The Lipschitz continuity of the distance function to the cut locus

Abstract:

Let $N$ be a closed submanifold of a complete smooth Riemannian manifold $M$ and $U\nu$ the total space of the unit normal bundle of $N$. For each $v \in U\nu$, let $\rho (v)$ denote the distance from $N$ to the cut point of $N$ on the geodesic $\gamma _v$ with the velocity vector $\dot \gamma _v(0)=v.$ The continuity of the function $\rho$ on $U\nu$ is well known. In this paper we prove that $\rho$ is locally Lipschitz on which $\rho$ is bounded; in particular, if $M$ and $N$ are compact, then $\rho$ is globally Lipschitz on $U\nu$. Therefore, the canonical interior metric $\delta$ 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.