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  "body": "\n## Summary\n\nThe cut locus of a point $p$ in a complete [[Riemannian manifold|7CBT4SM83Y]] is the set of points where [[geodesics|NSYMQMHPDN]] from $p$ stop being shortest paths.\nIts complement is the diffeomorphic image, under the [[exponential map|Y7Z0AGY1A7]], of an open star-shaped region of the tangent space.\nSo the complement of the cut locus is contractible, and the topology of the manifold is concentrated in the cut locus.\nThe distance from $p$ to its cut locus is the [[injectivity radius|Y6ZSKVAE3V]] at $p$.\n\n## Explanations\n\n### L1 human\n\nWalk from $p$ along a geodesic.\nAt first you are on a shortest route from $p$.\nAt some moment, a different route becomes just as short, and from then on your path is no longer shortest.\nThe place where this happens is a cut point.\nOn the unit sphere, the only cut point of the north pole is the south pole.\nMake a flat torus from a square centred at a point $p$ by gluing opposite sides.\nThen the cut locus of $p$ is the image of the boundary of the square.\n",
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