The Sard problem in step 2 and in filiform Carnot groups,

Abstract

Abstract. We study the Sard problem for the endpoint map in some well-known classes of Carnot groups. Our first main result deals with step 2 Carnot groups, where we provide lower bounds (depending only on the algebra of the group) on the codimension of the abnormal set; it turns out that our bound is always at least 3, which improves the result proved in [12] and settles a question emerged in [15]. In our second main result we characterize the abnormal set in filiform groups and show that it is either a horizontal line, or a 3-dimensional algebraic variety.

Carnot rectifiability of sub-Riemannian manifolds with constant tangent

“…where γ u is defined as in (4). In this case, the abnormal set can be described using Grassmannians.…”

“…In this context, abnormal geodesic are length-minimizing curves with a control that is singular with respect to the end-point map (16), see Section 4. In Carnot groups of step 2 it is known that the abnormal set Abn, i.e., the set of points reached by abnormal curves passing through the identity element of the group, is contained in an algebraic variety of co-dimension 3, see [4]. Free-Carnot groups of step 2 are a class for which this bound cannot be improved, in this sense they have the largest possible amount of abnormal curves within Carnot groups of step 2.…”

“…Nonetheless, a strong version of the Sard property (as in [11]) is needed in other to get a result analog to Proposition 4.3. However, this kind of results are known only for Carnot groups of small rank and/or step, see [4,11,12], even though no example of Carnot groups where the abnormal set has co-dimension smaller than 3 is known. A possibly harder phenomenon to study in wider classes of Carnot groups is the asymptotic local behaviour of the Lipschitz constant of d cc p0 G , ¨q, with respect to the distance from the abnormal set.…”