no. 6
Sard property for the endpoint map on some Carnot groups
Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 6, pp. 1639-1666.

In Carnot–Carathéodory or sub-Riemannian geometry, one of the major open problems is whether the conclusions of Sard's theorem holds for the endpoint map, a canonical map from an infinite-dimensional path space to the underlying finite-dimensional manifold. The set of critical values for the endpoint map is also known as abnormal set, being the set of endpoints of abnormal extremals leaving the base point. We prove that a strong version of Sard's property holds for all step-2 Carnot groups and several other classes of Lie groups endowed with left-invariant distributions. Namely, we prove that the abnormal set lies in a proper analytic subvariety. In doing so we examine several characterizations of the abnormal set in the case of Lie groups.

DOI : 10.1016/j.anihpc.2015.07.004
Classification : 53C17, 22F50, 22E25, 14M17
Mots clés : Sard's property, Endpoint map, Abnormal curves, Carnot groups, Polarized groups, Sub-Riemannian geometry 
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     title = {Sard property for the endpoint map on some {Carnot} groups},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1639--1666},
     publisher = {Elsevier},
     volume = {33},
     number = {6},
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Le Donne, Enrico; Montgomery, Richard; Ottazzi, Alessandro; Pansu, Pierre; Vittone, Davide. Sard property for the endpoint map on some Carnot groups. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 6, pp. 1639-1666. doi : 10.1016/j.anihpc.2015.07.004. http://archive.numdam.org/articles/10.1016/j.anihpc.2015.07.004/

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