Abstract
As we shall see throughout this chapter, the geometry of the “ordinary sphere” S2 – two dimensional in a space of three dimensions – harbors many pitfalls. It’s much more subtle than we might think, given the nice roundness and all the symmetries of the object. Its geometry is indeed not made easier – at least for certain questions – by its being round, compact, and bounded , in contrast to the Euclidean plane. Sect. III.3 will be the most representative in this regard; but, much simpler and more fundamental, we encounter the “impossible” problem of maps of Earth, which we will scarcely mention, except in Sect. III.3; see also 18.1 of [B]. One of the reasons for the difficulties the sphere poses is that its group of isometries is not at all commutative, whereas the Euclidean plane admits a commutative group of translations.
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