Characterization of a differentiable point of the distance function to the cut locus

Dedicated to Professor Takashi Sakai on the occasion of his sixtieth birthday

Abstract

We give a necessary and sufficient condition for a given point on the unit normal bundle of a closed submanifold N of a 2-dimensional complete Riemannian manifold M to be a differentiable point of the distance function to the cut locus of N.