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?