It is discussed in this question whether a simply-connected closed Riemannian manifold with non-negative Ricci curvature admits positive Ricci curvature, and the answer appears to be "no, there are counter-examples and known obstructions".

My question is this: are there any known examples of closed, simply-connected manifolds of non-negative **sectional** curvature that do not admit positive Ricci curvature? Are there known obstructions?