Open AccessIntegrable geodesic flows and Multi-Centre versus Bianchi A metrics
Author(s) -
Galliano Valent,
Hamed Ben Yahia
Publication year - 2007
Publication title -
hal (le centre pour la communication scientifique directe)
Language(s) - English
DOI - 10.1088/0264-9381/24/2/00
Subject(s) - geodesic , integrable system , geodesic flow , geodesic map , pure mathematics , mathematical physics , mathematics , computer science , mathematical analysis
latex 2e, 23 pagesInternational audienceIt is shown that most, but not all, of the four dimensional metrics in the Multi-Centre family with integrable geodesic flow may be recognized as belonging to the spatially homogeneous Bianchi type A metrics. We show that any diagonal bi-axial Bianchi II metric has an integrable geodesic flow, and that the simplest hyperkahler metric in this family displays a finite dimensional W-algebra for its observables. Our analysis puts also to light non-diagonal Bianchi VI0 and VII0 metrics which seem to be new. We conclude by showing that the elliptic coordinates advocated in the literature do not separate the Hammilton-Jacobi equation for the tri-axial Bianchi IX metric