TITLE: Riemannian manifolds that are scalar flat but not Ricci flat
QUESTION [7 upvotes]: 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?

REPLY [2 votes]: On a compact connected manifold that does not carry a metric of positive scalar curvature, every scalar flat metric is Ricci-flat. Thus on such manifolds there are
no such metrics.
If a manifold carries a metric of positive scalar curvature then it also carries a metric of zero scalar curvature. I assume that for dimension at least 3 one could extend this statement to saying that it even admits a scalar flat metric with non-zero Ricci curvature, but I have no proof at hand currently. What I know for sure is: there are many manifolds having an obstruction against Ricci-flat metrics and admitting a metric of positive scalar curvature. On Ricci-flat manifolds, the first Betti number is at most the dimension, and if it is the dimension then the manifold is flat. This yields manifolds without Ricci-flat metrics and many of them carry a metric of positive scalar curvature.