While teaching a course in differential geometry, I came up with the following problem, which I think is cool.

Assume $\gamma$ is a closed geodesic on a sphere $\Sigma$ with positive Gauss curvature. Can it look like one of the following curves in a parallelization of $\Sigma\backslash\{\text{point}\}$ by the plane?

I expect that there is a theorem that answers all questions like this. Is it indeed so?