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 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 $M \times M$.

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 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 $M \times M$.

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 $M \times M$.

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 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 $M \times M$.