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

2025/05/20 by Speight, Gareth, Zimmerman, Scott · 1 citation
#53C17 #58C25 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.14678

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 C¹ horizontal curve in a set of positive measure.

Citations

• On the equivalence of derivatives for maps between Carnot groups
• On the converse of Pansu's Theorem
• A C^k Lusin Approximation Theorem For Real-Valued Functions on Carnot Groups
• Whitney's Extension Theorem and the finiteness principle for curves in\n the Heisenberg group
• A C^m Whitney Extension Theorem for Horizontal Curves in the\n Heisenberg Group
• Lusin approximation for horizontal curves in step 2 Carnot groups
• The Whitney Extension Theorem for C¹, horizontal curves in the Heisenberg group
• Lusin Approximation and Horizontal Curves in Carnot Groups
• Metriques de Carnot-Caratheodory et Quasiisometries des Espaces Symetriques de rang un

Cited by

• A note on pliability and the openness of the multiexponential map in Carnot groups

Related

• Lusin approximation for horizontal curves in step 2 Carnot groups
• Lusin Approximation and Horizontal Curves in Carnot Groups
• A C^m Whitney Extension Theorem for Horizontal Curves in Free Step 2 Carnot Groups
• A note on pliability and the openness of the multiexponential map in Carnot groups
• Higher order Whitney extension and Lusin approximation for Horizontal curves in the Heisenberg group