(Anti-)Self-Dual Einsteinian Metrics of Zero Signature, Their Petrov Classes and Connection with Kähler and Para-Kähler Structures

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 I 0 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 I 0 . We left open the question of the existence of the type I 0 . 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.