Journal of Samara State Technical University, Ser. Physical and Mathematical SciencesJournal of Samara State Technical University, Ser. Physical and Mathematical Sciences1991-86152310-7081Samara State Technical University2056510.14498/vsgtu1562Research ArticleYang-Mills equations on conformally connected torsion-free 4-manifolds with different signaturesKrivonosovLeonid Nhttp://orcid.org/0000-0002-3533-9595 Cand. Phys. & Math. Sci.; Associate Professor; Dept. of Applied Mathematicsl.n.krivonosov@gmail.comLuk’yanovVyacheslav Ahttp://orcid.org/0000-0002-7294-0232 Cand. Phys. & Math. Sci.; Associate Professor; Dept. of Applied Mathematicsoxyzt@ya.ruNizhny Novgorod State Technical University1512201721463365014022020Copyright © 2017, Samara State Technical University2017In this paper we study spaces of conformal torsion-free connection of dimension 4 whose connection matrix satisfies the Yang-Mills equations. Here we generalize and strengthen the results obtained by us in previous articles, where the angular metric of these spaces had Minkowski signature. The generalization is that here we investigate the spaces of all possible metric signatures, and the enhancement is due to the fact that additional attention is paid to calculating the curvature matrix and establishing the properties of its components. It is shown that the Yang-Mills equations on 4-manifolds of conformal torsion-free connection for an arbitrary signature of the angular metric are reduced to Einstein's equations, Maxwell's equations and the equality of the Bach tensor of the angular metric and the energy-momentum tensor of the skew-symmetric charge tensor. It is proved that if the Weyl tensor is zero, the Yang-Mills equations have only self-dual or anti-self-dual solutions, i.e the curvature matrix of a conformal connection consists of self-dual or anti-self-dual external 2-forms. With the Minkowski signature (anti)self-dual external 2-forms can only be zero. The components of the curvature matrix are calculated in the case when the angular metric of an arbitrary signature is Einstein, and the connection satisfies the Yang-Mills equations. In the Euclidean and pseudo-Euclidean 4-spaces we give some particular self-dual and anti-self-dual solutions of the Maxwell equations, to which all the Yang-Mills equations are reduced in this case.manifolds with conformal connectioncurvaturetorsionYang-Mills equationsEinstein's equationsMaxwell's equationsHodge operator(anti)self-dual 2-formsWeyl tensorBach tensorмногообразия конформной связностикривизнакручениеуравнения Янга-Миллсауравнения Эйнштейнауравнения Максвеллаоператор Ходжа(анти)автодуальные 2-формытензор Вейлятензор Баха[Картан Э. Пространства аффинной, проективной и конформной связности. Казань: Казан. ун-т, 1962. 210 с.][Кривоносов Л. Н., Лукьянов В. А. Связь уравнений Янга-Миллса с уравнениями Эйнштейна и Максвелла // Журн. СФУ. Сер. Матем. и физ., 2009. Т. 2, № 4. С. 432-448.][Atiyah M. F., Hitchin N. J., Singer I. M. Self-duality in four-dimensional Riemannian geometry // Proc. Roy. Soc. London. Series A, 1978. vol. 362, no. 1711. pp. 425-461.doi: 10.1098/rspa.1978.0143.][Singerland I. M., Thorpe J. A. The curvature of 4-dimensional Einstein spaces / Global Analysis: Papers in Honor of K. Kodaira (PMS-29). Princeton: Princeton University Press, 2015. pp. 355-365. doi: 10.1515/9781400871230-021.][Sucheta Koshti, Naresh Dadhich The General Self-dual solution of the Einstein Equations, 1994, arXiv: gr-qc/9409046.][Ландау Л. Д., Лифшиц Е. М. Теория поля. М.: Наука, 1973. 504 с.][Фиников С. П. Метод внешних форм Картана в дифференциальной геометрии. М.: ГИТТЛ, 1948. 432 с.][Кривоносов Л. Н., Лукьянов В. А. Уравнения Эйнштейна на четырехмерном многообразии конформной связности без кручения // Журн. СФУ. Сер. Матем. и физ., 2012. Т. 5, № 3. С. 393-408.][Korzyjński M., Levandowski J. The Normal Conformal Cartan Connection and the Bach Tensor // Class. Quant. Grav., 2003. vol. 20, no. 16. pp. 3745-3764, arXiv: gr-qc/0301096v3. doi: 10.1088/0264-9381/20/16/314.]