Hello! In my GR class we were introduced to the parallel transport as the way in which 2 tensors can be compared with each other at different points (and how one reaches the curvature tensor from here). I was wondering why can't one use Lie derivatives, instead of parallel transport. As far as I understand, both define the transport of a tensor along a vector field, so why is one used instead of the other i.e. why is Lie derivative not good to define directional derivatives on a manifold? Thank you!