Singular gradient flow of the distance function and homotopy equivalence

Abstract

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