This article is cited in 1 paper

Differential Equations and Mathematical Physics

Hermitian metrics with (anti-)self-dual Riemann tensor

L. N. Krivonosov, V. A. Lukyanov

Nizhny Novgorod State Technical University, Nizhnii Novgorod, 603600, Russian Federation

Abstract: Equations of (anti-)self-duality for the components of the Levi–Civita connection of the Hermitian positive definite metric (not for the Riemann tensor) are compiled. With this well-known method, a simpler system of partial differential equations is obtained, which implies the (anti-)self-duality of the Riemann tensor. This system is of the 1st order, while the (anti-)self-duality conditions of the Riemann tensor are expressed by equations of the 2nd order. However, this method can obtain only particular solutions of the (anti-)self-duality equations of the Riemann tensor. The constructed equations turned out to be significantly different in the self-dual and anti-self-dual cases. In the case of self-duality, the equations are divided into three classes, for each of which a general solution is found. In the anti-self-dual case, we did not find the general solution, but gave two series of particular solutions. The connection between our solutions and Kähler metrics is shown. In the case of the (anti-)self-duality of the Levi–Civita connection for the Hermitian metric, a general form of parallel almost complex metric-preserving structures is obtained. These structures are all torsion free. For an arbitrary positive definite 4-metric, a general form of almost complex structures preserving this metric is found.

Keywords: (anti-)self-duality, Hodge operator, Einstein vacuum equations of gravitation, Riemann tensor, Hermitian, Kähler, hyper–Kähler metric.

UDC: 514.756

MSC: 53B30, 58A14, 53C18

Received: June 16, 2021
Revised: September 18, 2021
Accepted: October 12, 2021
First online: November 16, 2021

DOI: 10.14498/vsgtu1867

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