Holonomy of Sub-Riemannian Manifolds

Abstract

A sub-Riemannian manifold is a smooth manifold which carries a distribution equipped with a metric. We study the holonomy and the horizontal holonomy (i.e. holonomy spanned by loops everywhere tangent to the distribution) of sub-Riemannian manifolds of contact type relative to an adapted connection. In particular, we obtain an Ambrose–Singer type theorem for the horizontal holonomy and we classify the holonomy irreducible sub-Riemannian symmetric spaces (i.e. homogeneous sub-Riemannian manifolds admitting an involutive isometry whose restriction to the distribution is a central symmetry).