Cut locus
= Cut locus
{title2=$\operatorname{Cut}(p)$}
On a complete connected <Riemannian manifold>, let $c(v)$ for a unit tangent vector $v$ be the supremum of times for which $t\mapsto\exp_p(tv)$ minimizes <Riemannian distance> from $p$. The cut locus consists of the endpoints $\exp_p(c(v)v)$ with $c(v)<\infty$. Radial minimization fails beyond this first cut time; distance spheres can cease to be smooth there.