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.
KeywordsLine Segment Elliptic Function Orbit Space Euclidean Plane Klein Bottle
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