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Tringles on a sphere

  1. Oct 23, 2005 #1
    Hey I got this problem. Any help wil be great.

    Suppose you have a triangle on a sphere. Show that the amount by which a vector is rotated by a parallel transport around such a triangle equals the excess of the sum of the angles over 180 degrees.

    Thanks!!
     
  2. jcsd
  3. Oct 24, 2005 #2
    Anyone?? If you know some differential geometry, Im interested in your opinion!!
     
  4. Oct 24, 2005 #3
    Hey Tzar. What you need to use is the Gauss Bonet theorem. Once you've shown the excess angle is A/r^2 all you need to do is show that the covariant derivative is equal to this value.
     
  5. Oct 25, 2005 #4
    Start off with a co-ordinate system. Write down the metric for your 2D space. Then calculate the covariant derivative (using the Christoffel connection), then stick it in the equation for parallel transports. Surely there's an easier way, but this method is most general.
     
  6. Oct 27, 2005 #5
    doing the phys3550 assignment i take it :p
     
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