Generalisations and applications

Abstract

In the preceding chapters we have classified all 2-dimensional locally Euclidean geometries, that is, the worlds in which all properties of Euclidean geometry in the plane are satisfied in sufficiently small regions. An inhabitant of such a world who always remains within some distance r of a fixed point (home, for example) could not detect in his world any contradictions to Euclidean plane geometry. But the real space in which we live is 3-dimensional. Thus, it is of course more interesting to describe the 3-dimensional locally Euclidean geometries, that is, the worlds in which all properties of Euclidean geometry in 3-space are satisfied in sufficiently small regions; we can think of the description of the 2-dimensional geometries as just a model for this more interesting problem. In this section, we will concern ourselves with the description and some of the properties of 3-dimensional locally Euclidean geometries.