GENERALIZED Z-RECURRENT MANIFOLD WITH APPLICATIONS TO RELATIVITY
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Abstract
In this paper, we investigate generalized Z-recurrent manifolds and explore their relevance in the context of relativity. Several geometric characteristics of generalized Z-recurrent space-times are examined under specific curvature constraints. Furthermore, we demonstrate that for a perfect fluid generalized recurrent space-time satisfying Einstein’s field equations without a cosmological constant, the Ricci tensor fulfills the time-like convergence condition. As a consequence, in such a matter-free space-time, the pressure of the fluid must be positive.
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Pandey, G. (2026). GENERALIZED Z-RECURRENT MANIFOLD WITH APPLICATIONS TO RELATIVITY. Journal of the Tensor Society, 20(01). Retrieved from https://tensorsociety.com/journal/index.php/JTS/article/view/261
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Reveiw Article
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References
REFRENCES
[1] Arslan, K., De, U. C., Murathan, C., and Yildiz, A. (2009): “On generalized recurrent Riemannian manifolds”, Acta. Math. Hung., 123, 27-39.
[2] Basse, A. L. (1987): “Einstein manifolds”, Ergeb. Math. Grenzgeb.3, Floge, Bd 10, Springer, Berlin.
[3] Cartan, E. (1926): “Sur uneclasse remarkable d'espaces de Riemannian”, Bull Soc. Math Fr. 54,214-264.
[4] Chaki, M. C. and Gupta, B. (1963): “On conformally symmetricspaces”, Indian J. Math, 5,113-122.
[5] Chaki, M. C. (1987): “On pseudo-symmetric manifold”, An. Stiint. Univ. AL. Cuza. lasi 33,53-58.
[6] Chaki, M. C. and Maity, R. K. (2000): “On quasi-Einstein manifold”, Publ. Math. Debr. 57,297-306.
[7] De, U. C. and Kamilya, D. (1994): “On generalized conhormonically recurrent manifolds”, Indian J. Math. 36,49-54.
[8] De, U. C., Guha, N., Kamilya, D. (1995): “On generalized Ricci-recurrent manifolds”, Tensor (NS), 56,312-317.
[9] De, U. C. and Ghosh, G. C. (2004): “On quasi-Einstein and special quasi-Einstein manifolds”, Int. Proc. Of the International conference on math. and Appl., Kuwait University, 48,178-191.
[10] De, U.C. and Gazi, A. K. (2009): “On generalized concircularlly recurrent”, Stud. Sci. Math. Hung, 46, 287-296.
[11] De,U. C. and Pal, P. (2017): “On generalized z-recurrent manifolds”, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math., 24, 53-68.
[12] Kowalski, O. (1996): “An explicit classification of 3-dimensional Riemannian space satisfying R(X,Y).R=0”, Czechoslov Math. J., 46(121), 427-474.
[13] Mantica, C. A. and Suh, Y. J. (2011): “Conformally symmetric manifolds and quasi-conformally recurrent Riemannian manifolds”, Balk. J. Geom. Appl. 16,66-77.
[14] Mantica, C. A. and Suh, Y. J. (2012): “Pseudo symmetric Riemannian manifold with harmonic curvature tensor”, Int. J. Geom. Meth. Mod. Phys.,9(1), 1250004.
[15] Mantica, C. A. and Molinari L.G. (2012): “Weakly symmetric manifold”, Acta, Math. Hungar., 135(1-2), 80-96.
[16] Mallick, S. and De, U. C.(2013): “On generalized Ricci- recurrent manifold with applications to relativity”, Proc. Natl. Acad. Sci. India, Sec. A Phys., 83, 143-152.
[17] Maurya S. P. and Singh, R. N. (2020): "Space-time admitting generalized conformal curvature tensor", SEAJMMS, 16(2), 305-316.
[18] O'Neill, B. (1983): “Semi-Riemannian geometry with applications to the relativity”, Academic Press, New York.
[19] Patterson, E. M. (1952): “Some theorems on Ricci-recurrent space”, J. Lond. Math. Soc. 27, 287-295.
[20] Roter,W. (1974):“On conformally recurrent Ricci-recurrent manifold”, Colloq. Math., 31,287-295.
