Abstract
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 j i , 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.
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References
Cartan E. Spaces With an Affine, Projective and Conformai Connection (Kazan Univ. Press, 1962) [in Russian].
Stolyarov, A. V. A Space With Conformai Connection, Russian Mathematics 50, No. 11, 40–51 (2006).
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).
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).
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.
Sucheta Koshti, Naresh Dadhich “The General Self-Dual Solution of the Einstein Equations”, arXiv:gr-qc/9409046 (1994).
Arsen’eva, O. E. “Selfdual Geometry of Generalized Kahlerian Manifolds”, Russian Acad. Sci. Sb. Math. 79, No. 2, 447–457 (1994).
Dunajski, M. “Anti-Self-Dual Four-Manifolds With a Parallel Real Spinor”, arXiv:math/0102225 (2001).
Dunajski, M., Ferapontov, E., Kruglikov, B. “On the Einstein-Weyl and Conformal Self-Duality Equations”, arXiv:1406.0018 (2014).
Akivis, M. A. “Completely Isotropic Submanifolds of a Four-Dimensional Pseudoconformal Structure”, Soviet Math. 27, No. 1, 1–11 (1983).
Akivis, M. A., Konnov, V. V. “Some Local Aspects of the Theory of Conformal Structure”, Russian Math. Surveys 48, No. 1, 1–35 (1993).
Petrov, A. Z. New Methods in General Relativity Theory (Nauka, Moscow, 1966) [in Russian].
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].
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].
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Russian Text © L.N. Krivonosov, V.A. Luk’yanov, 2019, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2019, No. 2, pp. 29–38.
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Krivonosov, L.N., Luk’yanov, V.A. The Main Theorem for (Anti-)Self-Dual Conformal Torsion-Free Connection on a Four-Dimensional Manifold. Russ Math. 63, 25–34 (2019). https://doi.org/10.3103/S1066369X1902004X
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DOI: https://doi.org/10.3103/S1066369X1902004X