Does bounded scalar curvature imply bounded Ricci curvature?

Question

Does bounded scalar curvature imply bounded Ricci curvature?

It is trivial to show the converse, but I do not know whether the above is true. Inspired by a vaguely similar question, I am thinking that the product of a manifold with lower-unbounded and one with upper-unbounded Ricci curvature (provided the metrics be chosen carefully) might have bounded scalar curvature, but I am not sure.

Answer 1

Yes, your idea is okay. Let $(M,g)$ and $(N,h)$ be arbitrary Riemannian manifolds, then the Ricci curvature of the product manifold $(M\times N, g\oplus h)$ takes the form

$$ \mathrm{Ric}_{g\oplus h} = \begin{pmatrix} \mathrm{Ric}_g & 0 \\ 0 & \mathrm{Ric}_h \end{pmatrix} $$

So in particular if you arrange for $M$ to have constant scalar curvature $\lambda$ and $N$ to have constant scalar curvature $-\lambda$ you would have a product manifold of 0 scalar curvature.

So let $(M,g)$ be the standard sphere and $(N,h)$ be the hyperbolic sphere, and rescale, you immediately get counterexamples.