Abstract
We prove that the complex manifold of the superposition Eguchi-Hanson metric plus the pseudo-Fubini-Study metric is equal to the total space of the holomorphic line bundle of degree −n on the Riemann sphere. The apparent singularities of the metric can be resolved only if the Eguchi-Hanson parameter satisfies a 4=4(n−2)2(n+1)/3Λ2, n≥3. We give a geometrical explanation of the fact that we need n≥3. Finally, we generalize the metric of Gegenberg and Das to obtain a triaxial vacuum metric.
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Pedersen, H., Nielsen, B. On some euclidean einstein metrics. Letters in Mathematical Physics 12, 277–282 (1986). https://doi.org/10.1007/BF00402660
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DOI: https://doi.org/10.1007/BF00402660