TITLE: Are there any compact embedded 2-dimensional surfaces in $\mathbb R^3$ that are also flat?
QUESTION [8 upvotes]: 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.

REPLY [4 votes]: If I have understood your notation and the question correctly, the answer is Yes under some assumptions on smoothness and No under other assumptions. Jason's No-proof assumes smoothness.  But if one loosens the smoothness to $C^1$, then the answer is Yes.
See my MO question entitled,
$C^1$ isometric embedding of flat torus into $\mathbb{R}^3$,
and the references and answers supplied there.
See in particular my citing of Zalgaller's proof that the flat torus may be isometrically
embedded in $\mathbb{R}^3$, an amazing result that seems little known.