Integrable sub-Riemannian geodesic flows on the special orthogonal group

2024/11/13 by Bravo-Doddoli, Alejandro, Arathoon, Philip, Bloch, Anthony M. · 1 citation
#17B80 #37J35 #53C17 #70G65 #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2411.09008

Abstract

We analyse the geometry of the rubber-rolling distribution on the special orthogonal group and show that almost all the normal geodesics of any right-invariant sub-Riemannian metric defined on this distribution are completely integrable. Our argument is an adaptation of the method used to establish integrability of the Riemannian metric arising from the n-dimensional rigid body: namely, by exhibiting a Lax pair and bi-Hamiltonian structure for the reduced equations of motion.

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• Normal sub-Riemannian geodesics related to filtrations of Lie algebras

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