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I am looking for an introductory textbook to the geometry of the hyperbolic space $\mathbb{H}^n$. The book should include explicit description of geodesics and horospheres in various models (hyperboloid, Poincaré, Klein).

Apologies if the question is not appropriate for this site.

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    $\begingroup$ This is a nice introduction. math.brown.edu/~rkenyon/papers/cannon.pdf $\endgroup$
    – Deane Yang
    Commented Sep 10, 2019 at 17:41
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    $\begingroup$ An introductio to Geoetric Topology by Bruno Matelli is very good (Part 1 of the ook for Hyperbolic Geometry) $\endgroup$
    – Anubhav Mukherjee
    Commented Sep 10, 2019 at 17:54

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I really like Ratcliffe’s account of the 3 models ($H^n$, $U^n$, $B^n$) in Foundations of Hyperbolic Manifolds (2006, Chap. 3–5). It has what you ask for, and also copious exercises and historical notes.

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    $\begingroup$ I second this. It's a large book, but I remember it being very readable. $\endgroup$
    – Greg Friedman
    Commented Sep 17, 2019 at 21:42
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W. Thurston, Three-dimensional geometry and topology.

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Prasolov, V. V.; Tikhomirov, V. M., Geometry. Transl. from the Russian by O. V. Sipacheva. Transl. edited by A. B. Sossinski, Translations of Mathematical Monographs. 200. Providence, RI: American Mathematical Society (AMS). xi, 257 p. (2001). ZBL0977.51001.

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You might try the following:

  1. Jürgen Richter-Gebert: Perspectives on Projective Geometry,
  2. David Mumford, Caroline Series, David Wright: Indra´s Pearls.
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