Further results on non-diagonal Bianchi type III vacuum metrics

Details Further results on non-diagonal Bianchi type III vacuum metrics Further results on non-diagonal Bianchi type III vacuum metrics

2011-02-22

arxiv.org/abs/1102.4538v1

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

21 pages, no figures

Scientific paper

We present the derivation, for these vacuum metrics, of the Painlev\'e VI equation first obtained by Christodoulakis and Terzis, from the field equations for both minkowskian and euclidean signatures. This allows a complete discussion and the precise connection with some old results due to Kinnersley. The hyperk\"ahler metrics are shown to belong to the Multi-Centre class and for the cases exhibiting an integrable geodesic flow the relevant Killing tensors are given. We conclude by the proof that for the Bianchi B family, excluding type III, there are no hyperk\"ahler metrics.

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