#### Document Type

Article

#### Publication Date

7-13-2004

#### Publication Title

Indiana University Mathematics Journal

#### Department

Department of Mathematics

#### Abstract

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 C^1_N 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 T_x of H and r >0 so that f^{-1}(t) intersected with B_H(x,r) is T_x-approximable at x and an approximate tangent cone isomorphic to T_x at x.

#### DOI

10.1512/iumj.2004.53.2293

#### Dartmouth Digital Commons Citation

Pauls, Scott D., "A Notion of Rectifiability Modeled on Carnot Groups" (2004). *Dartmouth Scholarship*. 3847.

https://digitalcommons.dartmouth.edu/facoa/3847