Tubes in sub-Riemannian geometry and a Weyl's invariance result for curves in the Heisenberg groups

TL;DR

This paper establishes optimal regularity for sub-Riemannian distance from submanifolds, analyzes tubular volume asymptotics, and proves a Weyl invariance for curves in Heisenberg groups based on Reeb angle.

Contribution

It provides new regularity results, asymptotic volume formulas, and a Weyl invariance theorem for curves in Heisenberg groups, without volume computation.

Findings

Optimal regularity results for sub-Riemannian distances.

Asymptotic formulas for tubular neighborhood volumes.

Weyl's invariance of tube volume depending only on Reeb angle.

Abstract

The purpose of the paper is threefold: first, we prove optimal regularity results for the distance from $C^k$ submanifolds of general rank-varying sub-Riemannian structures. Then, we study the asymptotics of the volume of tubular neighbourhoods around such submanifolds. Finally, for the case of curves in the Heisenberg groups, we prove a Weyl's invariance result: the volume of small tubes around a curve does not depend on the way the curve is isometrically embedded, but only on its Reeb angle. The proof does not need the computation of the actual volume of the tube, and it is new even for the three-dimensional Heisenberg group.