In the study of manifolds with positive sectional curvature, I guess the following statement is true:



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> 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)
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I cannot find any counterexample for this statement. Can anybody give a counterexample for this statement? 



Manifolds with positive sectional curvature
Perhaps you are thinking of a conjecture of Heinz Hopf: no Riemannian metric on $S^2 \times S^2$ has positive sectional curvature. This conjecture is still open, as far as I can tell.


For more information see [Manifold with a quasi-positive curvature](https://mathoverflow.net/questions/144451/manifold-with-a-quasi-positive-curvature/191795#191795) and also see Wolfgang Ziller's survey paper: <https://www.math.upenn.edu/~wziller/papers/SurveyMexico.pdf>


If you take the product metric on $M \times M$, and take any two geodesics $C\_1, C\_2 \subset M$ then the sectional curvature along $C\_1 \times C\_2$ is zero, while, for any surface $S \subset M$ and point $p \in M$, the sectional curvature along $S \times \{p\}$ is positive by hypothesis. The sectional curvature is not negative along any surface in $M \times M$.


