Deforming metrics from non-negative to positive Ricci curvature Given a closed Riemannian manifold $(M,g)$ with non-negative Ricci curvature and $dim\geq 3$, when can we deform the metric to a positive Ricci curved one?
I know it's impossible in general due to the flat factor in the universal covering. But what about we add some topological restrictions on $M$ like simply connectedness? Are there any positive or negative results on this problem? 
( Besides, are there now any examples of simply connected closed manifold with positive scalar curved metric by do not admit a positive Ricci curved metric? )
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Thanks to the answer by Robert, I may simplify the problem in the following sense,
Given a simply connected flat Einstein manifold $(M,g)$ with $\hat{A}$-genus non-vanishing, and set $(\mathbb{S}^2,h)$ be the standard unit sphere, can $(M\times \mathbb{S}^2, g+h)$ be perturbed to a Ricci-positive manifold? $\ \ $In general, what about changing $(\mathbb{S}^2,h)$ to an arbitrary closed Ricci-positive manifold?
 A: This is not a complete answer but would be helpful. Here are a few facts:
Theorem (T. Aubin 1970 and P. Ehrlich 1976). If the Ricci curvature of a compact Riemannian manifold is
non-negative and positive somewhere, then the manifold carries a metric with
positive Ricci curvature.
Also vanishing the first Betti number is a necessary condition in compact case for admitting strictly
positive Ricci curvature.
Relation with scalar curvature:

*

*There are still no known examples of simply connected manifolds that admit
positive scalar curvature but not positive Ricci curvature


*If a manifold $M$ cannot have a metric with positive (or zero) Scalar curvature, then it certainly does not admit a metric with positive (zero res.) Ricci curvature.
This paper of G. Perelman is also useful: "Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers". Comparison Geometry. 30: 157–163 Click here for pdf
A: There are obstructions.  Perhaps the most famous comes from the theorem that, if a compact spin manifold has a metric of positive scalar curvature, then its $\hat A$-genus must vanish.
If you take a compact Riemannian spin manifold $(M,g)$ with special holonomy $\mathrm{G}_2$ (in dimension $7$), $\mathrm{Spin}(7)$ (in dimension $8$), or holonomy in $\mathrm{SU}(n)$ (in dimension $2n$) whose $\hat A$-genus is nonzero (and there are lots of these, even simply-connected ones), then $g$ will be Ricci-flat and hence will have non-negative Ricci curvature.  However, by the above theorem, it cannot carry any metric with positive scalar curvature, let alone a metric with positive Ricci curvature.
