Hyperbolic Surfaces

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.