i\rho_j.
\ee
Rewrite the Einstein metric (\ref{main_metric}) as
\[
g=g_W+\frac{1}{\Lambda} P_S+\Lambda \theta\odot \theta,
\]
where $g_W$ is the Walker metric (\ref{walker}), $P_S$ is the symmetrised part
of the Schouten tensor, and finally $\theta=\xi_idx^i$. We shall also
write ${\mathcal K}=\tilde{K}+ K_\rho$, where
$\tilde{K}$ is the complete lift (\ref{complete_lift}) 
and $K_\rho:=\rho_i\partial/\partial \xi_i$. Then
\begin{eqnarray*}
{\mathcal L}_{\mathcal K} g&=& {\mathcal L}_{\widetilde{K}} g_W+
\Lambda {\mathcal L}_{\widetilde{K}} \theta\odot\theta
+\frac{1}{\Lambda} {\mathcal L}_{K} P_S+
\frac{1}{\Lambda}  {\mathcal L}_{K_\rho} g_W+
{\mathcal L}_{K_\rho} (\theta\odot\theta)\\
&=& -\xi_k{\mathcal L}_{{K}} (\Gamma_{ij}^k) dx^i\odot dx^j
+0-\frac{1}{\Lambda}(\nabla_i\rho_j) dx^i\odot dx^j
+\frac{1}{\Lambda} dx^i\odot d\rho_i\\
&&
-\frac{1}{\Lambda} \Gamma_{ij}^k\rho_k dx^i \odot dx^j +
(\rho_idx^i)\odot (\xi_j dx^j)=0,
\end{eqnarray*}
where we have used (\ref{formula}), (\ref{projective_field})
and (\ref{rho_integrability}).

Now verify the symplectic condition
\begin{eqnarray*}
{\mathcal L}_{\mathcal K} \Omega&=&
{\mathcal L}_{\widetilde{K}} (d\xi_i\wedge dx^i)
+\frac{1}{\Lambda}{\mathcal L}_{K_\rho} (d\xi_i\wedge dx^i)
+\frac{1}{\Lambda}{\mathcal L}_{K} (\Rho_{ij}dx^i\wedge dx^j)\\
&=&\frac{1}{\Lambda}(d\rho_i\wedge dx^i- d\rho_i\wedge dx^i)=0
\end{eqnarray*}
as the complete lift $\widetilde{K}$ is Hamiltonian with respect to
$d\xi_i\wedge dx^i$, and we have used the skew part of the integrability conditions (\ref{rho_integrability}).

To prove the converse, consider a general vector field 
${\mathcal K}=K^i\partial/\partial x^i
+Q_i\partial/\partial \xi_i$ on $M$, and impose the Killing equations.
The $d\xi_i\odot d\xi_j$ components of these equations imply that $K^j=K^j(x^1, x^2)$.
The $d\xi_i\odot d x^j$ components yield the general form  (\ref{symp_kil}),
where $\rho_i$ are some unspecified functions on $N$. Finally 
the $dx^i\odot dx^j$ components imply that the vector field 
$K^i\p/\p x^i$ on $N$ is projective.
\end{proof}
\section{Examples}
\subsection{Homogeneous model $M=\mathrm{SL}(3, \R)/\mathrm{GL}(2, \R)$.}
\label{sec_homo}
Consider the flat projective structure on $(\mf=\RP^2, [\nabla])$, and choose
${\Gamma_{ij}}^k=0$. The resulting four manifold is the complement of 
an $\RP^1$ sub-bundle in the projective tractor bundle of $\RP^2$ which can be identified with with $M=\mathrm{SL}(3, \R)/\mathrm{GL}(2, \R)$. We shall establish this result in arbitrary dimension. Consider $\mf=\RP^n$, with
its flat projective structure, and an $\mathrm{SL}(n+1)$ action on the projective
tractor bundle $\PP(E)$ minus the diagonal   
\[
\mathrm{SL}(n+1):\R^{n+1}\times \R_{n+1}/\Delta\longrightarrow \R^{n+1}\times \R_{n+1}/\Delta
\]
where the `diagonal' $\Delta$ consists of all incident pairs of vectors $[v]\in \R^{n+1}$ and forms $[f]\in \R_{n+1}$ s.t. the corresponding point $v\in \RP^n$
belongs to  the hyperplane $f\in \RP_n$. This action is simply 
$(v, f)\rightarrow(Av, fA^{-1})$. It  is transitive, and clearly a subgroup stabilising a pair (point,  hyperplane) is $\mathrm{GL}(n)$ which sits in $\mathrm{SL}(n+1)$ as a lower diagonal
block.

