Cut locus (Riemannian manifold)

In Riemannian geometry, the cut locus of a point ${\displaystyle p}$ in a manifold is roughly the set of all other points for which there are multiple minimizing geodesics connecting them from ${\displaystyle p}$, but it may contain additional points where the minimizing geodesic is unique, under certain circumstances. The distance function from p is a smooth function except at the point p itself and the cut locus.