A Notion of Rectifiability Modeled on Carnot Groups

We introduce a notion of rectifiability modeled on Carnot groups. Precisely, we say that a subset E of a Carnot group M and N is a subgroup of M , we say E is Nrectifiable if it is the Lipschitz image of a positive measure subset of N . First, we discuss the implications of N-rectifiability, where N is a Carnot group (not merely a subgroup of a Carnot group), which include N-approximability and the existence of approximate tangent cones isometric to N almost everywhere in E. Second, we prove that, under a stronger condition concerning the existence of approximate tangent cones isomorphic to N almost everywhere in a set E, that E is N-rectifiable. Third, we investigate the rectifiability properties of level sets of C N functions, f : N → R, where N is a Carnot group. We show that for almost every t ∈ R and almost every noncharacteristic x ∈ f(t), there exists a subgroup Tx of H and r > 0 so that f (t) ∩ BH(x, r) is Txapproximable at x and an approximate tangent cone isomorphic to Tx at x.