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I started reading about Hyperbolic manifolds here: https://en.m.wikipedia.org/wiki/Hyperbolic_manifold and I didn't understand the following paragraph in the first section of Rigourous definition:

Every complete, connected, simply-connected manifold of constant negative curvature -1 is isometric to a real Hyperbolic space $\mathbb{H}^n.$ Thus, every such $M$ can be written as $\mathbb{H}^n/ \Gamma$ where $\Gamma$ is a torsion free discrete group of isometries on $\mathbb{H}^n.$

Could anyone explain to me why does every such $M$ can be written as $\mathbb{H}^n/ \Gamma$ where $\Gamma$ is a torsion free discrete group of isometries on $\mathbb{H}^n$ ?

user32415
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    See https://math.stackexchange.com/q/222066 – psl2Z Feb 03 '24 at 09:24
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    You should explain your background, for instance, are you familiar with the theory of covering spaces? Also, the Wikipedia article you are reading is poorly written. You are better off reading a textbook on the subject, say, Ratcliffe's book. – Moishe Kohan Feb 03 '24 at 10:19
  • @MoisheKohan, yes I have some knowledge about covering spaces, not very deep but I do have some. Thanks for recommending the book. – user32415 Feb 03 '24 at 10:30

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