My riemannian geometry professor gave us the following exercise.
Let $M = \mathbb{R P}^n \times \mathbb{R P}^n$, where $\mathbb{R P}^n$ is the $n$-dimensional projective space. Show that $M$ does not admit a metric with positive sectional curvature.
I was able to prove the result in the case $n$ is odd: In this case, $\mathbb{R P}^n$ is orientable and therefore $M$ is orientable. If $M$ admits a metric with positive sectional curvature then by the Synge theorem $M$ must be simply connected, which is not the case because $\pi^1(M) = \mathbb Z / 2\mathbb Z\times\mathbb Z / 2\mathbb Z$.
How do I prove the result if $n$ is even?
Suppose $n$ even. Then $M$ is non-orientable and has positive sectional curvature. Consider the orientable double cover $\hat M \rightarrow M$ and pull back the Riemannian metric to $\hat M$. Then $\hat M$ has positive sectional curvatures and by Synges theorem $\hat M$ is simply connected. This implies that $\pi_1(M)=\mathbb{Z}/2$, which is wrong.