On a complete connected Riemannian manifold, let for a unit tangent vector be the supremum of times for which minimizes Riemannian distance from . The cut locus consists of the endpoints with . Radial minimization fails beyond this first cut time; distance spheres can cease to be smooth there.
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In differential geometry, the cut locus of a point on a manifold is a critical concept, particularly in the study of Riemannian manifolds. The cut locus of a point \( p \) in a Riemannian manifold is the set of points where geodesics emanating from \( p \) cease to be minimizing geodesics.