Volume geodesic distortion and Ricci curvature for Hamiltonian dynamics
  We study the variation of a smooth volume form along extremals of a variational problem with nonholonomic constraints and an action-like Lagrangian. We introduce a new invariant describing the interaction of the volume with the dynamics and we study its basic properties. We then show how this invariant, together with curvature-like invariants of the dynamics, appear in the expansion of the volume at regular points of the exponential map. This generalizes the well-known expansion of the Riemannian volume in terms of Ricci curvature to a wide class of geometric structures, including all sub-Riemannian manifolds. 
Andrei Agrachev, Davide Barilari, Elisa Paoli
math.DG math.DS math.MG math.OC