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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 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?

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  • $\begingroup$ Perhaps some of the ideas in this question might be helpful mathoverflow.net/questions/295927/… $\endgroup$ – j.c. Aug 29 '18 at 18:53
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    $\begingroup$ @DeaneYang I'm not sure I follow. I'm not looking for the geodesics of the cylinder, I'm interested in a general characterisation of closed cylindrical curves (and I can't see how your comment relates to that). $\endgroup$ – Matheus Andrade Aug 29 '18 at 19:18
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    $\begingroup$ Sorry. I misunderstood. $\endgroup$ – Deane Yang Aug 29 '18 at 20:19
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    $\begingroup$ The map from the plane to the cylinder is $(\theta, z) \mapsto (\cos\theta,\sin\theta,z)$. Therefore, two points $(\theta_1,z_1)$ and $(\theta_2,z_2)$ on the plane map to the same point on the cylinder if and only if $\theta_2 = \theta_1 + 2\pi k$, for some integer $k$, and $z_1 = z_2$. $\endgroup$ – Deane Yang Aug 29 '18 at 22:02
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    $\begingroup$ Since you can develop any curve on a cylinder to a curve in the plane, it might be fruitful to first look for conditions for closed curves in the plane, and then roll them up to a cylinder. (I see this is related to Deane Yang's suggestion.) For example, simple planar curves have winding number $+1$. Nonsimple planar curves might lead to Gauss codes. $\endgroup$ – Joseph O'Rourke Aug 29 '18 at 23:24

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