RESEARCH

The main topic of my PhD studies is the investigation of some questions related to the geometry of submanifolds in sub-Riemannian geometry.

The study of the geometry of submanifolds of an ambient manifold, with a given geometric structure is a classical subject. In Riemannian geometry a submanifold inherits its natural Riemannian structure by restricting the metric tensor to the tangent space. Things are less straightforward in sub-Riemannian geometry. Indeed, there is no evident sub-Riemannian structure induced on a submanifold.

The aim of the project is to study problems related to sub-Riemannian curvature objects. In particular, trough the asymptotic analysis of the volume of tubular neighbothoods over non-characteristic submanifolds, the goal is to obtain the geometric information carried by the coefficients of such an expansion as the size of the tube 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.