Does there exist a closed, regular, non-self-intersecting developable surface in $\mathbb{R}^3$? I have had only minimal exposure to differential geometry in my education thus far. I am currently searching for a surface that has the following properties: 1. Surface in $\mathbb{R}^3$ 2. Zero Gaussian curvature (aka developable) 3. No self intersections 4. Closed 5. Regular (or at least $C^2$? I'm a little confused about how to phrase this constraint, but I don't want an infinitely "scrunched" surface like a flat torus or a surface with a cusp or vertex, like a sphericon. I haven't been able to find a surface that satisfies these qualities online, nor do I have the expertise to prove or disprove its existence, nor create it if it does exist. Any further reading on the subject would be also appreciated.

Any (real analytic) developable surface must be (a subset of) a plane, a cylinder, a cone, or a tangent developable (the surface of tangent lines to a smooth curve in space — this surface will have a cuspidal edge along the curve). If by closed you mean a compact surface without boundary, there is none. Every developable surface must be ruled, and ruled surfaces are never compact.

See page 61 and Exercise 12 on page 42 of my differential geometry text.