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 $Ω$ of a riemannian manifold $M$ is encoded into the properties of the distance, $d_{\partialΩ}$, %, $d:Ω\rightarrow [0,\infty [$, from the boundary of $Ω$. 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 $d_{\partialΩ}$. As an application, we deduce that such a singular set has the same homotopy type as $Ω$.