# 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, $d_{\partial\Omega}$, %, $d:\Omega\rightarrow [0,\infty [$, 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 $\Omega$.
Document type :
Journal articles
Domain :

https://hal.inria.fr/hal-00724729
Contributor : Estelle Bouzat <>
Submitted on : Wednesday, August 22, 2012 - 2:33:56 PM
Last modification on : Monday, October 5, 2020 - 11:22:01 AM

### Citation

Paolo Albano, Piermarco Cannarsa, Khai Tien Nguyen, Carlo Sinestrari. Singular gradient flow of the distance function and homotopy equivalence. Mathematische Annalen, Springer Verlag, 2013, 356 (1), pp.23-43. ⟨10.1007/s00208-012-0835-8⟩. ⟨hal-00724729⟩

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