# Riemannian manifolds that are scalar flat but not ricci flat

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?

• 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
• See this answer math.stackexchange.com/questions/47323/scalar-flat-metrics – user21574 Jun 8 '17 at 14:53
• 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 Kähler-Einstein metric. See Proposition 2.12 in the book of Gang Tian springer.com/in/book/9783764361945 – user21574 Jun 8 '17 at 15:30
• 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 – 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 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.