Singular gradient flow of the distance function and homotopy equivalence

Abstract

It is a generally shared opinion that significant information about the topology of a bounded domain $\Omega$ of a riemannian manifold $M$ is encoded into the properties of the distance, $d_{\partial \Omega }$, %, $d:\Omega \to [0,\infty [$, from the boundary of $\Omega$. To confirm such an idea we propose an approach based on the invariance of the singular set of the distance function with respect to the generalized gradient flow of of $d_{\partial \Omega }$. As an application, we deduce that such a singular set has the same homotopy type as $\Omega$.