Einstein considered a linear connection $ \nabla $ with torsion $ T $ on a differentiable manifold equipped with a nonsymmetric (0, 2)-tensor $ G = g+F $, where $ g $ is a pseudo-Riemannian metric associated with gravity, and $ F\ne0 $ is a skew-symmetric tensor associated with electromagnetism, such that $ (\nabla_X\, G)(Y, Z) = -G(T(X, Y), Z) $. In this paper, we explicitly present the Einstein connection of a nonsymmetric pseudo-Riemannian manifold with non-degenerate $ F $, satisfying the $ f^2 $-torsion condition $ T(f^2X, Y) = T(X, f^2Y) = f^2 T(X, Y) $, where $ g(X, fY) = F(X, Y) $, and show that in the almost Hermitian case, it reduces to the Prvanović's (1995) solution. We also explicitly present the Einstein connection of almost contact metric manifolds satisfying the $ f^2 $-torsion condition, discuss special Einstein connections, and give example in terms of the weighted product of almost Hermitian manifolds.
Citation: Vladimir Rovenski, Milan Zlatanović. Einstein connection of nonsymmetric pseudo-Riemannian manifold with the $ f^2 $-torsion condition[J]. AIMS Mathematics, 2026, 11(6): 18481-18501. doi: 10.3934/math.2026751
Einstein considered a linear connection $ \nabla $ with torsion $ T $ on a differentiable manifold equipped with a nonsymmetric (0, 2)-tensor $ G = g+F $, where $ g $ is a pseudo-Riemannian metric associated with gravity, and $ F\ne0 $ is a skew-symmetric tensor associated with electromagnetism, such that $ (\nabla_X\, G)(Y, Z) = -G(T(X, Y), Z) $. In this paper, we explicitly present the Einstein connection of a nonsymmetric pseudo-Riemannian manifold with non-degenerate $ F $, satisfying the $ f^2 $-torsion condition $ T(f^2X, Y) = T(X, f^2Y) = f^2 T(X, Y) $, where $ g(X, fY) = F(X, Y) $, and show that in the almost Hermitian case, it reduces to the Prvanović's (1995) solution. We also explicitly present the Einstein connection of almost contact metric manifolds satisfying the $ f^2 $-torsion condition, discuss special Einstein connections, and give example in terms of the weighted product of almost Hermitian manifolds.
| [1] |
D. Blair, A survey of Riemannian contact geometry, Complex Manifolds, 6 (2019), 31–64. https://doi.org/10.1515/coma-2019-0002 doi: 10.1515/coma-2019-0002
|
| [2] | A. Einstein, The Meaning of Relativity, Princeton: Princeton University Press, 1953. https://doi.org/10.1007/978-94-011-6022-3 |
| [3] |
A. Gray, L. M. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl., 123 (1980), 35–58. https://doi.org/10.1007/BF01796539 doi: 10.1007/BF01796539
|
| [4] |
R. T. Hammond, The necessity of torsion in gravity, Gener. Relat. Gravit., 42 (2010), 2345–2348. https://doi.org/10.1007/s10714-010-1045-x doi: 10.1007/s10714-010-1045-x
|
| [5] |
S. Ivanov, M. L. Zlatanović, Connections on Einstein's (generalized) Riemannian manifold and gravity, Class. Quant. Grav., 33 (2016), 075016. https://doi.org/10.1088/0264-9381/33/7/075016 doi: 10.1088/0264-9381/33/7/075016
|
| [6] |
T. Janssen, T. Prokopec, Problems and hopes in nonsymmetric gravity, J. Phys. A, Math. Theor., 40 (2007), 7067–7074. https://doi.org/10.1088/1751-8113/40/25/S63 doi: 10.1088/1751-8113/40/25/S63
|
| [7] |
J. W. Moffat, A new nonsymmetric gravitational theory, Phys. Lett. B, 355 (1995), 447–452. https://doi.org/10.1016/0370-2693(95)00670-G doi: 10.1016/0370-2693(95)00670-G
|
| [8] | M. Prvanović, Einstein connection of almost Hermitian manifold, Bull. Cl. Sci. Math. Nat. Sci. Math., 109 (1995), 51–59. |
| [9] |
V. Rovenski, Weak almost contact structures: A survey, Facta Univ. Ser. Math. Inform., 39 (2024), 821–841. https://doi.org/10.22190/FUMI240826055R doi: 10.22190/FUMI240826055R
|
| [10] |
V. Rovenski, M. L. Zlatanović, Weak metric structures on generalized Riemannian manifolds, J. Geom. Phys., 221 (2026), 105741. https://doi.org/10.1016/j.geomphys.2025.105741 doi: 10.1016/j.geomphys.2025.105741
|
| [11] |
M. I. Wanas, S. Nabil, K. ElAbd, N. E. Abdelhamid, New path equations in Einstein non-symmetric geometry, Gravit. Cosmol., 30 (2024), 489–495. https://doi.org/10.1134/S0202289324700385 doi: 10.1134/S0202289324700385
|
| [12] | K. Yano, On a structure $f$ satisfying $f^3+f = 0$, Technical Report No. 12, University of Washington, 1961. |
| [13] |
M. L. Zlatanović, V. Rovenski, Applications of weak metric structures to non-symmetrical gravitational theory, Filomat, 40 (2026), 1253–1269. https://doi.org/10.2298/FIL2604253Z doi: 10.2298/FIL2604253Z
|
Vladimir Rovenski, Milan Zlatanović. Einstein connection of nonsymmetric pseudo-Riemannian manifold with the $ f^2 $-torsion condition[J]. AIMS Mathematics, 2026, 11(6): 18481-18501. doi: 10.3934/math.2026751
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