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Are any two geodesics the same after applying an isometry?

  1. Jan 22, 2014 #1
    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.
  2. jcsd
  3. Jan 22, 2014 #2


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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.
  4. Jan 22, 2014 #3


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