A More Precise Definition
To be more precise, every Riemann surface has a universal covering map that is either the Riemann sphere, the complex plane or the upper half-plane. Given a covering map, where H is the upper half-plane.
The Fuchsian model of R is the quotient space . R. Note that is a complete 2D hyperbolic manifold.
Read more about this topic: Fuchsian Model
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