Starshapedness and convexity in carnot groups and geometry of hormander vector fields

Abstract

Sub-Riemannian geometries are very important, not only for a pure mathe-matics point of view but also for the many applications in physics (see [73]), in economics (see e.g [76]), in biology and image processing (for example visual vertex model by Citti-Sarti [33]). Sub-Riemannian geometries are manifolds where the Riemannian metric is de-fined only on subbundle of the tangent bundle, and this leads to constrains on the allowed directions when moving on the manifold. In particular, we look at the case when the distribution generating the subbundle satisfies the bracket generating condition (see Definition 2.1.1). This implies that, even if some directions are forbidden, we can still move everywhere on the manifold. Un-like the Riemannian case, these geometries are not equivalent to the Euclidean space at any scaling, and presents substantial differences w.r.t. more known Riemannian case (see Section 2.2).