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.