= Cut and conjugate times in round real projective space
{title2=$c(v)=\pi/2,\quad t_{\mathrm{conj}}=\pi$}
In unit-curvature <Real projective space> of dimension at least two, <Riemannian distance> is $d([v],[u])=\arccos|\langle v,u\rangle|$ for unit lifts. A unit-speed <geodesic> has lift $\cos t\,v+\sin t\,w$, with $w$ unit and perpendicular to $v$, and therefore ceases to minimize after $\pi/2$. Its normal <Jacobi fields> with zero initial value are $\sin t$ times <parallel vector fields along a curve>, giving first <conjugate point> at time $\pi$. The two times differ in every direction.
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