It is known that geodesics on a cylinder are helical lines (helix). As a special case of a helix--- straight line. But one can take two points so cleverly that you can draw two helixes between them (see picture), one of which is a straight line. Both satisfy geodesics differential equation. The question: is, can they both be considered as geodesics or only the shortest one is "true" geodesic?
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Depending on the number of rotations executed between the end points $(A,B)$ we have an infinite number of local length minimizing geodesics as helices shown on their cylindrical development.
There is one shortest among them running vertically down that you have sketched.
In development all helices are straight lines.
They are inclined to the generator line $L$ as
$$\tan^{-1}\dfrac{n\cdot 2 \pi R}{L}$$
whose periodicity can be visualized as below for $n=0,1,2,3$ in a development of multiple winding numbers associated with periodic wraps.
I think the only reasonable definition of a geodesic in such situations involves locally optimal (shorter) paths. After all, consider the case where the two end points are on precisely opposite positions on the cylinder. There are two equivalent arcs (geodesics). (There are an infinite number of geodesics on a sphere linking anti-podes, such as the north and south poles.) For a cylinder there are, then, an infinite number of geodesics in the arbitrary case, indexed by the number of rotations of the helix around the cylinder (like a winding number).
"A geodesic is a locally length-minimizing curve."