Complete Integrability of Subriemannian Geodesic Flows on S⁷

verfasst von

Wolfram Bauer, Abdellah Laaroussi, Daisuke Tarama

Abstract

Four subriemannian (SR) structures over the Euclidean sphere are considered in accordance to the previous literature. The defining bracket generating distribution is chosen as the horizontal space in the Hopf fibration, the quaternionic Hopf fibration, or spanned by a suitable number of canonical vector fields. In all cases the induced SR geodesic flow on is studied. Adapting a method by A. Thimm in [36], a maximal set of functionally independent and Poisson commuting first integrals are constructed, including the corresponding SR Hamiltonian. As a result, the complete integrability in the sense of Liouville is proved for the SR geodesic flow. It is observed that these first integrals arise as the symbols of commuting second-order differential operators one of them being a (not necessarily intrinsic) sublaplacian. On the way one explicitly derives the Lie algebras of all SR isometry groups intersected with.

Organisationseinheit(en)

Institut für Analysis

Externe Organisation(en)

Ritsumeikan University

Typ

Artikel

Journal

International Mathematics Research Notices

Band

2025

ISSN

1073-7928

Publikationsdatum

02.2025

Publikationsstatus

Veröffentlicht

Peer-reviewed

Ja

ASJC Scopus Sachgebiete

Allgemeine Mathematik

Elektronische Version(en)

https://doi.org/10.1093/imrn/rnaf002 (Zugang: Offen)