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Levi-Civita connection

  1. Sep 1, 2009 #1
    Can any one help me on this question:
    Under what relation between vector fields X and Y, the Levi-Civita connection of X with respect to Y, \nabla_{Y}X is 0?
    Any answers or suggestion will be highly appreciated.
  2. jcsd
  3. Sep 1, 2009 #2
    What would be the geometric interpretation of [tex]\nabla_{Y}X = 0[/tex]?
  4. Sep 1, 2009 #3
    The geometric interpretation is as follows: think of the flow lines of Y as paths in the manifold. What this condition is saying is that X does not change along these flow lines (with respect to the connection, or equivalently with respect to the metric in the case of the L-C connection). This is the idea of parallel transport, which is a very geometric concept. Given a curve c(t) and a vector at c(0), there is a unique extension of that vector to a parallel vector field along the curve.

    It might be helpful to draw some curves in the plane and find out how a parallel vector field must behave.
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