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Integrable geodesic flows and Multi-Centre versus Bianchi A metrics

Galliano Valent·Hamed Ben Yahia
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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.

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Galliano Valent, Hamed Ben Yahia. 2006-05-29. Integrable geodesic flows and Multi-Centre versus Bianchi A metrics. https://doi.org/10.1088/0264-9381%2F24%2F2%2F001

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