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 showingthat each non-conjugate cut point has a neighbourhood on whichthe distance function is the minimum of finitely many smooth‘radial functions’. This enables us to generaliseto an arbitrary complete manifold a number of recent results,mainly from stochastic analysis, which were either limited ornot valid on the cut locus because of the lack of differentiabilitythere of the distance function.