(Anti) self-dual Einstein metrics of zero signature, their Petrov classes and connection with Kahler and para-Kahler structures
L. N. Krivonosov,
V. A. Lukyanov Nizhny Novgorod State Technical University n.a. R.E. Alekseev, 24 Minina str., Nizhny Novgorod, 603950 Russia
Abstract:
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.
UDC:
514.756 Received: 13.11.2021
Revised: 03.06.2022
Accepted: 29.06.2022
DOI:
10.26907/0021-3446-2022-9-39-53
Fulltext:
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