In the previous chapters we showed that a surface serving as a model of non-Euclidean geometry must be geodesically complete and have constant negative curvature. We next search for surfaces in ℝ3 satisfying these conditions. One candidate, the surface of rotation of the tractrix, has constant negative curvature, but it fails to be complete. In fact, we will find that these conditions cannot be met by any surface in ℝ3! This leads us to consider a more abstract notion of a surface, that is, an object that need not be a subset of some Euclidean space. The material in this chapter is a bit out of the historical sequence. The development of the definition of an abstract surface between Riemann and Poincaré is another story (told in detail in Scholz (1980)); we give the modern version here. Theorem 14.4 frames our story, which picks up the historical thread again in Chapter 15.

To begin, let us suppose we have a surface S ⊂ ℝ3 of constant Gaussian curvature, K ≡ –C2. Recall from Chapter 10 that for each point p in S there is a neighborhood of p with asymptotic lines as coordinate curves. We first show that these coordinates have some even nicer properties.