Let $(M, g\_M)$ where $M= B \times\_f F$ and $g\_M=g\_B + f^2g\_F$, an Einstein warped product manifold (i.e., $Ric\_M= \lambda g\_M$), with Ricci flat fiber-manifold $F$, i.e., $Ric\_F=0$.
Then $M$ can admit only constant negative Ricci curvature or zero Ricci curvature (i e., $\lambda \le 0$) or $M$ could also have positive constant Ricci curvature (i.e., $\lambda >0$)?
In other words, $Ric\_F = 0$ necessarily implies $\lambda \le 0$, or can solutions be obtained with $\lambda > 0$?



Possible sign of scalar curvature for Einstein warped product manifold with Ricci-flat
It can not have constant positive Ricci curvature. By Bonnet-Myers constant positive Ricci curvature implies that $M$ is compact.


If $V$ is a vertical vector then by the formula for Ricci curvature of warped product (page 266 in Besse's book)


$$
Ric(V,V)=Ric\_F(V,V) -|V|^2(\frac{\Delta f}{f}+(p-1)\frac{|\nabla f|^2}{f^2})
$$
where $p=\dim F$.
If the fiber is Ricci flat then
at the point on the base where $f$ achieves minimum (which exists by compactness) it holds that $Ric(V,V)\le 0$


