Curvature: A variational approach

The curvature discussed in this paper is a far reaching generalisation of the Riemannian sectional curvature. We give a unified definition of curvature which applies to a wide class of geometric structures whose geodesics arise from optimal control problems, including Riemannian, sub-Riemannian, Finsler and sub-Finsler spaces. Special attention is paid to the sub-Riemannian (or Carnot-Carathéodory) metric spaces. Our construction of curvature is direct and naive, and similar to the original approach of Riemann. In particular, we extract geometric invariants from the asymptotics of the cost of optimal control problems. Surprisingly, it works in a very general setting and, in particular, for all sub-Riemannian spaces.