Tubular neighborhoods in the sub-Riemannian Heisenberg groups

In the present paper we consider the Carnot-Caratheodory distance delta(E), to a closed set E in the sub-Riemannian Heisenberg groups IHn, n >= 1. The IH-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 >= 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 derivative E and of the function < N, T >/vertical bar N-h vertical bar and its tangent derivatives.