I was wondering if it is possible to construct a surface with such a property. I am able to construct a surface such that the shortest geodesic loop passing through a particular point is not periodic, but there are shorter geodesic loops that exist elsewhere on the surface.
2026-05-15 16:01:20.1778860880
Shortest geodesic loop on surface diffeomorphic to $\mathbb S^2$ not periodic.
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