Is there a simple intuitive description of what the Ricci tensor and scalar represent? I have what seems to me a straightforward understanding of what the Riemann tensor Rabcd represents, as follows. If you parallel transport a vector b around a tiny rectangle, the sides of which are determined by two other vectors c and d, the change in the transported vector when it arrives back at the start will have component in direction a given by the application of the Riemann tensor to vectors b, c, d and one-form a. The Ricci tensor is a contraction of the Riemann tensor, and the Ricci scalar is a contraction of the Ricci tensor. However I can't think of a physical interpretation of these items that is similarly intuitive to the one above for the Riemann tensor. Does anybody have such an interpretation?