Hi, I've begun learning about General Relativity, though I've already had some exposure to differential geometry. In particular, I understand that Lie Differentiation is a more "primitive" process than Covariant Differentiation (in that the latter requires some sort of connection). My question is this: parallel transport can be used to understand how a vector changes when you drag in along a curve on a certain surface. To be sure, you institute local coordinates, compute the metric, and then the connection (here, the connection being used, in this coordinate basis, are the Christoffel symbols), and then solve the differential equation. In this way, you can find out, for instance, how much the vector changes its direction under a certain curve. But, is this information only encoded in the connection? That is to say, to find out how much the vector deviates, must I employ parallel transport, or is there some procedure, using only Lie Derivatives, to examine the change?