Abstract
The concept of hyperbolic surface is completely analogous to the concept of euclidean surface (Section 2.7). A hyperbolic surface is a set S with a realvalued distance function ds such that each P ∈ S has an ε-neighborhood isometric to a disc of ℍ2. The proof of the Killing-Hopf theorem (Section 2.9) carries over word-for-word (provided “line”, “distance” etc., are understood in the hyperbolic sense), showing that any complete, connected hyperbolic surface is of the form ℍ2/Γ, where Γ is a discontinuous, fixed point free group of ℍ2-isometries.
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© 1992 Springer Science+Business Media New York
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Stillwell, J. (1992). Hyperbolic Surfaces. In: Geometry of Surfaces. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0929-4_5
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DOI: https://doi.org/10.1007/978-1-4612-0929-4_5
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-97743-0
Online ISBN: 978-1-4612-0929-4
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