Tubes in the Heisenberg group

In Euclidean geometry, one can extract geometric invariants of a submanifold by studying the volume of a tube around it. The coefficients appearing in the corresponding expansion are given by the total masses of curvature measures. In this talk, we discuss how this picture changes in the Heisenberg group,

which is a simple, yet non-trivial, example of a sub-Riemannian manifold. In this setting, tubes around submanifolds can still be defined, but their geometry is more subtle. We investigate to what extent curvature measures can be introduced and which of their classical properties survive. In some cases, the classical notion of curvature measure must be replaced by the notion of curvature distribution. Based on a joint work with D. Barilari (UniPD) and E. Bellini (UniPD).