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What are the examples of Riemannian manifolds that have zero scalar curvature but non zero ricci curvature? Is there any sort of classification of such manifolds?

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There are a LOT of examples, first thing comes to mind is a products of unit sphere and surface of constant curvature $-1$. This condition is too soft (opposite of rigid), you can not expect to have a classification. –  Anton Petrunin Jun 8 '11 at 6:54
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To generalize Anton's comment a little, I should add that with the appropriate choice of $l$ and $k$, the product manifold $S^l \times N^k$ will have the property that you are looking for, where $N^k$ has hyperbolic $k$-dimensional half-space space as its cover. You can find the formulas for all of the geometric quantities related to these sorts of products in Chang, Han, Yang "On a class of locally conformally flat manifolds". This particular combination of manifolds can be used to construct many examples of manifolds with interesting curvature.

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Also, keep in mind that in three dimensions Einstein implies constant curvature, so the three sohere carries a scalar flat metric that is not Ricci flat. –  Viktor Bundle Jun 20 '11 at 2:28
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