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I notice that $T^2=S^1\times S^1$ can be embedded in $\mathbb{R}^3$ as a hypersurface (submnaifolds of codimension 1).

In general,

(1). could the product of spheres $S^{m_1}\times\cdots\times S^{m_n}$ be embedded in Euclidean space as a hypersurface?

(2). could $T^n=\prod_n S^{1}$ be embedded in Euclidean space as a hypersurface?

(3). could $S^m\times S^n$ be embedded in Euclidean space as a hypersurface?

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    $\begingroup$ You've asked 55 questions on MO so far (72 if I count your other two accounts that I know of, and 86 if I count math.SE), but answered 0 (everywhere). The SE system doesn't work unless people also answer questions in addition to asking them. Please consider answering some questions in the future and giving back to the community. $\endgroup$
    – Qiaochu Yuan
    Commented Nov 19, 2015 at 18:53
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    $\begingroup$ Shouldn't this question be closed? This is a homework-type question. $\endgroup$
    – Ryan Budney
    Commented Nov 20, 2015 at 0:03
  • $\begingroup$ @QiaochuYuan Yes. Once I meet a question that I have a solution, I will answer. $\endgroup$
    – QSR
    Commented Nov 20, 2015 at 6:05

1 Answer 1

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This is asked on MSE, and answered (see Jim Belk's answer, which is NOT the accepted answer).

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  • $\begingroup$ What about the minimum dimension at which the embedding can be made isometric? $\endgroup$
    – John Jiang
    Commented Nov 19, 2015 at 15:21
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    $\begingroup$ If the category is $C^1,$ then the same dimension as the topological embedding dimension :) $\endgroup$
    – Igor Rivin
    Commented Nov 19, 2015 at 17:00
  • $\begingroup$ This is really cool; I didn't know $\mathcal{C}^1$ is so powerful: mathoverflow.net/questions/31222/… $\endgroup$
    – John Jiang
    Commented Nov 19, 2015 at 19:18

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