On the completeness of the Manakov integrals

TL;DR

This paper provides simple proofs demonstrating the completeness of Manakov's integrals for rigid body motion in n-dimensional space and for geodesic flows on certain homogeneous spaces, advancing understanding of integrable systems.

Contribution

It introduces straightforward proofs confirming the completeness of Manakov's integrals in higher-dimensional rigid body and homogeneous space contexts, extending prior results.

Findings

Proves completeness of Manakov's integrals for n-dimensional rigid body motion.

Establishes completeness for geodesic flows on specific homogeneous spaces.

Simplifies previous proofs of integrability in these systems.

Abstract

The aim of this note is to present simple proofs of the completeness of Manakov's integrals for a motion of a rigid body fixed at a point in $\mathbb R^n$, as well as for geodesic flows on a class of homogeneous spaces $SO(n)/SO(n_1)\times\dots\times SO(n_r)$.