I thought to know very well the answer to the classification problem for developable surfaces, so I sought for some confirmation in the literature. However, what I encountered are seemingly inequivalent statements that I am not sure how to interpret.
Recall that a smooth surface $S \subset \mathbb{R}^{3}$ is said to be developable if it is locally isometric to $\mathbb{R}^{2}$ (or, equivalently, if its Gaussian curvature vanishes).
First off, what I thought was true is the following statement, present for example in Pressley's Elementary differential geometry, 2nd edition, at page 203:
If $S$ is developable, then it is the union of (generalized) cylinders, (generalized) cones, and tangent developables, joined together along segments of straight lines.
However, if one checks Do Carmo's Differential geometry of curves and surfaces (2nd edition, pp. 197–198), then one gets a seemingly different answer. After listing cylinders, cones, and tangent surfaces, he writes:
"Of course, the above cases do not exhaust all possibilities. As usual, if there is a clustering of zeros of the functions involved, the analysis may become rather complicated. At any rate, away from these cluster points, a developable surface is a union of pieces of cylinders, cones, and tangent surfaces."
So he affirms that Pressley's claim is only valid away from cluster points. The same conclusion is also drawn in Gray's Modern differential geometry of curves and surfaces with Mathematica (3rd edition, p. 448).
As if that weren't enough, another seemingly different statement is given in Kühnel's Differential geometry. Curves—surfaces—manifolds (3rd edition, p. 88):
"An open and dense subset of every torse [= developable surface] consists of pieces of planes, cones, cylinders or tangent developable".
So my questions are the following. Are all these statements really equivalent? If so, why?