In first course differential geometry you learn, that Ricci-curvature is something like a mean-value of the curvature endomorphism, because it's a trace, and the scalar curvature is again a mean-value of the Ricci curvature, again because it's a trace. I'm now interested, what examples of manifolds can be given, with big Ricci-curvature but small scalar curvature, i.e. how can one describe in a picture what scalar curvature forgets what Ricci curvature still sees. The same question one can ask for ricci curvature and the curvature endomorphism.
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