Complex coordinates and Kähler potential for the Atiyah-Hitchin metric (Q1179156)

Let \(M^0_2\) be the configuration space of two interacting \(\mathrm{SU}(2)\) magnetic monopoles with fixed center. \textit{M. F. Atiyah} and \textit{N. Hitchin} [Phys. Lett., A 107, 21--25 (1985; Zbl 1177.53069)] introduced a Riemannian metric (called \(A\)-\(H\) metric) on \(M^0_2\) and developed a complex geometry of the space \(M^0_2\), equipped with the \(A\)-\(H\) metric in their book [The geometry and dynamics of magnetic monopoles. Princeton, NJ: Princeton University Press (1988; Zbl 0671.53001)]. The author gives explicit expressions of the complex coordinates on \(M^0_2\) and the Kähler potential for the \(A\)-\(H\) metric by making use of the three-dimensional formulation of the Einstein equations by \textit{R. Geroch} [J. Math. Phys. 12, 918--924 (1971; Zbl 0214.49002)] and the results by \textit{C. P. Boyer} [J. Math. Phys. 23, 1126--1130 (1982; Zbl 0484.53051)] on Killing vectors.