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Let $\overline{g}$ be the flat metric on $\mathbb{R}^3$.

I would like to know if there is any compact embedded 2-dimensional surface $M$ in $\mathbb{R}^3$ (without boundary) such that $\iota^*\overline{g}$ is flat, where $\iota: M \hookrightarrow \mathbb{R}^3$ is the inclusion.

It appears that the answer is "no", but I am having a hard time coming up a with a rigorous proof, which does not use any results about the existence or non-existence of isometric embeddings of abstract surfaces into $\mathbb{R}^3$. Any suggestions would be appreciated.

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    I imagine you forbid manifolds with boundaries ?2011-12-06
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    Also I think this kind of questions are often non trivial2011-12-06
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    If you allow boundary then of course there are flat surfaces -- the disc, the cylinder, Moebius band, etc.2011-12-07
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    @Glougloubarbaki: You're right -- the surface is required to have no boundary.2011-12-07
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    Kaloyan, I put a comment at your other question. I edited my answer to that one within an hour of answering, essentially to match what Jason answers below. You were right to be skeptical about what I first wrote.2011-12-07

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