Mathematics

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.