Singular gradient flow of the distance function and homotopy equivalence

Let M be a Riemannian manifold and let Omega be a bounded open subset of M. It is well known that significant information about the geometry of Omega is encoded into the properties of the distance, d , from the boundary of Omega. Here, we show that the generalized gradient flow associated with the distance preserves singularities, that is, if x_0 is a singular point of d then the generalized characteristic starting at x_0 stays singular for all times. As an application, we deduce that the singular set of d has the same homotopy type as Omega.