Non-commutative Integrability, Moment Map and Geodesic Flows

TL;DR

This paper explores the relationship between commutative and non-commutative integrability in Hamiltonian systems, providing new examples of integrable geodesic flows on Riemannian manifolds, especially on bi-quotients of compact Lie groups.

Contribution

It demonstrates that geodesic flows on bi-quotients of compact Lie groups are integrable in non-commutative and classical senses using polynomial and smooth integrals.

Findings

Geodesic flow of bi-invariant metrics on bi-quotients is non-commutatively integrable.

Such flows are also integrable in the classical sense with smooth integrals.

New examples of integrable geodesic flows are constructed on Riemannian manifolds.

Abstract

The purpose of this paper is to discuss the relationship between commutative and non-commutative integrability of Hamiltonian systems and to construct new examples of integrable geodesic flows on Riemannian manifolds. In particular, we prove that the geodesic flow of the bi-invariant metric on any bi-quotient of a compact Lie group is integrable in non-commutative sense by means of polynomial integrals, and therefore, in classical commutative sense by means of $C^\infty$--smooth integrals.