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

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 23 (2019) no. 2, pp. 207-228.

Résumé

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.

Détail

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

Bibliographie

[1] Luk'yanov V. A., Krivonosov L. N., “Yang–Mills equations on conformally connected torsion-free 4-manifolds with different signatures”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 21:4 (2017), 633–650 (In Russian) | DOI | Zbl

[2] Akivis M. A., Konnov V. V., “Some local aspects of the theory of conformal structure”, Russian Math. Surveys, 48:1 (1993), 1–35 | DOI | MR | Zbl

[3] Akivis M. A., “On completely isotropic submanifolds of a four-dimensional pseudoconformal structure”, Soviet Math. (Iz. VUZ), 27:1 (1983), 1–11 | MR | Zbl | Zbl

[4] Konnov V. V., “The asymptotic pseudoconformal structure on a four-dimensional hypersurface and its totally isotropic two-dimensional submanifolds”, Russian Math. (Iz. VUZ), 36:6 (1992), 67–74 | MR | Zbl

[5] Cartan É., “Les espaces à connexion conforme”, Ann. Soc. Pol. Math., 2 (1923), 171–221

[6] Kobayashi Sh., “Automorphisms of G-Structures”, Transformation Groups in Differential Geometry, Classics in Mathematics, 70, Springer, Berlin, Heidelberg, 1995, 1–38 | DOI

[7] Atiyah M. F., Hitchin N. J., Singer I. M., “Self-duality in four-dimensional Riemannian geometry”, Proc. Roy. Soc. London. Series A, 362:1711 (1978), 425–461 | DOI

[8] Krivonosov L. N., Luk'yanov V. A., “The main theorem for (anti)self-dual conformal torsion-free connection”, Izv. Vyssh. Uchebn. Zaved. Mat., 2019, no. 2, 29–38 (In Russian) | DOI

[9] Krivonosov L. N., Luk'yanov V. A., “Structure of the Main Tensor of Conformally Connected Torsion Free Space. Conformal Connections on Hypersurfaces of Projective Space”, J. Math. Sci., 231:2 (2018), 189–205 | DOI | DOI

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

[11] Finikov S. P., Metod vneshnih form Kartana v differencial'noj geometrii [Cartan's method of exterior forms in differential geometry], Moscow, Leningrad, 1948, 432 pp. (In Russian)