A notion of rectifiability modeled on Carnot groups

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-N(1) functions, f : N --> R, where N is a N Carnot group. We show that for almost every t is an element of R and almost every noncharacteristic x is an element of f(-1) (t), there exist a subgroup T-x of H and r > 0 so that f (-1) (t) boolean AND B-H (x, r) is T-x-approximable at x and an approximate tangent cone isomorphic to T-x at x.