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\begin{theorem}
    For each $g \ge 2$, there exists an abstract
    Riemannian metric on the compact orientable
    surface of genus $g$ with curvature $\kappa
    \equiv -1$ (locally isometric to $\HH$).
\end{theorem}

\noindent
Recall
\begin{center}
    \includegraphics[width=0.6\linewidth]
    {images/799fedd6c18f11ed.png}
\end{center}
$g$ is the number of holes, $\chi = 2 - 2g$. $g =
0$, round sphere, $\kappa = 1$. $g = 1$, torus,
$\RR^2 / \ZZ^2$ curvature $\kappa \equiv 0$.

\begin{proof}[Sketch proof]
    This will be an outline of the key points.
    Recall that:
    \begin{center}
        \includegraphics[width=0.6\linewidth]
        {images/d7a1e768c18f11ed.png}
    \end{center}
    Analogously, a $4g$-gon with side labels $a_1
    b_1 a_1^{-1} b_1^{-1} a_2 b_2 a_2^{-1}
    b_2^{-1} \ldots a_g b_g a_g^{-1} b_g^{-1}$
    would give (after gluing) an orientable
    surface $\Sigma_g$ of genus $g$.

    \myskip
    Observation: Let's say that a \emph{flag}
    comprises:
    \begin{enumerate}[(i)]
        \item An oriented hyperbolic line.
        \item A point on that line.
        \item A choice of side to the line.
    \end{enumerate}
    Given two such flags, there is a hyperbolic
    isometry taking one to the other: we can swap
    sides using inversions if \emph{needed}. Now
    consider the regular $4g$-gon from Lemma 4.11
    with internal angle $\frac{\pi}{2g}$.
    \begin{center}
        \includegraphics[width=0.6\linewidth]
        {images/b4ce2ad4c19011ed.png}
    \end{center}
    For each paired set of 2 edges, there is a
    hyperbolic isometry taking one to the other
    (respecting orientation) and taking ``inside''
    of the polygon at $e_1$ to the ``outside'' of
    its paired edge $e_2$. Now we'll give an atlas
    for $\Sigma_g$ as follows:
    \begin{itemize}
        \item If $p \in
            \operatorname{Interior}(P)$, just take
            a small disc contained in interir of
            $P$ and include it into $D \subset
            \RR^2$.
            \begin{center}
                \includegraphics[width=0.3\linewidth]
                {images/9ed76f46c19111ed.png}
            \end{center}
        \item If $p \in \operatorname{edge}(P)$
            (not a vertex), say $e_1$ and $\hat{p}
            \in e_2$ and $\gamma_{\hat{p}} = p$,
            we know that $\gamma$ exchanges sides.
            \begin{center}
                \includegraphics[width=0.6\linewidth]
                {images/17d1637ac19211ed.png}
            \end{center}
            \[ [p] = [\hat{p}] \in \Sigma_g =
            \operatorname{Polygon} / \sim \]
            Define $U \cup \tilde{U} \to D$ to be
            the inclusion in $D$ and $\gamma$ on
            $\tilde{U}$. These descend to maps on
            $[U] \subset \Sigma$, $[\tilde{U}]
            \subset \Sigma$ which agree on $[U
            \cap \tilde{U}]$ (projection to
            $\Sigma_g$).
        \item In our gluing pattern all $4g$
            vertices are identified to one point
            on $\Sigma_g$ and we'd like a chart
            there.
            \begin{center}
                \includegraphics[width=0.4\linewidth]
                {images/488b6be6c19211ed.png}
            \end{center}
            \begin{center}
                \includegraphics[width=0.6\linewidth]
                {images/687dbe7cc19211ed.png}
            \end{center}
            Condition that internal angles sum to
            $2\pi$ means that we have a
            neighbourhood of $[v] \in \Sigma_g$
            that defines a chart at $[v] \in
            \Sigma_g$.
    \end{itemize}
    All chrarts are obtained either from inclusion
    into $D$ or the composite of inclusion and
    some hyperbolic isometry (so smooth) and we
    got an atlas that defines a Riemannian metric
    on $\Sigma_g$ that is locally isometric to
    $\HH$.
\end{proof}

\begin{remark*}
    Really what is going on is that $\Sigma_g =
    \HH / \Gamma$ where $\Gamma = \langle
    \gamma_1, \ldots, \gamma_{2g} \rangle$ Fuchsian
    group. ($T^2 = \RR / \ZZ^2$). Algebraic
    topology: $\HH \to \HH / \Gamma$.
\end{remark*}

\noindent
Hyperbolic \emph{right-angled hexagons} are also
very useful for similar constructions.

\begin{lemma}
    For each $l_\alpha, l_\beta, l_\gamma$
    positive numbers, there exists a right-angle
    hexagon with side lengths $l_\alpha, ?,
    l_\beta, ?, l_\gamma, ?$ in cyclic order:
    \begin{center}
        \includegraphics[width=0.6\linewidth]
        {images/5eb3835cc19411ed.png}
    \end{center}
\end{lemma}

\begin{proof}
    Take a pair of ultraparallel lines. Example
    sheet 4 implies there exists a unique common
    perpendicular geodesic
    \begin{center}
        \includegraphics[width=0.6\linewidth]
        {images/a4d1e2acc19411ed.png}
    \end{center}
    Given $l_\alpha$ and $l_\beta > 0$ we shoot
    off new geodesics $\sigma$ and
    $\tilde{\sigma}$ orthogonal to the originals
    and having travelled $l_\alpha$ and $l_\beta$
    from common perpendicular. In fact given $t >
    0$ there exists an original ultraparallel pair
    distance exactly $t$ apart. There exists a
    threshold value $t_0$, by continuity when the
    new common perpendicular has length
    $l_\gamma$.
    \begin{center}
        \includegraphics[width=0.6\linewidth]
        {images/8019f282c19511ed.png}
    \end{center}
    This is our right-angled hexagon.
\end{proof}

\myskip
Gluing two identical right-angled hexagons gives
rise to a \emph{pair of pants}:
\begin{center}
    \includegraphics[width=0.6\linewidth]
    {images/c8bfef5ac19511ed.png}
\end{center}
$g$, $\kappa \equiv -1$.

\myskip
Interesting things to look up: Teichm\"uller space, Moduli space.
