Volume geodesic distortion and Ricci curvature for Hamiltonian dynamics

Agrachev, Andrei A.; Barilari, Davide; Paoli, Elisa

Annales de l'Institut Fourier, Tome 69 (2019) no. 3, pp. 1187-1228.

On considère la variation d’un volume lisse le long des extrema d’un problème variationnel avec contraintes non holonômes et lagrangien de type action. On introduit un nouvel invariant, appelé derivée canonique du volume, qui décrit l’interaction entre la forme volume et la dynamique. On montre comment cet invariant, avec des invariants de type courbure associés à la dynamique, apparaissent dans le développement asymptotique du volume. Cela généralise le développement classique du volume riemannien le long du flot géodésique en termes de la courbure de Ricci à une vaste classe de flots hamiltoniens, notamment tous les flots géodésiques sous-riemanniens.

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, called volume geodesic derivative, 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 asymptotic expansion of the volume. This generalizes the well-known expansion of the Riemannian volume in terms of Ricci curvature to a wide class of Hamiltonian flows, including all sub-Riemannian geodesic flows.

Classification : 53C17, 53B21, 53B15
Keywords: volume, geodesics, Ricci curvature, Hamiltonian systems, sub-Riemannian geometry
Mot clés : volume, géodésiques, courbure de Ricci, systèmes Hamiltoniens, géométrie sous-riemannienne

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