Appears in collection : Sub-Riemannian manifolds : from geodesics to hypoelliptic diffusion / Géométrie sous-riemannienne : des géodésiques aux diffusions hypoelliptiques

A sub-Riemannian distance is obtained when minimizing lengths of paths which are tangent to a distribution of planes. Such distances differ substantially from Riemannian distances, even in the simplest example, the 3-dimensional Heisenberg group. This raises many questions in metric geometry: embeddability in Banach spaces, bi-Lipschitz or bi-Hölder comparison of various examples. Emphasis will be put on Gromov's results on the Hölder homeomorphism problem, and on a quasisymmetric version of it motivated by Riemannian geometry.