Sard property for the endpoint map on some Carnot groups

Abstract

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.

On the codimension of the abnormal set in step two Carnot groups

“…An immediate consequence of Proposition 2.12, proved in [5,Theorem 1.4], is the following result. Theorem 2.13.…”

“…See also [1], [3] and [7]. In [5], the authors of this note and others proved the Sard Property in a number of special cases, and they also obtained the following first result concerning the interesting problem of obtaining finer estimates on the size of the abnormal set.…”

“…In this section we establish the notation we are going to use and we recall some preliminary facts that were proved in [5].…”

“…Theorem ( [5,Theorem 3.15]). In any free nilpotent group of step 2 the abnormal set is an algebraic subvariety of codimension 3.…”

A dynamical approach to the Sard problem in Carnot groups

“…A similar result has been obtained in [10] by D. Barilari and the first author in the more general case of control systems that are affine in the control, i.e., admitting a drift. In [28], the authors prove the negligibility of Abn G in Carnot groups of step 2 as well as in some other cases, some of which will be mentioned below. A detailed study of the singular set has been carried out in [13,12] for 3-dimensional analytic sub-Riemannian manifolds with 2-dimensional analytic horizontal distributions: it turns out that such a set has Hausdorff dimension 1 and, actually, it is a semi-analytic curve.…”