I have encountered this problem:
Show that a geodesic of $(\mathbb RP_n, g_0)$ with $g_0$ being the metric given by the canonical metric on $\mathbb S^n$ via the $2:1$ Riemainnian covering, is minimal if and only if its length is less than or equal to $\pi/2$.
I think it is a standard exercise. But I have no idea how to crack it. Any commnet is appreciated.
Lift to $S^2$.
Suppose that your geodesic in $\mathbb RP^2$ has length $L$ larger than $\pi/2$. Lift to a geodesic segment in $S^2$ between points $A$ and $B$. Let $A' = -A$. Then the extension of the great-circle arc from $A$ to $B$ onwards to $A'$ will give you a segment $BA'$ that's shorter than $\pi/2$. Its projection to $\mathbb RP^2$ will be a (shorter) geodesic between your original two points.
The other halfof this -- that if it's not minimal, then its length is greater than $\pi/2$ -- follows a similar argument: lift to $S^2$, and you've got a not-minimal "short" geodesic between two points $A$ and $B$; the "shorter" geodesic might run from $B$ to $A'$ rather than $A$, but evn if it does, you end up with a split geodesic between $A$ and $A'$ whose length is less than $\pi$, which is impossible. If the "shorter" geodesic ran between $A$ and $B$, you'd have a contradiction, because the raduis of injectivity of the exponential map on $S^2$ is $\pi$ (i.e., all geodesic paths of length less than $\pi$ on $S^2$ are in fact shortest paths).