I do not get the conceptual difference between Riemann and Ricci tensors. It's obvious for me that Riemann have more information that Ricci, but what information? The Riemann tensor contains all the informations about your space. Riemann tensor appears when you compare the change of the sabe vector(or other tensor) when it takes two different paths. You can see it comutanting two differents cov. derivatives of a vector ou computing the parallel displacement. Studying general relativity I saw : "If Riemann tensor is zero, the space is flat; if the Ricci tensor is zero the space is empty". Someone knows some mathematical proof of this affirmation? And what the Ricci scalar say to us? It's always directly proporcional to curvature?