# Tubular neighborhoods in the sub-Riemannian Heisenberg groups

## TL;DR

This paper studies the regularity and geometric properties of tubular neighborhoods around sets in the sub-Riemannian Heisenberg groups, providing explicit formulas and regularity results based on curvature and boundary smoothness.

## Contribution

It establishes regularity of the Carnot-Carathéodory distance function and derives explicit volume formulas for tubular neighborhoods in the Heisenberg groups, extending geometric analysis in sub-Riemannian geometry.

## Key findings

- Proved $ ext{H}$-regularity of the distance function under mild conditions.
- Showed that for $C^k$ boundaries, the distance function is $C^k$ outside singular points.
- Derived explicit volume formulas involving horizontal curvatures and boundary functions.

## Abstract

We consider the Carnot-Carath\'eodory distance $\delta_E$ to a closed set $E$ in the sub-Riemannian Heisenberg groups $\mathbb{H}^n$, $n\ge 1$. The $\mathbb{H}$-regularity of $\delta_E$ is proved under mild conditions involving a general notion of singular points. In case $E$ is a Euclidean $C^k$ submanifold, $k\ge 2$, we prove that $\delta_E$ is $C^k$ out of the singular set. Explicit expressions for the volume of the tubular neighborhood when the boundary of $E$ is of class $C^2$ are obtained, out of the singular set, in terms of the horizontal principal curvatures of $\partial E$ and of the function $\langle N,T\rangle/|N_h|$ and its tangent derivatives.