For a problem I'm working on, I'm trying to find curves in $\mathbb{S}^1 \times \mathbb{R} = \{(x, y, z) \in \mathbb{R}^3 \ \vert   x^2 + y^2 = 1 \}$ that satisfy certain properties (not relevant enough to write here). I have only found non closed curves and I'm interested in finding closed ones, but it's proven to be a hard task, so I looked for references on the subject and all I was able to find was [this article](https://link.springer.com/article/10.1007/s00605-014-0705-4) on the characterisation of cylindrical curves. At the end the authors give a sufficient and necessary condition for curves contractible to a point on the surface of the cylinder to close up, but it depends on the geodesic curvature being periodic, so it turns out to be too weak of a proposition to help me. Naturally I'm now interested in knowing whether or not there exists any more work along these lines (or is this as far as people have gotten so far?), that is: *what are sufficient and/or necessary conditions for a cylindrical curve of non periodic geodesic curvature to close up?*