A Notion of Rectifiability Modeled on Carnot Groups

• Published in: Indiana University mathematics journal 2004-01, Vol.53 (1), p.49-81
• Main Author: Pauls, Scott D.
• Format: Article
• Language: English
• Subjects: Algebra; Calculus of variations and optimal control; Density; Differential geometry; Euclidean space; Exact sciences and technology; Geometry; Hausdorff dimensions; Hausdorff measures; Lie groups; Mathematical analysis; Mathematical theorems; Mathematics; Sciences and techniques of general use; Tangents
• ISSN: 0022-2518; 1943-5258
• DOI: 10.1512/iumj.2004.53.2293

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 Tx of H and r > 0 so that f−1(t) ∩ BH(x,r) is Tx-approximable at x and an approximate tangent cone isomorphic to Tx at x.