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Abstract
Extending the concept of Petrov tensor, in this article we attempt to introduce generalised space matter tensor [1],[2], [3], [4]. In the Riemannian manifold, it is found that the second Bianchi identity for the generalized space-matter tensor is satisfied if the energy-momentum tensor is of Codazzi type [5]. We study the nature of Riemannian manifolds by imposing curvature restrictions like symmetry, recurrent, weakly symmetry [6], [7], [8] etc. on this generalized Petrov space-matter tensor. We obtain the eigen values of the Ricci tensor S corresponding to the vector fields associated with the various 1− forms.
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References
- S. Jana, K. Baishya, and B. Das, “A study on generalized space matter tensor,” Scientific Studies and Research Series Mathematics and Informatics, no. 2, pp. 33–62, 2023.
- F. Defever, R. Deszcz, M. Hotlo´s, M. Kucharski, and Z. Sent¨urk, “Generalisations of robertson-walker spaces,” Annales Univ. Sci. Budapest. E¨otv¨os Sect. Math., vol. 43, pp. 13–24, 2000.
- S. K. Jana and A. A. Shaikh, “On quasi-einstein spacetime with space-matter tensor,” Lobachevski Journal of Mathematics, vol. 33, no. 3, pp. 255–261, 2012. https://doi.org/10.1134/S1995080212030122.
- A. Debnath, S. Jana, F. Nurcan, and J. Sengupta, “On quasi-einstein manifolds admitting space-matter tensor,” in Conference Proceedings of Science and Technology, vol. 2, pp. 104–109, 2019. https://dergipark.org.tr/en/pub/cpost/issue/50294/604945.
- D. Ferus, “A remark on codazzi tensors on constant curvature space,” in Lecture Notes in Mathematics, vol. 838, New York: Springer-Verlag, 1981. https://doi.org/10.1007/BFb0088868.
- T. Q. Binh, “On weakly symmetric riemannian spaces,” Publ. Math. Debrecen, vol. 42, pp. 103–107, 1993.
- A. Ghosh, “On the non-existence of certain types of weakly symmetric manifold,” Sarajevo Journal of Mathematics, vol. 2, no. 15, pp. 223–230, 2006.
- L. Tam´assy and T. Q. Binh, “On weakly symmetric and weakly projective symmetric riemannian manifolds,” in Coll. Math. Soc. J. Bolyai, vol. 50, pp. 663–670, 1989.
- A. Z. Petrov, Einstein Spaces. Oxford: Pergamon Press, 1949.
References
S. Jana, K. Baishya, and B. Das, “A study on generalized space matter tensor,” Scientific Studies and Research Series Mathematics and Informatics, no. 2, pp. 33–62, 2023.
F. Defever, R. Deszcz, M. Hotlo´s, M. Kucharski, and Z. Sent¨urk, “Generalisations of robertson-walker spaces,” Annales Univ. Sci. Budapest. E¨otv¨os Sect. Math., vol. 43, pp. 13–24, 2000.
S. K. Jana and A. A. Shaikh, “On quasi-einstein spacetime with space-matter tensor,” Lobachevski Journal of Mathematics, vol. 33, no. 3, pp. 255–261, 2012. https://doi.org/10.1134/S1995080212030122.
A. Debnath, S. Jana, F. Nurcan, and J. Sengupta, “On quasi-einstein manifolds admitting space-matter tensor,” in Conference Proceedings of Science and Technology, vol. 2, pp. 104–109, 2019. https://dergipark.org.tr/en/pub/cpost/issue/50294/604945.
D. Ferus, “A remark on codazzi tensors on constant curvature space,” in Lecture Notes in Mathematics, vol. 838, New York: Springer-Verlag, 1981. https://doi.org/10.1007/BFb0088868.
T. Q. Binh, “On weakly symmetric riemannian spaces,” Publ. Math. Debrecen, vol. 42, pp. 103–107, 1993.
A. Ghosh, “On the non-existence of certain types of weakly symmetric manifold,” Sarajevo Journal of Mathematics, vol. 2, no. 15, pp. 223–230, 2006.
L. Tam´assy and T. Q. Binh, “On weakly symmetric and weakly projective symmetric riemannian manifolds,” in Coll. Math. Soc. J. Bolyai, vol. 50, pp. 663–670, 1989.
A. Z. Petrov, Einstein Spaces. Oxford: Pergamon Press, 1949.