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Define a hyperbolic surface $H$ as a metric space locally isometric to the Poincaré upper half-plane $\mathbb{H}^2$ with the appropriate metric. Note that the quotient $\mathbb{H}^2/\Gamma$ is a hyperbolic surface, where $\Gamma$ is a Fuchsian group acting properly discontinuously without fixed point on $\mathbb{H}^2$.

Every closed, connected Riemann surface $S$ of genus greater than $2$ admits the structure of a hyperbolic surface : this is a (fairly) direct consequence of uniformization. Indeed the universal cover $\tilde{S}$ is $\mathbb{H}^2$, and $Aut(\mathbb{H}^2)=PSL(2,\mathbb{R})$, so $S=\tilde{S}/\Gamma$ where $\Gamma$ is Fuchsian, and we conclude by the previous remark.

We have the following result:

Theorem : Let $H$ be a closed, connected, oriented hyperbolic surface of genus $\geq 2$. Then $S$ is isometric to $\mathbb{H}^2/\Gamma$ for some Fuchsian $\Gamma$. In particular, such hyperbolic surface carries the structure of a Riemann surface.

In the proof we do use the fact that $H$ is oriented to show that transition maps are holomorphic. However the following question arises : what are some interesting examples of non-orientable hyperbolic surfaces ? How much work is needed to find such a surface ? I cannot think of one just yet. Moreover, if every closed hyperbolic surface can be obtained by gluing pairs of pants, how do we even find one ? Where does the construction degenerate ?

Many thanks for reading.

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  • $\begingroup$ Can you mimic the standard Klein-bottle construction using e.g. the (6,4) tiling of the hyperbolic plane to glue hexagon edges in some appropriately orientation-reversing fashion? $\endgroup$ – Steven Stadnicki Jan 10 '18 at 19:54
  • $\begingroup$ @StevenStadnicki Is this true ? It seems like a good construction but my intuition in hyperbolic geometry is not strong enough to accept it on good faith. $\endgroup$ – Bass Jan 10 '18 at 20:32
  • $\begingroup$ That's my status as well, I'm afraid - that's why I put it as a comment rather than an answer. It seems as though it should be correct, since even though 'in principle' the tiling is built off of a reflection group one can use (hyperbolic) translations to construct a quotient surface, so in principle we should be able to glue and then use one orientation-reversing gluing. But my intuition for these things is not incredibly strong. $\endgroup$ – Steven Stadnicki Jan 10 '18 at 20:43
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Embed a surface $S$ of odd genus like in the picture: enter image description here so that it is centered and symmetric with respect to the origin (in the obvious sense) Then the antipodal map $\sigma$ is a smooth self-diffeomorphism of order two which reverses the orientation of the surface. The quotient $S/\sigma$ is a non-orientable surface. If we put on $S$ a hyperbolic metric, then one can arrange to have $\sigma$ be an isometry, and then $S/\sigma$ inherits a hyperbolic metric.

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