Cut locus

Summary

The cut locus of a point $p$ in a complete Riemannian manifold is the set of points where geodesics from $p$ stop being shortest paths. Its complement is the diffeomorphic image, under the exponential map, 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 at $p$.