To finish the proof we need to argue that $\R^{n+1}\times \R_{n+1}/\Delta$ projects down to a complement of an $\RP_{n-1}$ sub-bundle in $\PP(E)$. This
sub-bundle is just $\PP(T^*\mf)$ and it has an injection into $\PP(E)$ 
given by $f\rightarrow (0, f)$. A point in $\mf$ with homogeneous coordinates $[1, 0, ..., 0]$
(corresponding to our  choice of an affine chart) is not incident with
any tractor in $\PP(E)/\RP_{n-1}$, so removing a diagonal is equivalent to
looking at the complement of this sub-bundle.

The Einstein metric
(\ref{einstein_lift}) on this manifold admits a Kerr-Schild form
\be
\label{sl3gl2}
g=d\xi_i\odot dx^i +\Lambda (\xi_jdx^j)^2
\ee
with eight dimensional isometry group $\mathrm{SL}(3, \R)$ \cite{CDT13}
in agreement with Theorem \ref{theo_kill}).
This metric is a neutral signature analog of the Fubini--Study metric
on $\CP^2$. Both metrics arise as different real forms
of $\mathrm{SL}(3, \C)/\mathrm{GL}(2, \C)$. The limit $\Lambda=0$ 
in (\ref{sl3gl2}) gives the flat metric.
\subsection{Ricci--flat limits.} Motivated by the previous example
let us now consider the general case of projective structures which admit a connection with skew-symmetric
Schouten tensor. In this case one can always choose local coordinates
on $N$ and a connection $\nabla\in [\nabla]$ such that
\cite{Wong}
\[
\Gamma_{11}^1=-\frac{\partial f}{\partial x^1}, \quad 
\Gamma_{22}^2=\frac{\partial f}{\partial x^2},
\]
where $f:\mf\longrightarrow \R$ is an arbitrary function, and all other components of $\nabla$
vanish\footnote{An alternative characterisation of the 
corresponding projective structures is that they arise from second-order ODEs point equivalent to derivatives of first order ODEs \cite{DW}.
These projective structures where further characterised in 
\cite{randal} and \cite{krynski}.}. In this case
\[
\Rho=\frac{1}{3}\frac{\p^2 f}{\partial x^1\partial x^2} dx^1\wedge dx^2,
\]
and the metric is given by
\be
\label{skew_metrics}
g=d\xi_i\odot dx^i +\xi_1\frac{\partial f}{\partial x^1} (dx^1)^2
-\xi_2\frac{\partial f}{\partial x^2} (dx^2)^2+
\Lambda (\xi_jdx^j)^2.
\ee
Setting $\Lambda=0$ gives an ASD Ricci-flat 
metric which has a form of the Walker lift (\ref{walker}) 
and has  appeared in the work of Derdzinski \cite{derdzinski}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Cohomogeneity--one examples}
The dimension of the Lie algebra $\mathfrak{g}$ of projective vector fields for a given  projective structure on a surface $N$ can be $8, 3, 2, 1$ or $0$
(see \cite{Lie}, and also \cite{romanovskii,bryant_mano,dumitriescu}). If 
the dimension is maximal and equal to 8 then $\mathfrak{g}=\mathfrak{sl}(3, \R)$, and
the projective structure is flat. We have shown that in this case the resulting
metric (\ref{main_metric}) is given by (\ref{sl3gl2}), and admits $8$ Killing 
vectors in agreement with Theorem \ref{theo_kill}. We shall now consider the 
submaximal case, where $\mathfrak{g}=\mathfrak{sl}(2, \R)$. There are two one--parameter families of non--flat projective structures with this symmetry. Their unparametrised geodesics are integral curves of a second order ODE
\[
y''=c(xy'-y)^3,
\]
where $c\neq 0$. We compare this to the general second order ODE defining a projective structure (see e.g. \cite{BDE})
