Let N be a closed submanifold of a complete smooth Riemannian manifold M and Uν the total space of the unit normal bundle of N. For each v ∈ Uν, let ρ(v) denote the distance from N to the cut point of N on the geodesic γv with the velocity vectorγv(0) = v. The continuity of the function ρ on Uν is well known. In this paper we prove that ρ is locally Lipschitz on which ρ is bounded; in particular, if M and N are compact, then ρ is globally Lipschitz on Uν. Therefore, the canonical interior metric δ may be introduced on each connected component of the cut locus of N, and this metric space becomes a locally compact and complete length space.

No abstract

In this note we prove that the HausdorfT dimension of a cut locus on a smooth Riemannian manifold is an integer.

The cut locus from a point on the surface of a convex polyhedron is a tree containing a line segment beginning at every vertex. In the limit of infinitely small triangles, the cut locus from a point on a triangulation of a smooth surface therefore tends to become dense in the smooth surface, whereas the cut locus from the same point on the smooth surface is also a tree, but of finite length. We introduce a method for avoiding this problem. The method involves introducing a minimal angular resolution and discarding those points of the cut locus on the triangulation for which the angle measured between the shortest geodesic curves meeting at these points is smaller than the given angular resolution. We also describe software based upon this method that allows one to visualize the cut locus from a point on a surface of the form (x/a) n + (y/b) n + (z/c) n = 1, where n is a positive even integer. We use the software to support a new conjecture that the cut locus of a general ellipsoid is a subarc of a curvature line of the ellipsoid.