Steiner's and Weyl's tube formulae in sub-Riemannian Geometry

The objective of this thesis is to explore the geometric properties of submanifolds within sub-Riemannian structures through the study of the volume of their tubular neighborhoods. In particular, we aim at investigate the geometric information carried by the coefficients of the asymptotics of the volume of the tube as the size tends to zero. First of all, we consider the case of smooth surfaces embedded in a three-dimensional contact sub-Riemannian manifold and that do not contain characteristic points. We derive a Steiner-like formula computing the asymptotics of the volume of the half-tubular neighborhood up to the third order and with respect to any smooth measure. Subsequently, we extend the Weyl's tube formula for non-characteristic submanifolds of class C^2 with arbitrary codimension, and the Steiner's formula for non-characteristic hypersurfaces of class C^2 in any sub-Riemannian manifold equipped with a smooth measure. The volume of the tubular and half-tubular neighborhoods is a smooth or real-analytic function whenever the ambient sub-Riemannian structure and the assigned measure are smooth or real-analytic, respectively. Moreover, the coefficients of the Taylor expansion are written in terms of integrals of iterated divergences of the distance from the submanifold. Finally, we present a result concerning the local integrability of the sub-Riemannian mean curvature of a surface in a 3D contact sub-Riemannian manifold in presence of isolated characteristic points. In particular, the main result focuses on the integrability of the mean curvature around the so-called mildly degenerate points and with respect to the Riemannian induced measure.