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). 


-- 


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!




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


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



Embedding of flat surfaces
It may be that this theorem of Burago & Zalgaller (partially) answers your question?



> 
> **Theorem** (Burago-Zalgaller 1.7). 
>  Every polyhedron $M$ admits an isometric piecewise-linear $C^0$
>  immersion into $\mathbb{R}^3$.
>  If $M$ is orientable or has a nonempty
>  boundary, then $M$ admits an isometric piecewise-linear $C^0$
>  embedding into $\mathbb{R}^3$.
> 
> 
> 


This is from:


* Yu. D. Burago and V. A. Zalgaller. 
"Isometric piecewise linear immersions of two-dimensional manifolds with polyhedral metrics into $\mathbb{R}^3$.
*St. Petersburg Math. J.*, 7(3):369--385, 1996. Translated by S. G. Ivanov.
English translation: 
[Scanned PDF (15MB)](http://multires.caltech.edu/pubs/BZ96.pdf).


Here is their definition of a "polyhedron":



> 
> By a two-dimensional manifold with polyhedral metric (in brief,
>  a *polyhedron*), we mean a metric space endowed with the structure
>  of a connected compact two-dimensional manifold (possibly with
>  boundary) every point $x$ of which has a neighborhood isometric to
>  the vertex of a cone. ... The metric is locally 
>  flat everywhere except
>  for a finite collection of points; these points are the `true' vertices."
> 
> 
> 


A nice phrase they use in Lemma 2.2 to describe the mapping is that each
triangle becomes a *pleated surface* in $\mathbb{R}^3$.