\be
\label{projective_ode}
y''=\Gamma_{22}^1 (y')^3+(2\Gamma_{12}^1-\Gamma_{22}^2) (y')^2+(\Gamma_{11}^1-
2\Gamma_{12}^2)y'-\Gamma_{11}^2,
\ee
and chose the representative connection $\nabla$ by
\[
\Gamma_{11}^1= -\Gamma_{12}^2=-\Gamma_{21}^2 =cxy^2, \quad 
\Gamma_{22}^2=- \Gamma_{21}^1= -\Gamma_{12}^1= cx^2y, \quad \Gamma_{22}^1=cx^3, \quad \Gamma_{11}^2=cy^3.
\]
The corresponding ASD Einstein metric (\ref{main_metric}) is
\be
\label{sl_metric}
g=d\xi_i \odot dx^i +\Lambda(\xi_i dx^i)^2+\frac{4c}{\Lambda}(x^2dx^1-x^1dx^2)^2-\Gamma_{ij}^kdx^i\odot dx^j,
\ee
where $x^i=(x, y)$. This metric admits a three--dimensional isometry group
$SL(2, \R)$
generated by left--invariant vector fields $K_\alpha, \alpha=1, 2, 3$ given by
\[
K_1=x^1\frac{\p}{\p x^1}-x^2\frac{\p}{\p x^2}-
\xi_1\frac{\p}{\p \xi_1}+\xi_2\frac{\p}{\p \xi_2}, \quad
K_2=2x^1\frac{\p}{\p x^2}-2\xi_2\frac{\p}{\p \xi_1}, \quad
K_3= 2\xi_1\frac{\p}{\p \xi_2}- 2x^2\frac{\p}{\p x^1},
\]
and acting on $M=\R\times SL(2, \R)$ with three--dimensional orbits. We shall
use an invariant coordinate $r$ given by
 $r^2\equiv {(x^1\xi_1+x^2\xi_2)}$ 
which is constant on the orbits. Let $\sigma^{\alpha}$ be right--invariant one--forms
on $SL(2, \R)$ such that
\[
{\mathcal{L}}_{K_\alpha} \sigma^\beta=0, \quad \forall \alpha, \beta,\quad
\mbox{and}\quad
d\sigma^1+2\sigma^2\wedge\sigma^3=0, \quad
d\sigma^2+\sigma^2\wedge\sigma^1=0, \quad
d\sigma^3-\sigma^3\wedge\sigma^1=0.
\]
There is some freedom, measured by functions of $r$, in choosing these one--forms. 
If we chose $\Lambda<0$, and take
\[
\sigma^1= \frac{\xi_idx^i-x^id\xi_i}{r^2}+ \frac{2\Lambda rdr}{\Lambda r^2-1}, 
\quad \sigma^2=\frac{\Lambda r^2-1}{r^2}(x^1 dx^2-x^2dx^1),
\quad  \sigma^3=  
\frac{\xi_1d\xi_2-\xi_2 d\xi_1}{r^2(\Lambda r^2-1)}
\] 
then
the metric (\ref{sl_metric}) takes the form
\be
g=\label{homogeneous_metric}
\frac{dr^2}{1-\Lambda r^2}-\frac{1}{4}r^2(1-\Lambda r^2)(\sigma^1)^2-
\frac{c}{\Lambda}\frac{(\Lambda r^2-4)r^4}{(\Lambda r^2-1)^2}(\sigma^2)^2 +r^2\sigma^2\odot\sigma^3, \quad \Lambda<0.
\ee
This metric appears to be singular when $r=0$, but calculating the invariant 
norm of the Weyl curvature we find $|C|^2=96\Lambda^2$, which is regular. In fact near $r=0$ the metric
(\ref{homogeneous_metric}) approaches the space of constant curvature which is a 
neutral signature analogue of the hyperbolic space. To exhibit this space
in a standard form we neglect the small terms involving $r^4$, 
and set $r=2R/(1+\Lambda R^2)$. Then, near $R=0$,  the metric 
(\ref{homogeneous_metric}) becomes
\[
g\sim \frac{4}{(1+\Lambda R^2)^2}\Big( dR^2 -\frac{R^2}{4}\Big( (\sigma^1)^2-4\sigma^2\odot\sigma^3\Big)\Big).
\]
To this end, we note a curious Ricci--flat limit of (\ref{homogeneous_metric}).
Setting $c=m\Lambda$, and taking the limit $\Lambda\rightarrow 0$ yields
a Ricci--flat metric with 9--dimensional group of conformal isometries
\[
g=d\xi_i\odot dx^i+4m(x^2dx^1-x^1dx^2)^2.
\]
This is a submaximal  metric of neutral 
signature \cite{CDT13, the}: if the dimension of the conformal
isometry algebra $\mathfrak{g}$ exceeds 9, then $\mathfrak{g}=\mathfrak{sl}(4, \R)$,
and the metric is conformally flat.

