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Geodesic and the shortest path

  1. Feb 18, 2010 #1
    it comes from the calculus of variation that the shortest path between two points on a surface must be geodesic.
    then must the geodesic connected two points be the shortest path?
    if not, what about the example?
    Thanks for any reply!
  2. jcsd
  3. Feb 19, 2010 #2
    check out the example in wikipedia
  4. Feb 19, 2010 #3
    No. On a cylinder there are infinitely many geodesics between most points. The same is true of a flat torus.
  5. Feb 19, 2010 #4
    yeah, cylinder is really a good example!
  6. Feb 20, 2010 #5
    There is an example of a geodesic on a fluted surface of negative curvature that winds almost all of the way down the surface circling around it in a helical motion then turns around and comes back! The shortest geodesic though between two adjacent points is a simple arc. I will try to look this up. It is pretty incredible.
  7. Feb 21, 2010 #6
    it reminds me the magnetic lines of force in tokamak. they are all helical.
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