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Curvature Questions,

  1. Nov 16, 2008 #1
    Curvature Questions, Please Help!

    1. The problem statement, all variables and given/known data
    1) Prove that if M is locally symmetric (i.e. the Riemann tensor is constant), connected and 2 dimensional, then M has constant sectional curvature.
    2) Prove that if M has constant (sectional) curvature, then M is a locally
    symmetric space.

    2. Relevant equations
    R(X, Y) Z = constant along any geodesic, i.e. it is a parallel vector field.


    3. The attempt at a solution
    For the first part:
    Since M is two dim, the sectional curvature coincides with the actual curvature. Why do we need that M is connected? Am I suppose to use that the sectional curvature (hence the Riemann curvature) does not change along geodesics?

    * What exactly does constant sectional curvature mean? Does it mean that the sectional curvature K does not depend on the 2-dimensional space and that it does not change along any curve?

    For the second part:
    Not quite sure about this part. I see the idea and the picture, but what is the first step?

    Thanks guys!
     
  2. jcsd
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