Are any two geodesics the same after applying an isometry?

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The discussion centers on whether applying an isometry to a known geodesic on a Riemannian manifold can generate all geodesics on that manifold. It is suggested that this is true for homogeneous and isotropic manifolds, as well as for simple cases like the plane or sphere. However, the argument raises concerns about manifolds with geodesics of varying lengths, where the invariance of geodesic length under isometries complicates the situation. The possibility of closed versus non-closed geodesics further challenges the assumption that all geodesics can be derived from a single one through isometries. Overall, the conclusion is that the property may not hold in more complex manifolds.
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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