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.

everysurface is not evident. $\endgroup$ – MK7 Apr 8 at 14:08