Geodesic
(Anti) self-dual Einstein metrics of zero signature, their Petrov classes and connection with Kahler and para-Kahler structures
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2022), pp. 39-53 Cet article a éte moissonné depuis la source Math-Net.Ru

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For (anti) self-dual Einstein metrics, as well as for any (anti) self-dual metrics of zero signature, not $6$ Petrov types are logically possible, but $7$. In addition to the usual types I, D, O, II, III and N the type I$_{0}$ is also possible, described by characteristic root $0$ 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 Kahler and para-Kahler structures are found for the (anti) self-dual Einstein metric. For a metric of signature $0$, the notion of hyper-Kahler property is introduced for the first time. Its definition differs from the definition of hyper-Kahler Riemannian metrics, but is equivalent to it for dimension $4$. Each (anti) self-dual Einstein metric of zero signature is simultaneously hyper-Kahler and para-hyper-Kahler. Conversely, any hyper-Kahler (para-hyper-Kahler) $4$-metric of zero signature is (anti) self-dual and Einstein metric.
Keywords: (anti) self-duality, Hodge operator, vacuum Einstein equation, Riemann tensor, almost Hermitian, almost para-Hermitian, Kahler, para-Kahler, hyper-Kahler, para-hyper-Kahler metric. 
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     author = {L. N. Krivonosov and V. A. Lukyanov},
     title = {(Anti) self-dual {Einstein} metrics of zero signature, their {Petrov} classes and connection with {Kahler} and {para-Kahler} structures},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {39--53},
     year = {2022},
     number = {9},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2022_9_a3/}
}
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L. N. Krivonosov; V. A. Lukyanov. (Anti) self-dual Einstein metrics of zero signature, their Petrov classes and connection with Kahler and para-Kahler structures. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2022), pp. 39-53. http://geodesic.mathdoc.fr/item/IVM_2022_9_a3/

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