\appendix

\section{The construction for higher dimensions}

Of course, the definition of a projective structure makes sense in higher dimensions as well and hence it is natural to ask if the construction described in the main body of this article carries over to higher dimensions. Here we briefly show that this is indeed the case.  

As usual, let $\mathrm{PGL}(n+1,\R)$ denote the quotient of the general linear group $\mathrm{GL}(n+1,\R)$ by its center $Z$, so that
$$
\mathrm{PGL}(n+1,\R)\simeq \left\{ \begin{array}{cl} \mathrm{SL}(n+1,\R) &n\;\text{even},\\\mathrm{SL}_{\pm}(n+1,\R)/\{\pm\mathrm{I}_{n+1}\}& n\;\text{odd},\end{array}\right.
$$
where $\mathrm{SL}_{\pm}(n+1,\R)$ denotes the group of real $(n+1)$-by-$(n+1)$ matrices with determinant $\pm 1$.  

The projective linear group acts from the left on $\mathbb{RP}^n=\left(\R^{n+1}\setminus\{0\}\right)/\mathbb{R}^*$ by matrix multiplication. The stabiliser subgroup of the line spanned by ${}^t(1\;0\;\ldots\;0)$ will be denoted by $G\subset \mathrm{PGL}(n+1,\R)$. The elements of $G$ are matrices of the form
$$
\begin{pmatrix} \det a^{-1} & b \\ 0 & a\end{pmatrix}
$$
for $n$ even and
$$
\left[\begin{array}{cc} \pm\det a^{-1} & b \\ 0 & a\end{array}\right]
$$
for $n$ odd, where $b\in \R_n$ and $a \in \mathrm{GL}(n,\R)$. Here, the square brackets indicate that the matrix is only well defined up to an overall sign.

