Further results on non-diagonal Bianchi type III vacuum metrics

TL;DR

This paper derives the Painlevé VI equation for non-diagonal Bianchi type III vacuum metrics, connecting it with classical results, and classifies hyperkähler metrics within this family, showing their relation to integrable geodesic flows.

Contribution

It provides a complete derivation of Painlevé VI for these metrics and clarifies their connection to known solutions and hyperkähler structures, extending previous work.

Findings

Painlevé VI equation derived for Bianchi type III metrics

Hyperkähler metrics belong to the Multi-Centre class

No hyperkähler metrics exist for Bianchi B family excluding type III

Abstract

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.