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Cohomogeneity-one metrics with self-dual Weyl tensor. (English) Zbl 0827.53017

Huggett, Stephen (ed.), Twistor theory. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 169, 171-184 (1995).
The author reviews spatially homogeneous (or more generally cohomogeneity-one) four-dimensional metrics with self-dual Weyl tensor, with special attention to Bianchi-type IX Einstein metrics. In particular, this provides a test of the “integrability metatheorem” for systems of ODE derived from systems of PDE: a problem solvable by the twistor construction should have the (global) PainlevĂ© property.
For the entire collection see [Zbl 0803.00012].

MSC:

53B30 Local differential geometry of Lorentz metrics, indefinite metrics
83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory