# Embedding of flat surfaces

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.

• For genus 0, this is discussed in the monograph of Jürgen Richter-Gebert "Realization spaces of polytopes." It is published by Springer. There are still PDF copies archived on citeseer and other sites that you can find with google. – j.c. Sep 24 '13 at 14:05

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.

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 $\mathbb{R}^3$.

• Thanks a lot! Burango-Zalgallar theorem is quite impressive and is certainly the best answer that can be expected to the first part of Question 1. – Lucien from IHP Sep 16 '13 at 13:55
• It is an amazing theorem! I believe they do not attempt to bound the number of flat, noncreased triangles in the final construction, but it would be astronomical in most cases. – Joseph O'Rourke Sep 16 '13 at 14:18