Integrable geodesic flows and Multi-Centre versus Bianchi A metrics

Details Integrable geodesic flows and Multi-Centre versus Bianchi A metrics Integrable geodesic flows and Multi-Centre versus Bianchi A metrics

2006-05-29

arxiv.org/abs/math-ph/0605072v1

Class.Quant.Grav. 24 (2007) 255-276

Physics

Mathematical Physics

22 pages, latex2e, no figures

Scientific paper

10.1088/0264-9381/24/2/001

It 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 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 hyperk\"ahler metric in this family displays a finite dimensional W-algebra for its observables. Our analysis puts also to light non-diagonal Bianchi VI$_0$ and VII$_0$ metrics which seem to be new. We conclude by showing that the elliptic coordinates advocated in the literature do not separate the Hamilton-Jacobi equation for the tri-axial Bianchi IX metric.

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