Agrachev, Andrei A.Barilari, DavideRizzi, Luca2013-12-032013-12-032013-06-22https://openscience.sissa.it/handle/1963/722688 pages, 10 figures, (v2) minor typos corrected, (v3) added sections on Finsler manifolds, slow growth distributions, Heisenberg groupThe curvature discussed in this paper is a rather far going generalization of the Riemannian sectional curvature. We define it for a wide class of optimal control problems: a unified framework including geometric structures such as Riemannian, sub-Riemannian, Finsler and sub-Finsler structures; a special attention is paid to the sub-Riemannian (or Carnot-Caratheodory) metric spaces. Our construction of the curvature is direct and naive, and it is similar to the original approach of Riemann. Surprisingly, it works in a very general setting and, in particular, for all sub-Riemannian spaces.enPreprintCrurvature, subriemannian metric, optimal control problemMAT/03 GEOMETRIA