JournalsaihpcVol. 33, No. 6pp. 1639–1666

Sard property for the endpoint map on some Carnot groups

  • Alessandro Ottazzi

    Dipartimento di Matematica, Università di Trento, Trento, Italy, School of Mathematics and Statistics, University of New South Wales, Sydney, Australia

  • Pierre Pansu

    Département de Mathématiques, Université Paris-Sud, Orsay, France

  • Davide Vittone

    Dipartimento di Matematica Pura ed Applicata, Università di Padova, Padova, Italy, Institut für Mathematik, Universität Zürich, Zürich, Switzerland

  • Enrico Le Donne

    Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland

  • Richard Montgomery

    Mathematics Department, University of California, Santa Cruz, USA
Sard property for the endpoint map on some Carnot groups cover
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Abstract

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.

Cite this article

Alessandro Ottazzi, Pierre Pansu, Davide Vittone, Enrico Le Donne, Richard Montgomery, Sard property for the endpoint map on some Carnot groups. Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 6, pp. 1639–1666

DOI 10.1016/J.ANIHPC.2015.07.004