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I'm trying to define a mapping from $\mathbb{R}^4$ into $\mathbb{R}^3$ that takes the flat torus to a torus of revolution.

Where the flat torus is defined by $x(u,v) = (\cos u, \sin u, \cos v, \sin v)$. And the torus of revolution by $x(u,v) = ( (R + r \cos u)\cos v, (R + r \cos u)\sin v, r \sin u)$.

I think an appropriate map would be: $f(x,y,z,w) = ((R + r x)z, (R + r x)w, r y)$ where $R$, $r$ are constants greater than $0$.

But now I'm having trouble showing this is one-to-one.

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    Presumably the mapping is only going to be one-to-one when restricted to the torus?2011-11-30

3 Answers 3