Integrable sub-Riemannian geodesic flows on the special orthogonal group

Alejandro Bravo-Doddoli, Philip Arathoon, Anthony M. Bloch

Published 2024-11-13Version 1

One way to define a sub-Riemannian metric is as the limit of a Riemannian metric. Consider a Riemannian structure depending on a parameter $s$ such that its limit defines a sub-Riemannian metric when $s \to \infty$, assuming that the Riemannian geodesic flow is integrable for all $s$. An interesting question is: Can we determine the integrability of the sub-Riemannian geodesic flow as the limit of the integrals of motion of the Riemannian geodesic flow? The paper's main contribution is to provide a positive answer to this question in the special orthogonal group. Theorem 1.1 states that the Riemannian geodesic flow is Liuville integrable: The Manakov integrals' limit suggests the existence of a Lax pair formulation of the Riemannian geodesic equations, and the proof of Theorem 1.1 relies on this Lax pair.