[21] Szabo, Z. I. (1982): “Structure theorems on Riemannian spaces satisfying R(X,Y).R = 0”, The local version, J. Diff. Geom. 17,531-582.
[22] Tamassy, L. And Binh, T. Q. (1989): “On weakly symmetric and weakly projectively symmetric Riemannian manifold”, Colloq. Math. Soc. Janos Bolyai, 65, 663-670.
[23] Walker, A. G. (1950): “On Ruse's space of recurrent curvature”, Proc. London Math. Soc. 52(2), 36-64.
[1] Arslan, K., De, U. C., Murathan, C., and Yildiz, A. (2009): “On generalized recurrent Riemannian manifolds”, Acta. Math. Hung., 123, 27-39.
[2] Basse, A. L. (1987): “Einstein manifolds”, Ergeb. Math. Grenzgeb.3, Floge, Bd 10, Springer, Berlin.
[3] Cartan, E. (1926): “Sur uneclasse remarkable d'espaces de Riemannian”, Bull Soc. Math Fr. 54,214-264.
[4] Chaki, M. C. and Gupta, B. (1963): “On conformally symmetricspaces”, Indian J. Math, 5,113-122.
[5] Chaki, M. C. (1987): “On pseudo-symmetric manifold”, An. Stiint. Univ. AL. Cuza. lasi 33,53-58.
[6] Chaki, M. C. and Maity, R. K. (2000): “On quasi-Einstein manifold”, Publ. Math. Debr. 57,297-306.
[7] De, U. C. and Kamilya, D. (1994): “On generalized conhormonically recurrent manifolds”, Indian J. Math. 36,49-54.
[8] De, U. C., Guha, N., Kamilya, D. (1995): “On generalized Ricci-recurrent manifolds”, Tensor (NS), 56,312-317.
[9] De, U. C. and Ghosh, G. C. (2004): “On quasi-Einstein and special quasi-Einstein manifolds”, Int. Proc. Of the International conference on math. and Appl., Kuwait University, 48,178-191.
[10] De, U.C. and Gazi, A. K. (2009): “On generalized concircularlly recurrent”, Stud. Sci. Math. Hung, 46, 287-296.
[11] De,U. C. and Pal, P. (2017): “On generalized z-recurrent manifolds”, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math., 24, 53-68.
[12] Kowalski, O. (1996): “An explicit classification of 3-dimensional Riemannian space satisfying R(X,Y).R=0”, Czechoslov Math. J., 46(121), 427-474.
[13] Mantica, C. A. and Suh, Y. J. (2011): “Conformally symmetric manifolds and quasi-conformally recurrent Riemannian manifolds”, Balk. J. Geom. Appl. 16,66-77.
[14] Mantica, C. A. and Suh, Y. J. (2012): “Pseudo symmetric Riemannian manifold with harmonic curvature tensor”, Int. J. Geom. Meth. Mod. Phys.,9(1), 1250004.
[15] Mantica, C. A. and Molinari L.G. (2012): “Weakly symmetric manifold”, Acta, Math. Hungar., 135(1-2), 80-96.
[16] Mallick, S. and De, U. C.(2013): “On generalized Ricci- recurrent manifold with applications to relativity”, Proc. Natl. Acad. Sci. India, Sec. A Phys., 83, 143-152.
[17] Maurya S. P. and Singh, R. N. (2020): "Space-time admitting generalized conformal curvature tensor", SEAJMMS, 16(2), 305-316.
[18] O'Neill, B. (1983): “Semi-Riemannian geometry with applications to the relativity”, Academic Press, New York.
[19] Patterson, E. M. (1952): “Some theorems on Ricci-recurrent space”, J. Lond. Math. Soc. 27, 287-295.
[20] Roter,W. (1974):“On conformally recurrent Ricci-recurrent manifold”, Colloq. Math., 31,287-295.
[21] Szabo, Z. I. (1982): “Structure theorems on Riemannian spaces satisfying R(X,Y).R = 0”, The local version, J. Diff. Geom. 17,531-582.
[22] Tamassy, L. And Binh, T. Q. (1989): “On weakly symmetric and weakly projectively symmetric Riemannian manifold”, Colloq. Math. Soc. Janos Bolyai, 65, 663-670.
[23] Walker, A. G. (1950): “On Ruse's space of recurrent curvature”, Proc. London Math. Soc. 52(2), 36-64.