Given a smooth manifold with no other structure (like a metric), one can define a derivative for a vector field called the Lie derivative. One can also define a Lie derivative for any tensor, including covectors. Incidentally, with antisymmetric covectors (differential forms) one can define another type of derivative called 'd' that doesn't seem to have any direction (not a directional derivative). If you add a metric tensor, then you can define another derivative for a vector field called (at least in physics) the covariant derivative, which is a partial derivative plus a connection term. My question is which derivative is really the directional derivative of a vector field: the Lie derivative, or the covariant derivative? Also, probably related, is the definition of a directional derivative unique? And how can there be a derivative 'd' that has no direction? That would have to imply, roughly speaking, that 'd' is a derivative averaged around a circle at a point?