- #36
PeterDonis
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davidge said:can't they even be defined in terms of ##\mathbb{R}^3##?
No. For someone who said they had concerns about using the notion of embedding, you seem to have a big problem getting rid of the conceptual crutch of trying to visualize everything in terms of embeddings/subspaces/whatever using Euclidean spaces. You need to stop doing that, and learn how to deal with these various spaces as metric spaces in their own right, using only their intrinsic properties.
To restate the one part of this that does have some relationship to ##\mathbb{R}^3##: of the three possible cases, the ##k = 0## case and the ##k = -1## case can be completely covered by a single coordinate chart (for the ##k = 0## case this is trivial since it is just ##\mathbb{R}^3##; for the ##k = -1## case constructions are easy to find), which means there is a one-to-one mapping between those spaces and ##\mathbb{R}^3##. For the ##k = 1## case, however, this is not possible--which should also be obvious because ##\mathbb{S}^3## is compact and ##\mathbb{R}^3## is not, so there can't possibly be a one-to-one mapping between them. It takes at least two coordinate charts to cover ##\mathbb{S}^3##.