Are any two geodesics the same after applying an isometry?

[nonequilibrium]

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.

 

[jgens]

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.

 

[jcsd]

Sure on a manifold where some geodesics are closed(?) and some are not this can't be true