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.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{IVM_2022_9_a3,
     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},
     publisher = {mathdoc},
     number = {9},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2022_9_a3/}
}
TY  - JOUR
AU  - L. N. Krivonosov
AU  - V. A. Lukyanov
TI  - (Anti) self-dual Einstein metrics of zero signature, their Petrov classes and connection with Kahler and para-Kahler structures
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2022
SP  - 39
EP  - 53
IS  - 9
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2022_9_a3/
LA  - ru
ID  - IVM_2022_9_a3
ER  - 
%0 Journal Article
%A L. N. Krivonosov
%A V. A. Lukyanov
%T (Anti) self-dual Einstein metrics of zero signature, their Petrov classes and connection with Kahler and para-Kahler structures
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2022
%P 39-53
%N 9
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2022_9_a3/
%G ru
%F IVM_2022_9_a3
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/

[1] Krivonosov L.N., Lukyanov V.A., “Spetsifika klassifikatsii Petrova (anti)avtodualnykh metrik nulevoi signatury”, Izv. vuzov. Matem., 2020, no. 9, 56–67 | DOI | MR | Zbl

[2] Dunajski M., Ferapontov E.V., Kruglikov B., “On the Einstein–Weyl and conformal self-duality equation”, J. Math. Phys., 56:8 (2015), 083501 | DOI | MR | Zbl

[3] Plebanski J.F., “Some solutions of complex Einstein equations”, J. Math. Phys., 16:12 (1975), 2395–2402 | DOI | MR

[4] Eguchi T., Gilkey P.B., Hanson A.J., “Gravitation, Gauge Theories and Differential Geometry”, Phys. Reports, 66:6 (1980), 213–393 | DOI | MR

[5] Kirichenko V.F., “Metody obobschennoi ermitovoi geometrii v teorii pochti kontaktnykh struktur”, Itogi nauki i tekhn., Ser. Probl. geom., 18, VINITI, M., 1986, 25–71

[6] Arseneva O.E., “Avtodualnaya geometriya obobschennykh kelerovykh mnogoobrazii”, Matem. sb., 184:8 (1993), 137–148 | DOI | Zbl

[7] Atya M., Khitchin N., Geometriya i dinamika magnitnykh monopolei, Mir, M., 1991 | MR

[8] Krivonosov L.N., Lukyanov V.A., “Ermitovy metriki s (anti)avtodualnym tenzorom Rimana”, Vestn. Samarsk. gos. tekhn. un-ta, Ser. Fiz.-matem. nauki, 25:4 (2021), 616–633 | DOI | Zbl

[9] Akivis M.A., Konnov V.V., “Nekotorye lokalnye aspekty teorii konformnykh struktur”, UMN, 48:1 (1993), 3–40 | MR | Zbl

[10] Atiyah M.F., Hitchin N.J., Singer I.M., “Self-duality in four-dimensional Riemannian geometry”, Proc. Roy. Sos. London Ser. A, 362 (1978), 425–461 | DOI | MR | Zbl

[11] Vinogradov A.M., Krasilschik I.S., Lychagin V.V., Vvedenie v geometriyu nelineinykh differentsialnykh uravnenii, Nauka, M., 1986 | MR