2000/04/30 by Scott D. Pauls
Mathematics · #math.CA #math.DG
published as Indiana Univ. Math. J. 53 (2004), 49-82 · 27 pages
arxiv created 2004/07/13 · arxiv updated 2009/11/30
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 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 C1N functions, where N is a Carnot group. We show that for almost every real number t and almost every noncharacteristic point x in a level set of f, there exists a subgroup Tx of H and r >0 so that f-1(t) intersected with BH(x,r) is Tx-approximable at x and an approximate tangent cone isomorphic to Tx at x.