Sard property for the endpoint map on some Carnot groups

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• 作者：Enrico Le Donne^a ; ^{enrico.ledonne@jyu.fi" class="auth_mail" title="E-mail the corresponding author} ; Richard Montgomery^b ; ^{rmont@ucsc.edu" class="auth_mail" title="E-mail the corresponding author} ; Alessandro Ottazzi^c ; ^d ; ^{alessandro.ottazzi@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Pierre Pansu^e ; ^{Pierre.Pansu@math.u-psud.fr" class="auth_mail" title="E-mail the corresponding author} ; Davide Vittone^f ; ^g ; ^{vittone@math.unipd.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：53C17 ; 22F50 ; 22E25 ; 14M17
• 刊名：Annales de l'Institut Henri Poincare (C) Non Linear Analysis
• 出版年：2016
• 出版时间：November-December 2016
• 年：2016
• 卷：33
• 期：6
• 页码：1639-1666
• 全文大小：504 K

文摘

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.