Abstract
For (anti-)self-dual Einsteinian metrics, as well as for any (anti-)self-dual metrics of zero signature, the number of logically possible Petrov types is seven rather than six. In addition to the usual types I, D, O, II, III, and N, type I0 is also possible with a characteristic zero root of multiplicity 4. A system of anti-self-duality equations for the Riemann tensor is compiled for a metric that is universal in the class of anti-self-dual zero-signature metrics. Particular solutions are found for all types except I0. We left open the question of the existence of the type I0. For an arbitrary metric of zero signature, all almost-Hermitian and almost para-Hermitian structures are found. All Kähler and para-Kähler structures are found for the (anti-)self-dual Einsteinian metric. For a metric of zero signature, the notion of hyper-Kählerianity is introduced for the first time. Its definition differs from that of hyper-Kählerianity for Riemann metrics, but is equivalent to it for dimension 4. Each (anti-)self-dual Einsteinian metric of zero signature is simultaneously hyper-Kählerian and para-hyper-Kählerian. Conversely, any hyper-Kählerian (para-hyper-Kählerian) four-metric of zero signature is (anti-)self-dual and Einsteinian.
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Translated by V. Arutyunyan
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Krivonosov, L.N., Luk’yanov, V.A. (Anti-)Self-Dual Einsteinian Metrics of Zero Signature, Their Petrov Classes and Connection with Kähler and Para-Kähler Structures. Russ Math. 66, 33–45 (2022). https://doi.org/10.3103/S1066369X22090043
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DOI: https://doi.org/10.3103/S1066369X22090043