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arXiv · 1703.01592

Tubular neighborhoods in the sub-Riemannian Heisenberg groups

Manuel Ritoré
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Abstract

We consider the Carnot-Carathéodory distance $δ_E$ to a closed set $E$ in the sub-Riemannian Heisenberg groups $\mathbb{H}^n$, $n\ge 1$. The $\mathbb{H}$-regularity of $δ_E$ 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 $δ_E$ 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.

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Manuel Ritoré. 2017-09-21. Tubular neighborhoods in the sub-Riemannian Heisenberg groups. https://doi.org/10.1515/acv-2017-0011

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