Cartan's construction carries over to higher dimensions so that we canonically obtain a Cartan geometry $(\pi : B_{[\nabla]} \to N,\theta)$ of type $(\mathrm{PGL}(n+1,\R),G)$ for every projective structure $[\nabla]$ on a smooth $n$-manifold $N$. Again, we write
$$
\theta=\left(\begin{array}{cc} -\tr \phi & \eta \\ \omega & \phi\end{array}\right)
$$
for an $\R_n$-valued $1$-form $\eta$, an $\R^n$-valued $1$-form $\omega$ and a $\mathfrak{gl}(n,\R)$-valued $1$-form $\phi$. The curvature $2$-form $\Theta$ satisfies
$$
\Theta=\d \theta+\theta\wedge\theta=\left(\begin{array}{cc} 0 & L(\omega\wedge\omega) \\ 0 & W(\omega\wedge\omega)\end{array}\right),
$$
for smooth curvature functions 
\[L : B_{[\nabla]} \to \mathrm{Hom}\left(\R^n\wedge\R^n,\R_n\right)
\] and 
$$
W : B_{[\nabla]} \to \mathrm{Hom}\left(\R^n\wedge\R^n,\R_n\otimes \R^n\right).
$$ 
The function $W$ represents the~\textit{Weyl projective curvature tensor} of $(N,[\nabla])$ and the function $L$ represents the~\textit{projective Cotton tensor} of $(N,[\nabla])$. Note that we also have the Bianchi-identity
$$
\d \Theta=\Theta\wedge\theta-\theta\wedge\Theta,
$$
the algebraic part of which reads
\begin{equation}\label{algbianchi:again}
0=L(\omega\wedge\omega)\wedge\omega\quad \text{and}\quad 0=W(\omega\wedge\omega)\wedge\omega. 
\end{equation}
We have a Lie group embedding defined by
$$
\chi : \mathrm{GL}(n,\R) \to G, \quad a \mapsto \begin{pmatrix} \det a^{-1} & 0 \\ 0 & a\end{pmatrix},
$$
for $n$ even and defined by 
$$
\chi : \mathrm{GL}(n,\R) \to G, \quad a \mapsto 
\left[\begin{array}{cc} |\det a^{-1}| & 0 \\ 0 & a\end{array}\right],
$$
for $n$ odd. 

Recall that $\theta$ satisfies the equivariance property 
$$
R_g^*\theta=\mathrm{Ad}(g^{-1})\circ\theta,
$$
for all $g \in G$, where $\mathrm{Ad}$ denotes the adjoint representation of $G$. Identifying $\mathrm{GL}(n,\R)$ with its image under $\chi$, the equivariance property of $\theta$ implies that the tensor field $\eta\otimes \omega=\eta_i\otimes\omega^i$ is invariant under the $\mathrm{GL}(n,\R)$ right action. Furthermore, since $\omega$ and $\eta$ are both semi-basic for the quotient projection $B_{[\nabla]} \to B_{[\nabla]}/\mathrm{GL}(n,\R)$, it follows that the smooth $2n$-manifold $M=B_{[\nabla]}/\mathrm{GL}(n,\R)$ carries a unique signature $(n,n)$ metric $g$ and a unique non-degenerate $2$-form $\Omega$ having the property that $g$ pulls back to $B_{[\nabla]}$ to be the symmetric part of $\eta\otimes \omega$ and $\Omega$ pulls back to $B_{[\nabla]}$ to be the anti-symmetric part of $\eta\otimes \omega$. Moreover, we compute
$$
\aligned
0=&\;\d\left(\eta\wedge\omega\right)=\d \eta\wedge\omega-\eta\wedge\d\omega=\left[-\eta\wedge(\phi+\mathrm{Id}\tr \phi)+L(\omega\wedge\omega)\right]\wedge\omega\\
&-\eta\wedge\left[-(\phi+\mathrm{Id}\tr\phi)\wedge\omega\right]\\
=&\;L(\omega\wedge\omega)\wedge\omega,
\endaligned
$$ 
where we used~\eqref{algbianchi:again}. It follows that $\Omega$ is symplectic. 

We leave it to the interested reader to check that the pair $(g,\Omega)$ defines again a bi-La\-grang\-ian structure on $M$ whose symmetry vector fields are in one-to-one correspondence with the symmetry vector fields of $(N,[\nabla])$. Moreover, we may introduce local coordinates on $M$ so that $g$ and $\Omega$ take the form~\eqref{coordexpmetsymp}. In particular, the metric $g$ is still Einstein with non-zero scalar curvature, as can be verified by direct computation. 
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\end{document}
     
