The indefinite anti-self-dual metrics and the Painlevé equations. (English) Zbl 1062.83022
Summary: We classify the \(\text{SU}(2)\)-invariant anti-self-dual metrics with a signature \((+, +, -, -)\). The metrics are specified by a solution of Painlevé VI, V, III or II. Moreover, we show the geometric meaning of the metrics specified by each type of Painlevé function.
MSC:
| 83C20 | Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory |
| 34M55 | Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies |
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