Volume geodesic distortion and Ricci curvature for Hamiltonian dynamics
Differential Geometry
2018-03-15 v3 Dynamical Systems
Metric Geometry
Optimization and Control
Abstract
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.
Keywords
Cite
@article{arxiv.1602.08745,
title = {Volume geodesic distortion and Ricci curvature for Hamiltonian dynamics},
author = {Andrei Agrachev and Davide Barilari and Elisa Paoli},
journal= {arXiv preprint arXiv:1602.08745},
year = {2018}
}
Comments
31 pages, 5 figures. v3 Minor revision, to appear on Annales de l'Institut Fourier