Let $S$ be a orientable compact surface with a flat euclidean structure with conical singularities (cf. [T] for instance). Let also $\mathcal P$ be a polyhedral euclidean decomposition of $S$ (with vertices at the singular points of the euclidean structure of $S$).

**Question 1**: can $S$ be realized (as a polyhedral surface) in an euclidean space? If yes, what is known about the set of such `euclidean polyhedral realizations'?

**Question 2**: same questions than above, but for $\mathcal P$ (i.e. an euclidean polyhedral decomposition of $S$ is fixed).

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Remarks:

there is no assumption on the genus of $S$ (when $S$ has genus 0, an answer to the (first part of) Question 1 is given by Alexandrov theorem)

A similar question (but for Riemann surfaces) already appears in [B] (page 9). I guess that a lot has been done on this problem since. Any relevant reference would constitute an interesting answer.

Thanks for any help!

[B] Bers L., Riemann surfaces (1958)

[T] Troyanov M. -- *Les surfaces a singularités coniques*. Enseign. Math. **32** (1986), 79–94.