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Directional Pliability, Whitney Extension, and Lusin Approximation for Curves in Carnot Groups

Gareth Speight, Scott Zimmerman

TL;DR

The work addresses Whitney extension and Lusin approximation for horizontal curves in Carnot groups, introducing a directional pliability condition on a subset of the horizontal layer $\mathfrak{g}_1$. By proving a $C^1_H$ Whitney extension theorem on a compact set of pliable directions and applying it to the Engel group, the authors obtain partial Lusin-approximation results and show that no horizontal curve in the Engel group can avoid intersections with some $C^{1}$ horizontal curve on a set of positive measure. They classify pliable directions in the Engel group, showing that every horizontal direction except multiples of $X_2$ are pliable, which yields a Whitney-extension and Lusin-approximation framework in this group. The results provide a mechanism to construct Lusin approximations via Whitney extensions in non-Euclidean settings and give geometric insight into horizontal curves in step-3 Carnot groups, with concrete implications for the Engel group.

Abstract

We show that, in arbitrary Carnot groups, pliability in a subset of directions is sufficient to guarantee the existence of a Whitney-type extension and a Lusin approximation for curves with tangent vectors in the same set of directions. We apply this to show that every horizontal curve in the Engel group must intersect a $C^{1}$ horizontal curve in a set of positive measure.

Directional Pliability, Whitney Extension, and Lusin Approximation for Curves in Carnot Groups

TL;DR

The work addresses Whitney extension and Lusin approximation for horizontal curves in Carnot groups, introducing a directional pliability condition on a subset of the horizontal layer . By proving a Whitney extension theorem on a compact set of pliable directions and applying it to the Engel group, the authors obtain partial Lusin-approximation results and show that no horizontal curve in the Engel group can avoid intersections with some horizontal curve on a set of positive measure. They classify pliable directions in the Engel group, showing that every horizontal direction except multiples of are pliable, which yields a Whitney-extension and Lusin-approximation framework in this group. The results provide a mechanism to construct Lusin approximations via Whitney extensions in non-Euclidean settings and give geometric insight into horizontal curves in step-3 Carnot groups, with concrete implications for the Engel group.

Abstract

We show that, in arbitrary Carnot groups, pliability in a subset of directions is sufficient to guarantee the existence of a Whitney-type extension and a Lusin approximation for curves with tangent vectors in the same set of directions. We apply this to show that every horizontal curve in the Engel group must intersect a horizontal curve in a set of positive measure.

Paper Structure

This paper contains 13 sections, 13 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Carnot Groups
  4. Overview
  5. Coordinates
  6. Structure in exponential coordinates of the first kind
  7. Curves and distances in Carnot groups
  8. Smooth curves and pliability
  9. The Whitney Extension Property
  10. Engel Group
  11. Whitney extension and Lusin approximation in a subset of directions
  12. Pliability and Lusin approximation in the Engel Group
  13. Appendix

Key Result

Theorem 1.1

Suppose $\mathbb{G}$ is a Carnot group with horizontal layer $\mathfrak{g}_1$, and fix $\mathfrak{O} \subset \mathfrak{g}_1$. If every vector in $\mathfrak{O}$ is pliable, then $\mathbb{G}$ has the $C^1_H$ Whitney extension property on $\mathfrak{O}$.

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 21 more