Integrable geodesic flows and Multi-Centre versus Bianchi A metrics

Abstract

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 hyperkähler 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.

22 pages, latex2e, no figures

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Cited by in corpus (6)

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• The geodesic flow of the BGPP metric is Liouville integrable
• Further results on non-diagonal Bianchi type III vacuum metrics