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It is known that any closed orientable surface of genus $g \geq 2$ admits a hyperbolic metric, and the Teichmüller space of such metrics has dimension $6g - 6$. I was wondering if there is a corresponding statement for non-orientable closed surfaces of negative Euler characteristic. By the Gauss-Bonnet theorem and Ricci flow theory, for example, I can see that such surfaces admit many hyperbolic metrics, but I am not sure about the structure of the space of such metrics.

Also, on the Teichmüller space for a genus $g \geq 2$ closed orientable surface, there is a concept of distance (see, for example, Definition 6.4.1 of Buser's book "Geometry and spectra of compact Riemann surfaces"). Is there a similar concept in the non-orientable case? Thanks in advance!

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    $\begingroup$ Send me an email, I am preparing a nice paper about it, [email protected] $\endgroup$
    – user21574
    Commented Oct 19, 2016 at 18:35

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There is a nice paper by Pablo Ares Gastesi, where lots of standard results are generalized to that setting (there is an arxiv.org preprint, as well).

Gastesi, P.A., 1997. Some results on Teichmueller spaces of Klein surfaces. Glasgow Mathematical Journal, 39(01), pp.65-76.
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A very nice paper about the Teichm\"uller space of non-orientable surfaces, Fenchel-Nielsen coordinates and other generalizations of Thurston's theory to non-orientable surfaces is this one by Papadopoulos and Penner:

https://arxiv.org/pdf/1309.0383.pdf

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