Examples
On the unit sphere with the standard round Riemannian metric, every great circle is an example of a closed geodesic. Manifolds all of whose geodesics are closed have been thoroughly investigated in the mathematical literature. On a compact hyperbolic surface, whose fundamental group has no torsion, closed geodesics are in one-to-one correspondence with non-trivial conjugacy classes of elements in the Fuchsian group of the surface.
Read more about this topic: Closed Geodesic
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