A notion of rectifiability modeled on Carnot groups

Abstract

We introduce a notion of rectifiability modeled on Carnot groups. Precisely, for E a subset of a Carnot group M and N a subgroup of M, we say E is N-rectifiable 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 1 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 -1 (t), there exist a subgroup T x of H and r > 0 so that f -1 (t) n B H (x, r) is T x -approximable at x and an approximate tangent cone isomorphic to T x at x.