1509.04276.extracted_bib
{atiyah Atiyah, M.~F. (1970) Vector fields on manifolds. {\em Arbeitsgemeinschaft f\"ur Forschung des Landes Nordrhein Westfalen 200\/ , 7--24.
1509.04276.extracted_bib
{BEG Bailey, T. N., Eastwood, M. G. \and Gover, A. R. (1994) Thomas's structure bundle for conformal, projective and related structures. Rocky Mountain J. Math. {24, 1191--1217.
1509.04276.extracted_bib
{Gilkey Brozos-Vazquez, M., Garcia-Rio, E., Gilkey, P., Nikcevic, S. and Vazquez-Lorenzo, R. (2009) {\em The Geometry of Walker Manifolds. Synthesis Lectures on Mathematics and Statistics. Morgan and Claypool. %
1509.04276.extracted_bib
{walker_book %Brozos-Vazquez, M., Garcia-Rao, E., Gilkey, P., Nikevic, S. and %Vazquez-Lorenzo, R. (2009) %{\em The Geometry of Walker Manifolds Morgan \& Claypool.
1509.04276.extracted_bib
{bryantboch Bryant, R.~L. (2001) Bochner-K\"ahler metrics. J.~Amer.~Math.~Soc. {14, 623--715.
1509.04276.extracted_bib
{bryant_mano Bryant, R.~L., Manno, G. and Matveev, V. (2008) A solution of a problem of Sophus Lie: normal forms of two-dimensional metrics admitting two projective vector fields. Math. Ann. {340, 437.
1509.04276.extracted_bib
{BDE Bryant, R. L., Dunajski, M., and Eastwood, M. G. (2009) Metrisability of two-dimensional projective structures, J. Differential Geometry {83, 465--499.
1509.04276.extracted_bib
{Cal1 Calderbank, D. M. J. (2014) Selfdual 4-manifolds, projective structures, and the Dunajski-West construction. SIGMA {10, 034.
1509.04276.extracted_bib
{Cal2 Calderbank, D. M. J. (2014) Integrable Backround Geometries. SIGMA {10, 035.
1509.04276.extracted_bib
{parabook \v{Cap, A. and Slov\'ak, J. (2009) Parabolic Geometries I: Background and General Theory, American Mathematical Society 2009.
1509.04276.extracted_bib
{MR1504846 Cartan, E. (1924) Sur les vari\'et\'es a connexion projective. Bull. Soc. Math. France. {52, 205--241.
1509.04276.extracted_bib
{spanish Calvino-Louzao, E., Garcia-Rio, E, Gilkey, P. and Vazquez-Lorenzo, R. (2009) The geometry of modified Riemannian extensions. Proceedings of the Royal Society {A465.
1509.04276.extracted_bib
{CDT13 Casey, S., Dunajski, M. and Tod, K. P. (2013) Twistor geometry of a pair of second order ODEs Comm. Math. Phys. { 321, 681--701.
1509.04276.extracted_bib
{Chudecki Chudecki, A. and Przanowski, M. (2008) From hyperheavenly spaces to Walker and Osserman spaces: I Class.Quant.Grav. {25 145010.
1509.04276.extracted_bib
{derdzinski Derdzi\'nski, A. (2008) Connections with skew-symmetric Ricci tensor on surfaces. Results Math. {52, 223-245.
1509.04276.extracted_bib
{derdzinski2 Derdzi\'nski, A (2009) Non-Walker Self-Dual Neutral Einstein Four-Manifolds of Petrov Type III Journal of Geometric Analysis, {19 301-357
1509.04276.extracted_bib
{dumitriescu Dumitriescu, S. and Gulliot, A. (2013) Quasihomogeneous analytic affine connections on surfaces. J. Topol. Anal. {5 491.
1509.04276.extracted_bib
{DW Dunajski, M., and West, S. (2007) Anti-self-dual conformal structures from projective structures. { Comm. Math. Phys. {272, 85--118.
1509.04276.extracted_bib
{DT10 Dunajski, M. and Tod, K. P. (2010) Four Dimensional Metrics Conformal to K\"ahler, Math. Proc. Camb. Phil. Soc. {148, 485-503.
1509.04276.extracted_bib
