ar X iv : g r - qc / 9 81 00 25 v 1 7 O ct 1 99 8 Self - dual SU ( 2 ) invariant Einstein metrics and modular dependence of theta - functions
نویسندگان
چکیده
We simplify the Hitchin's description of SU (2)-invariant self-dual Einstein metrics, making use of the tau-function of related four-pole Schlesinger system. The SU (2) invariant self-dual Einstein metrics were studied in a number of papers [1, 2, 3]. The local classification of the metrics of this type was given in extensive paper by Hitchin [3]. However, the final form of the metric coefficients related to Painlevé 6 equation in Hitchin's description was rather complicated. It is the purpose of this note to give simpler expressions for the same metric, exploiting the formula for the tau-function of the algebro-geometric solutions of the Schlesinger system found in the paper [4]. Following Tod [5, 1], we start from the following form of SU (2)-invariant self-dual Einstein metric: g = F dµ 2 + σ 2
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