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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?

QSR
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    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. – Qiaochu Yuan Nov 19 '15 at 18:53
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    Shouldn't this question be closed? This is a homework-type question. – Ryan Budney Nov 20 '15 at 00:03
  • @QiaochuYuan Yes. Once I meet a question that I have a solution, I will answer. – QSR Nov 20 '15 at 06:05

1 Answers1

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

Community
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Igor Rivin
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  • What about the minimum dimension at which the embedding can be made isometric? – John Jiang Nov 19 '15 at 15:21
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    If the category is $C^1,$ then the same dimension as the topological embedding dimension :) – Igor Rivin Nov 19 '15 at 17:00
  • This is really cool; I didn't know $\mathcal{C}^1$ is so powerful: http://mathoverflow.net/questions/31222/c1-isometric-embedding-of-flat-torus-into-mathbbr3 – John Jiang Nov 19 '15 at 19:18