In the study of manifolds with positive sectional curvature, I guess the following statement is true:
Let $M$ be a manifold with positive (but not necessarily constant) sectional curvature. Then $M\times M$ admits a Riemannian metric with non-negative sectional curvature such that there exists a direction with $K=0$ and there exists a direction with $K=1$. ($K$= Sectional curvature)
I cannot find any counterexample for this statement. Can anybody give a counterexample for this statement?