Euclidean Surfaces

Abstract

The aim of this chapter is to answer the question: which unbounded surfaces look locally like the euclidean plane ℝ²? The question arises because ℝ² is intended to model “flat” surfaces in the real world; yet all physical flat surfaces are of finite extent and have boundaries. It is not clear that such a surface would resemble ℝ² when extended indefinitely, even if small parts of it matched small parts of ℝ² with absolute precision. Indeed, we may never know enough about the large-scale structure of the universe to say what an unbounded flat surface would really be like. What we can do, however, is find which flat surfaces are mathematically possible.