{EM Eastwood, M. and Matveev, V. S. (2007) Metric connections in projective differential geometry. In {\em Symmetries and Overdetermined Systems of Partial Differential Equations, IMA Vol. Math. Appl., 144.
1509.04276.extracted_bib
{hamerl Hammerl, M. and Sagerschnig, K. (2011) A non-normal Fefferman-type construction of split-signature conformal structures admitting twistor spinors. {\tt arXiv:1109.4231.
1509.04276.extracted_bib
{hamerl2 Hammerl, M., Sagerschnig, K., Silhan, J., Taghavi-Chabert, A. and Zadnik, V. A non-normal Fefferman-type construction of split-signature conformal structures from projective structures. {\em In preparation.
1509.04276.extracted_bib
{HH {Hirzebruch, F. and Hopf, H. (1958) Felder von {Fl\"achenelementen in 4-dimensionalen {Mannigfaltigkeiten. {\em Math. Ann. 136\/, 156--172.
1509.04276.extracted_bib
{kobanag Kobayashi, S. and Nagano, T. (1964) On projective connections, J. Math. Mech. {13, 215--235.
1509.04276.extracted_bib
{krynski Kry\'nski, W. (2014) Webs and projective structures on a plane. Diff. Geom. Appl. {37, 133.
1509.04276.extracted_bib
{the Kruglikov, B. and The, D. (2014) The gap phenomenon in parabolic geometries. J. reine angew. Math.
1509.04276.extracted_bib
{libermann Libermann, P. (1954) Sur le probl\`eme d'\'equivalence de certaines structures infinit\'esimales, Ann. Mat. Pura Appl. (4) {36, 27--120.
1509.04276.extracted_bib
{Lie Lie, S. (1882) Untersuchungen \"uber geod\"atische Kurven, Math. Ann. {20.
1509.04276.extracted_bib
{MW Mason, L.~J. and Woodhouse, N.~M.~J. (1996) {\em {Integrability, selfduality, and twistor theory. \newblock Oxford, UK: Clarendon (LMS monographs, new series: 15).
1509.04276.extracted_bib
{tm15 Mettler, T. (2015) Extremal conformal structures on projective surfaces, in preparation.
1509.04276.extracted_bib
{nakata Nakata, F. (2007) Self-dual Zollfrei conformal structures with alpha-surface foliation. J. Geom. Phys. {57, 2077-2097,
1509.04276.extracted_bib
{penrose Penrose, R. (1976) Nonlinear gravitons and curved twistor theory, Gen. Rel. Grav. {7, 31--52
1509.04276.extracted_bib
{randal Randal, M. (2014) {Local obstructions to projective surfaces admitting skew-symmetric Ricci tensor Jour. Geom. Phys. {76, 192.
1509.04276.extracted_bib
{romanovskii Romanovskii, Y. R. (1996) Calculation of local symmetries of second-order ordinary differential equations by Cartans equivalence method, Math. Notes {60 56.
1509.04276.extracted_bib
{tafel Tafel, J. Wojcik, D. (1998) Null Killing vectors and reductions of the self-duality equations, Nonlinearity {11 835.
1509.04276.extracted_bib
{yano Yano, K. and Ishihara, S. (1973) {\em Tangent and Cotangent Bundles. Differential Geometry Marcel Dekker, Inc. New York.
1509.04276.extracted_bib
{yano2 Yano, K. (1955) {\em The Theory Of Lie Derivatives And Its Applications. North Holland
1509.04276.extracted_bib
{Walker_1 Walker, A. G. (1950) Canonical form for a Riemannian space with a parallel field of null planes. Quart. J. Math. Oxford {1, 69 .
1509.04276.extracted_bib
{walker Walker, A. G. (1953) Riemann extensions of non-Riemannian spaces. In {\em Convegno di Geometria Differenziale. Venice.
1509.04276.extracted_bib
{Wong Wong, Y. C. (1964) Two dimensional linear connexions with zero torsion and recurrent curvature. Monatsh. Math. {68, 175.