I now study general relativity and have a few questions regarding the mathematical formulation: 1) What ist the relation between an connection and a covariant derivative? Can you explain the exact difference? 2) One a lorentzian manifold, what ist the relation between the levi-civita-connection and the used covariant derivative? Unrelated, but probably not important enough to justify a new thread: 3) In the einstein field equitations the ricci curvature tensor is used, the non trivial contraction of the riemann curvature tensor. My professor told me, you can understand this as averaging along all curves of the parellel transports. Is there an easy way to see this?