Calculus of Variations and Geometric Measure Theory

A. Agrachev - D. Barilari - E. Paoli

Volume geodesic distortion and Ricci curvature for Hamiltonian dynamics

created by barilari on 28 Feb 2016
modified on 16 Apr 2021


Published Paper

Inserted: 28 feb 2016
Last Updated: 16 apr 2021

Journal: Annales de l'Institut Fourier
Volume: 69
Number: 3
Pages: 1187-1228
Year: 2019


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.