Singular gradient flow of the distance function and homotopy equivalence
  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, $d_{\partial\Omega}$, %, $d:\Omega\rightarrow [0,\infty [$, 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 $\Omega$. 
Paolo Albano and Piermarco Cannarsa and Khai Tien Nguyen and Carlo   Sinestrari
math.AP cs.SY math.DG math.OC