Russian Mathematics

, Volume 63, Issue 2, pp 25–34 | Cite as

The Main Theorem for (Anti-)Self-Dual Conformal Torsion-Free Connection on a Four-Dimensional Manifold

  • L. N. KrivonosovEmail author
  • V. A. Luk’yanovEmail author


In this paper we obtain results that occur on a four-manifold with conformai torsion-free connection for all possible signatures of angular metric. It is proved that three of the four terms of the formula for the decomposition of the basic tensor are equidual, one is anti-dual. Based on this result we find conditions for (anti-)self-duality of exterior 2-forms Φ i j , i, j = 1, 2, 3, 4, which are part of components of the conformal curvature matrix. With the help of the latter result, the main theorem is proved: a conformal torsion-free connection on a four-manifold with the signatures of the angular metric s = ±4; 0 is (anti-)self-dual if and only if the Weyl tensor of the angular metric and the exterior 2-form Φ 0 0 are (anti-)self-dual and Einstein’s and Maxwell’s equations are satisfied. In particular, the normal conformal Cartan connection is (anti-)self-dual if and only if such is also the Weyl tensor of the angular metric.

Key words

conformal connection (anti-)self-duality the Weyl tensor conformal curvature Einstein’s equations Maxwell’s equations 


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  1. 1.
    Cartan E. Spaces With an Affine, Projective and Conformai Connection (Kazan Univ. Press, 1962) [in Russian].Google Scholar
  2. 2.
    Stolyarov, A. V. A Space With Conformai Connection, Russian Mathematics 50, No. 11, 40–51 (2006).MathSciNetGoogle Scholar
  3. 3.
    Krivonosov, L. N., Luk’yanov, V. A. “The Structure of the Main Tensor of Conformally Connected Torsion-Free Space. Conformai Connections on Hypersurfaces of Projective Space”, Sib. J. Pure and Appl. Math. 17, No. 2, 21–38 (2017).Google Scholar
  4. 4.
    Atiyah, M. F., Hitchin, N. J., Singer, I. M. “Self-Duality in Four-Dimensional Riemannian Geometry”, Proc. Roy. Soc. London Ser. A. 362, 421–457 (1978).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Singerland, I. M., Thorpe, J. A. “The curvature of 4-dimensional Einstein spaces”, Global Anal., Papers in Honour of K. Kodaira (Princeton Univer. Press, Princeton, 1969), pp. 355–365.Google Scholar
  6. 6.
    Sucheta Koshti, Naresh Dadhich “The General Self-Dual Solution of the Einstein Equations”, arXiv:gr-qc/9409046 (1994).Google Scholar
  7. 7.
    Arsen’eva, O. E. “Selfdual Geometry of Generalized Kahlerian Manifolds”, Russian Acad. Sci. Sb. Math. 79, No. 2, 447–457 (1994).MathSciNetGoogle Scholar
  8. 8.
    Dunajski, M. “Anti-Self-Dual Four-Manifolds With a Parallel Real Spinor”, arXiv:math/0102225 (2001).zbMATHGoogle Scholar
  9. 9.
    Dunajski, M., Ferapontov, E., Kruglikov, B. “On the Einstein-Weyl and Conformal Self-Duality Equations”, arXiv:1406.0018 (2014).Google Scholar
  10. 10.
    Akivis, M. A. “Completely Isotropic Submanifolds of a Four-Dimensional Pseudoconformal Structure”, Soviet Math. 27, No. 1, 1–11 (1983).MathSciNetzbMATHGoogle Scholar
  11. 11.
    Akivis, M. A., Konnov, V. V. “Some Local Aspects of the Theory of Conformal Structure”, Russian Math. Surveys 48, No. 1, 1–35 (1993).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Petrov, A. Z. New Methods in General Relativity Theory (Nauka, Moscow, 1966) [in Russian].Google Scholar
  13. 13.
    Krivonosov, L. N., Luk’yanov, V. A. “Einstein’s Equations on a 4-Manifold of Conformal Torsion-Free Connection”, J. Sib. Fed. Univ. Math. Phys. 5, No. 3, 393–408 (2012) [in Russian].Google Scholar
  14. 14.
    Krivonosov, L. N., Luk’yanov, V. A. Connection of Yang-Mills Equations with Einstein and Maxwell’s Equations, J. Sib. Fed. Univ. Math. Phys. 2, No. 4, 432–448 (2009) [in Russian].Google Scholar

Copyright information

© Allerton Press, Inc. 2019

Authors and Affiliations

  1. 1.Nizhny Novgorod State Technical UniversityNizhny NovgorodRussia

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