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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 $M \times M$.