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?

14$\begingroup$ 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. $\endgroup$ – Anton Petrunin Jun 8 '11 at 6:54

2$\begingroup$ See this answer math.stackexchange.com/questions/47323/scalarflatmetrics $\endgroup$ – user21574 Jun 8 '17 at 14:53

2$\begingroup$ To generalize your question in Kähler $M$, If $ω$ a Kähler metric of constant scalar curvature with $\pi c_1(M)=λ[\omega]$,, then $\omega$ is KählerEinstein metric. See Proposition 2.12 in the book of Gang Tian springer.com/in/book/9783764361945 $\endgroup$ – user21574 Jun 8 '17 at 15:30

1$\begingroup$ Let for symplectic manifold $(X,\omega)$ we have $ [ω]=λ⋅c_1(X)$ for some $λ∈R_{>0}$, such manifolds are called monotone symplectic manifold. Fukaya category of a monotone symplectic manifold are very important to verify HMS $\endgroup$ – user21574 Jul 20 '17 at 20:46
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 halfspace 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.

$\begingroup$ 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. $\endgroup$ – Viktor Bundle Jun 20 '11 at 2:28
On a compact manifold that does not carry a metric of positive scalar curvature, every scalar flat metric is Ricciflat. 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 nonzero Ricci curvature, but I have no proof at hand currently. What I know for sure is: there are many manifolds having an obstruction against Ricciflat metrics and admitting a metric of positive scalar curvature. On Ricciflat 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 Ricciflat metrics and many of them carry a metric of positive scalar curvature.

1$\begingroup$ Could you please give a reference about the statement "On a compact manifold that does not carry a metric of positive scalar curvature, every scalar flat metric is Ricciflat."? Thanks. $\endgroup$ – Bilateral Nov 15 '18 at 14:01

$\begingroup$ @Bilateral See mathoverflow.net/a/294346/394 $\endgroup$ – José FigueroaO'Farrill Apr 11 at 19:03