Abstract
This paper deals with Z-symmetric Riemannian manifolds with concircular Ricci symmetric tensor. In the first section, we give an introduction of Z-symmetric manifold. In the second section, the definition of concircular Ricci symmetric tensor is given. In the third section, we introduce Z-symmetric Riemannian manifold admitting concircular Ricci symmetric tensor and we examine some properties of these manifolds. In the last section, we study Z-symmetric spacetimes admitting concircular Ricci symmetric tensor and we give two examples for the existence of these manifolds.
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References
Cartan, E.: Surune classe remarquable d’ espaces de Riemannian. Bull. Soc. Math. Fr. 54, 214–264 (1926)
O’Neill, B.: Semi-Riemannian Geometry with Applications to the Relativity. Academic Press, New York (1983)
Chaki, M.C., Gupta, B.: On conformally symmetric spaces. Indian J. Math. 5, 113–295 (1963)
Walker, A.G.: On Ruse’ s space of recurrent curvature. Proc. Lond. Math. Soc. 52, 36–64 (1951)
Adati, T., Miyazawa, T.: On a Riemannian space with recurrent conformal curvature. Tensor (NS) 18, 348–354 (1967)
Chaki, M.C.: On pseudo symmetric manifolds. An. Stiint. Univ. Al. I. Cuza Iasi 33, 53–58 (1987)
Tamassy, L., Binh, T.Q.: On weakly symmetric and weakly projectively symmetric Riemannian manifolds. Colloq. Math. Soc. Janos Balyai 56, 663–670 (1989)
Mantica, C.A., Molinari, L.G.: Weakly Z symmetric manifolds. Acta Math. Hungar. 135, 80–96 (2012)
De, U.C., Mantica, C.A., Suh, Y.J.: On weakly cyclic Z symmetric manifolds. Acta Math. Hungar. 146(1), 153–167 (2015)
Mantica, C.A., Suh, Y.J.: Pseudo Z symmetric riemannian manifolds with harmonic curvature tensors. Int J Geometr Methods Modern Phys 9, 1 (2012)
De, U.C., Pal, P.: On almost pseudo-Z-symmetric manifolds. Acta Univ. Palacki. Fac. Rer. Nat. Math. 53(1), 25–43 (2014)
Besse, A.L.: Einstein Manifolds. Springer, New York (1987)
Gray, A.: Einstein-like manifolds which are not Einstein. Geom. Dedicat. 7, 259–280 (1978)
Roter, W.: On a generalization of conformally symmetric metrics. Tensor (NS) 46, 278–286 (1987)
Suh, Y.J., Kwon, J.H., Yang, H.Y.: Conformal symmetric semi-Riemannian manifolds. J. Geom. Phys. 56, 875–901 (2006)
Ruse, H.S.: On simply harmonic spaces. J. Lond. Math. Soc. 21, 243–247 (1946)
Ruse, H.S.: On simply harmonic ’kappa spaces’ of four dimensions. Proc. Lond. Math. Soc. 50, 317–329 (1948)
Ruse, H.S.: Three dimensional spaces of recurrent curvature. Proc. Lond. Math. Soc. 50, 438–446 (1948)
Chaki, M.C.: Some theorems on recurrent and Ricci recurrent spaces. Rend. Sem. Mat. Univ. Padova 26, 168–176 (1956)
Prakash, N.: A note on Ricci-recurrent and recurrent spaces. Bull. Calcutta Math. Soc. 54, 1–7 (1962)
Roter, W.: On conformally symmetric Ricci-recurrent spaces. Bull. Colloq. Math. Soc. 31, 87–96 (1974)
Yomaguchi, S., Matsumato, M.: On Ricci-recurrent spaces. Tensor (N.S.) 19, 64–68 (1968)
De, U.C., Guha, N., Kamilya, D.: On generalized Ricci-recurrent manifolds. Tensor (NS) 56, 312–317 (1995)
Yano, K.: Concircular geometry I. Imp. Acad. Sci. Jpn. 16, 165 (1940)
Yano, K.: Concircular geometry I, Concircular transformations. Proc. Imp. Acad. 16, 195–200 (1940)
Yano, K., Kon, M.: Structures of Manifolds. World Scientific Publishing, Singapore (1984)
De, U.C., Ghosh, G.C.: On weakly concircular Ricci symmetric manifolds, South East Asian. J. Math. Math. Sci. 3(2), 9–15 (2005)
Narlikar, J.V.: General Relativity and Gravitation. The Macmillan Co. of India, New York (1978)
Stephani, H.: General Relativity. An Introduction to the Theory of Gravitational Field. Cambridge University Press, Cambridge (1982)
Chaki, M.C., Roy, S.: Space-times with covariant-constant energy-momentum tensor. Int. J. Theor. Phys. 35, 1027–1032 (1996)
De, U.C., Velimirovic, L.: Spacetimes with semi-symmetric energy-momentum tensor. Int. J. Theor. Phys. 54(6), 1779–1783 (2015)
Ozen, F.: Zengin, M-projectively flat spacetimes. Math. Rep. 4(4), 363–370 (2012)
Mantica, C.A., Suh, Y.J.: Pseudo-Z symmetric spacetimes. J. Math. Phys. 55(4), 042502 (2014)
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Taşci, A.Y., Zengin, F.Ö. Z-symmetric manifold admitting concircular Ricci symmetric tensor. Afr. Mat. 31, 1093–1104 (2020). https://doi.org/10.1007/s13370-020-00782-5
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DOI: https://doi.org/10.1007/s13370-020-00782-5