Are any two geodesics the same after applying an isometry?

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SUMMARY

The discussion centers on the relationship between geodesics and isometries in Riemannian manifolds. It is established that pushing a known geodesic forward through an isometry of a manifold M results in another geodesic. However, this property is confirmed to hold true primarily for homogeneous and isotropic manifolds. In cases where these conditions do not apply, such as manifolds with geodesics of varying lengths, the assertion fails, indicating that not all geodesics can be generated through isometries.

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nonequilibrium
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Hello, I was wondering the following.

Suppose you start with a Riemannian manifold M. Say you know one geodesic.
Pushing this geodesic forward through an isometry M -> M gives again a geodesic.
Can this procedure give you all geodesics?

Thinking of the plane or the sphere it seems obviously true.
 
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This is certainly true for homogeneous and isotropic manifolds. If these conditions fail, then this property might not be recoverable.

Edit: Find a manifold with geodesics of different lengths and since the length of a geodesic should be invariant under isometries this provides an ample number of examples. Do let me know if some part of this argument does not check out though since I have not thought it through very thoroughly.
 
Sure on a manifold where some geodesics are closed(?) and some are not this can't be true
 

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