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Obtaining the connection from Parallel Transport

  1. Oct 11, 2008 #1
    How do I obtain the Levi-Civita connection from the concept of parallel transport?

    So Do Carmo asks to prove that for vector fields X, Y on M, and for c(t) an integral curve of X, i.e. c(t_0) = p and X(c(t)) = dc/dt, the covariant derivative of Y along X is the derivative of the parallel transport of Y(c(t)).

    Do I just prove that the derivative of the parallel transport of a vector field satisfies the definition of an affine connection, metric compatibility and symmetric properties?

    I tried doing this but I ran into some road blocks.

    Is there an elegant way prove this?
  2. jcsd
  3. Oct 11, 2008 #2
    I think I solved it. Thanks anyway.
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