  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 $\Omega$.
