Calculus of Variations and Geometric Measure Theory

A. Bravo-Doddoli - E. Le Donne - N. Paddeu

Sympletic reduction of the sub-Riemannian geodesic flow for metabelian nilpotent groups

created by ledonne on 28 Sep 2023



Inserted: 28 sep 2023
Last Updated: 28 sep 2023

Year: 2022

ArXiv: 2211.05846 PDF


We consider nilpotent Lie groups for which the derived subgroup is abelian. We equip them with subRiemannian metrics and we study the normal Hamiltonian flow on the cotangent bundle. We show a correspondence between normal trajectories and polynomial Hamiltonians in some euclidean space. We use the aforementioned correspondence to give a criterion for the integrability of the normal Hamiltonian flow. As an immediate consequence, we show that in Engel-type groups the flow of the normal Hamiltonian is integrable. For Carnot groups that are semidirect products of two abelian groups, we give a set of conditions that normal trajectories must fulfill to be globally length-minimizing. Our results are based on a symplectic reduction procedure.

Tags: GeoMeG