Abstract
The cut locus of a submanifold embedded properly in Euclidean space, hyperbolic space, or the sphere has an elementary description in terms of the set of maximal supporting balls of the submanifold. Three applications are given. The first is a proof that the topology of the cut locus of a submanifold is invariant when the submanifold is subjected to a Möbius transformation. The second is a simple method for constructing Riemannian manifolds which have a point whose cut locus is nontriangulable. The third is an investigation of the behavior of the cut locus when a submanifold of a sphere is subjected to a transformation of Lie Sphere Geometry.
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