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my aim is to show that the $n$-Torus $T^n = S^1 \times \ldots \times S^1$ can be embedded into $\mathbb{R}^{n+1}$ by giving a function $f: \mathbb{R}^{n+1} \rightarrow \mathbb{R}$ such that the Torus is described by all points satisfying $\ f(x_1,\ldots,x_n)=0$. This is quite simple in the two dimensional case with the equation $(\sqrt{x^2+y^2}-R)^2+z^2=r^2$. Trying to find a similar equation for higher dimensions i was starting with a parametric description of $T^3$ and then tried to eliminate the parameters in order to get a implicit description. But already in this case it got extremely messy with a lot of case distinctions (since $\sin$ and $\cos$ can't be inverted at once for the whole interval of the parameters $(0,2\pi)$).

So does anybody know a more elegant way to achieve an implicit description of $T^3$ and eventually $T^n$ or is there just none?

user9784
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1 Answers1

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Start with the equation $$x_1^2+x_2^2=r_1^2$$ which describes a centered circle in the place of radius $r_1$. Replace $x_2$ by $\sqrt{x_2^2+x_3^2}-r_2$, so get $$x_1^2+(\sqrt{x_2^2+x_3^2}-r_2)^2=r_1^2.$$ This is a $2$-torus with radii $r_1$ and $r_2$. Let's do this again: replace the last variable $x_3$ by $\sqrt{x_3^2+x_4^2}-r_3$, to get $$x_1^2+(\sqrt{x_2^2+(\sqrt{x_3^2+x_4^2}-r_3)^2}-r_2)^2=r_1^2.$$ This is a $3$-torus with radii $r_1$, $r_2$, $r_3$.

One can go on as long as one wants.

This is just using the procedure which goes from, say, a curve to the surface of revolution it generates, several times.

(Of course, the radii have to be such that the thing is an actual torus, without any self-intersections)

Mariano Suárez-Álvarez
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  • You seem to give the formula for a 2-Torus in $\mathbb{R}^{n+1}$ and not for a n-Torus, or am I getting it wrong? – user9784 Feb 18 '13 at 23:30
  • This gives a nice, round $n$-torus in $\mathbb R^{n+1}$. You should look up how to write down equations for hypersurfaces of revolution. – Mariano Suárez-Álvarez Feb 18 '13 at 23:38
  • But if its a n-Torus = $S^1 \times\ldots \times S^1$ there should be n radii to specify it and not just two. – user9784 Feb 19 '13 at 13:57
  • If you wanted that, it would have been useful to be explicit about it in the body of the question :-) – Mariano Suárez-Álvarez Feb 19 '13 at 18:28
  • I'm sorry, i thought there was some kind of convention about what an n-Torus is. I clarified it in the question above. – user9784 Feb 19 '13 at 19:21
  • Your edit is not that helpful, really. What you seem to want is equations for all possible $n$-tori within a certain class, which you did not specify. You seem to be defining a torus to be a specific thing, and you want all possibilities: explain what you want :-) – Mariano Suárez-Álvarez Feb 19 '13 at 19:32
  • I may be missing something, but it seems to me the only confusion is that you have described an embedding of $S^{n-1}\times S^1$, while the OP wants an embedding of $S^1\times\cdots\times S^1$. I did not know the former was also referred to as an $n$-torus. –  Feb 19 '13 at 20:05