# Existence of developable ribbonization of a surface

Let $$S$$ be a smooth compact surface embedded in $$\mathbb{R}^{3}$$. It is well-known that there exists a triangulation of $$S$$. I am considering an alternative way of approximating $$S$$, where instead of planar pieces one uses developable surfaces.

Definition: Let $$\gamma$$ be a smooth curve in $$S$$. A developable surface passing through $$\gamma$$ and having the same tangent plane as $$S$$ at all points of $$\gamma$$ is called a flat ribbon approximation of $$S$$ (along $$\gamma$$). A union $$\mathcal{R}$$ of multiple flat ribbon approximations of $$S$$, such that $$\mathcal{R}$$ and $$S$$ are homeomorphic, is called a flat ribbonization of $$S$$.

(Examples of flat ribbonizations can be found in this paper.)

My question is the following.

Question: Does every surface $$S$$ have a flat ribbonization?

Note that a sufficient condition for the existence of a flat ribbon approximation of $$S$$ along $$\gamma$$ is that the normal curvature of $$\gamma$$ is nowhere zero.

• The paper you cite says "We develop the concept of Cartan ribbons and a method by which they can be used to ribbonize any given surface in space by intrinsically flat ribbons." Without studying the paper, they seem to claim to answer your question. Is the issue that their method doesn't apply to every surface? – Joseph O'Rourke Apr 8 at 13:41
• @JosephO'Rourke I guess that such claim needs a proof, which is not in the paper. To me the applicability of the method to every surface is not evident. – MK7 Apr 8 at 14:08