---
tag: X5WKGC5Q6T
kind: concept
name: Cut locus
msc: [53C22]
sources:
  - url: https://en.wikipedia.org/wiki/Cut_locus
same_as:
  wikidata: Q55951116
  wikipedia: Cut locus
---

## Summary

The 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.
Its complement is the diffeomorphic image, under the [[exponential map|Y7Z0AGY1A7]], of an open star-shaped region of the tangent space.
So the complement of the cut locus is contractible, and the topology of the manifold is concentrated in the cut locus.
The distance from $p$ to its cut locus is the [[injectivity radius|Y6ZSKVAE3V]] at $p$.

## Explanations

### L1 human

Walk from $p$ along a geodesic.
At first you are on a shortest route from $p$.
At some moment, a different route becomes just as short, and from then on your path is no longer shortest.
The place where this happens is a cut point.
On the unit sphere, the only cut point of the north pole is the south pole.
Make a flat torus from a square centred at a point $p$ by gluing opposite sides.
Then the cut locus of $p$ is the image of the boundary of the square.