This article is cited in 1 paper

Differential Equations and Mathematical Physics

Duality equations on a 4-manifold of conformal torsion-free connection and some of their solutions for the zero signature

L. N. Krivonosov, V. A. Lukyanov

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

Abstract: On a 4-manifold of conformal torsion-free connection with zero signature $( --++) $ we found conditions under which the conformal curvature matrix is dual (self-dual or anti-self-dual). These conditions are 5 partial differential equations of the 2nd order on 10 coefficients of the angular metric and 4 partial differential equations of the 1st order, containing also 3 coefficients of external 2-form of charge. (External 2-form of charge is one of the components of the conformal curvature matrix.) Duality equations for a metric of a diagonal type are composed. They form a system of five second-order differential equations on three unknown functions of all four variables. We found several series of solutions for this system. In particular, we obtained all solutions for a logarithmically polynomial diagonal metric, that is, for a metric whose coefficients are exponents of polynomials of four variables.

Keywords: manifold of conformal connection, curvature, torsion, Hodge operator, self-duality, anti-self-duality, Yang–Mills equations.

UDC: 514.756.2

MSC: 53A30

Received: January 23, 2019
Revised: May 12, 2019
Accepted: June 10, 2019
First online: June 12, 2019

DOI: 10.14498/vsgtu1674

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