TY - RPRT T1 - The curvature: a variational approach Y1 - 2013 A1 - Andrei A. Agrachev A1 - Davide Barilari A1 - Luca Rizzi KW - Crurvature, subriemannian metric, optimal control problem AB - The 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. PB - SISSA UR - http://hdl.handle.net/1963/7226 N1 - 88 pages, 10 figures, (v2) minor typos corrected, (v3) added sections on Finsler manifolds, slow growth distributions, Heisenberg group U1 - 7260 U2 - Mathematics U4 - 1 U5 - MAT/03 GEOMETRIA ER -