The notion of an abstract surface frees us to seek models of non-Euclidean geometry without the restriction of finding a subset of a Euclidean space. A set, not necessarily a subset of some ℝn, with coordinate charts and a Riemannian metric determines a geometric surface. With this new freedom we can achieve our goal of constructing a realization of the geometry of Lobachevskiῐ, Bolyai, and Gauss. In this chapter we present the well-known models of non-Euclidean geometry due to E. Beltrami (1835–1906) and J. Henri Poincaré.

This chapter contains many computational details, like a lot of nineteenth-century mathematics. The foundations for these calculations lie in the previous chapters. It will be the small details that open up new vistas.

We begin by considering a problem Beltrami posed and partially solved in a paper of 1865. He asked for local conditions on a pair of surfaces, S1 and S2, that guarantee that there is a local diffeomorphism of S1 → S2 such that geodesies on S1 are taken to geodesies on S2. Such a mapping is called a geodesic mapping. Beltrami solved the problem when one of the surfaces is the Euclidean plane.