Tubular neighborhoods in the sub-Riemannian Heisenberg groups

Abstract

In the present paper we consider the Carnot–Carathéodory distance δE to a closed set E in the sub-RiemannianHeisenberg groupsℍn, n ⩾ 1. Theℍ-regularity of δE is proved under mild conditions involving a general notion of singular points. In case E is a Euclidean Ck submanifold, k ⩾ 2, we prove that δE is Ck out of the singular set. Explicit expressions for the volume of the tubular neighborhood when the boundary of E is of class C2 are obtained, out of the singular set, in terms of the horizontal principal curvatures of ∂E and of the function ⟨N, T⟩/|Nh| and its tangent derivatives.