SOME CONSEQUENCES OF THE NATURE OF THE DISTANCE FUNCTION ON THE CUT LOCUS IN A RIEMANNIAN MANIFOLD

We render the cut locus more accessible to analysis by showing that ea ch non-conjugate cut point has a neighbourhood on which the distance f unction is the minimum of finitely many smooth 'radial functions'. Thi s enables us to generalise to an arbitrary complete manifold a number of recent results, mainly from stochastic analysis, which were either limited or not valid on the cut locus because of the lack of different iability there of the distance function.