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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.
Keywords: Sard's property, Endpoint map, Abnormal curves, Carnot groups, Polarized groups, Sub-Riemannian geometry
@article{AIHPC_2016__33_6_1639_0,
author = {Le Donne, Enrico and Montgomery, Richard and Ottazzi, Alessandro and Pansu, Pierre and Vittone, Davide},
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},
year = {2016},
doi = {10.1016/j.anihpc.2015.07.004},
mrnumber = {3569245},
zbl = {1352.53025},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2015.07.004/}
}
TY - JOUR AU - Le Donne, Enrico AU - Montgomery, Richard AU - Ottazzi, Alessandro AU - Pansu, Pierre AU - Vittone, Davide TI - Sard property for the endpoint map on some Carnot groups JO - Annales de l'I.H.P. Analyse non linéaire PY - 2016 SP - 1639 EP - 1666 VL - 33 IS - 6 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2015.07.004/ DO - 10.1016/j.anihpc.2015.07.004 LA - en ID - AIHPC_2016__33_6_1639_0 ER -
%0 Journal Article %A Le Donne, Enrico %A Montgomery, Richard %A Ottazzi, Alessandro %A Pansu, Pierre %A Vittone, Davide %T Sard property for the endpoint map on some Carnot groups %J Annales de l'I.H.P. Analyse non linéaire %D 2016 %P 1639-1666 %V 33 %N 6 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2015.07.004/ %R 10.1016/j.anihpc.2015.07.004 %G en %F AIHPC_2016__33_6_1639_0
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
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