Singular Manakov Flows and Geodesic Flows on Homogeneous Spaces

TL;DR

This paper proves the complete integrability of certain geodesic flows on homogeneous spaces, including new proofs for Manakov symmetric rigid body motion and Einstein metrics, advancing understanding of integrable systems on manifolds.

Contribution

It provides new proofs of integrability for Manakov-type flows, Einstein metrics, and geodesic flows on specific homogeneous spaces, expanding the class of known integrable systems.

Findings

Complete integrability of SO(n)-invariant geodesic flows on specified homogeneous spaces.

New proof of integrability for Manakov symmetric rigid body motion.

Proof of integrability for Einstein metrics on certain homogeneous spaces.

Abstract

We prove complete integrability of the Manakov-type SO(n)-invariant geodesic flows on homogeneous spaces $SO(n)/SO(k_1)\times...\times SO(k_r)$, for any choice of $k_1,...,k_r$, $k_1+...+k_r\le n$. In particular, a new proof of the integrability of a Manakov symmetric rigid body motion around a fixed point is presented. Also, the proof of integrability of the SO(n)-invariant Einstein metrics on $SO(k_1+k_2+k_3)/SO(k_1)\times SO(k_2)\times SO(k_3)$ and on the Stiefel manifolds $V(n,k)=SO(n)/SO(k)$ is given.