Abstract In the present paper we consider the Carnot–Carathéodory distance δ E {\delta_{E}} to a closed set E in the sub-Riemannian Heisenberg groups ℍ n {{\mathbb{H}}^{n}} , n ⩾ 1 {n\geqslant 1} . The ℍ {{\mathbb{H}}} -regularity of δ E {\delta_{E}} is proved under mild conditions involving a general notion of singular points. In case E is a Euclidean C k {C^{k}} submanifold, k ⩾ 2 {k\geqslant 2} , we prove that δ E {\delta_{E}} is C k {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 {C^{2}} are obtained, out of the singular set, in terms of the horizontal principal curvatures of ∂ ⁡ E {\partial E} and of the function 〈 N , T 〉 / | N h | {\langle N,T\rangle/|N_{h}|} and its tangent derivatives.

中文翻译：

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亚黎曼海森堡群中的管状邻域

摘要 在本文中，我们考虑到子黎曼海森堡群中的闭集 E 的 Carnot–Carathéodory 距离 δ E {\delta_{E}} ℍ n {{\mathbb{H}}^{n}} , n ⩾ 1 {n\geqslant 1} 。δ E {\delta_{E}} 的 ℍ {{\mathbb{H}}} -正则性在涉及奇异点一般概念的温和条件下得到证明。如果 E 是欧几里得 C k {C^{k}} 子流形，k ⩾ 2 {k\geqslant 2} ，我们证明 δ E {\delta_{E}} 是 C k {C^{k}}的单数集。当 E 的边界属于类 C 2 {C^{2}} 时，管状邻域体积的显式表达式从奇异集合中获得，根据∂ ⁡ E {\partial E 的水平主曲率} 和函数 〈 N , T 〉 / | Nh | {\langle N,T\rangle/|N_{h}|} 及其切线导数。