On Z-Symmetric Manifold with Conharmonic Curvature Tensor in Special Conditions


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Authors: A. Y. TAşCı AND F. Ö. ZENGIN

DOI: 10.46793/KgJMat2501.065T

Abstract:

The object of the present paper is to study the Z-symmetric manifold with conharmonic curvature tensor in special conditions. In this paper, we prove some theorems about these manifolds by using the properties of the Z-tensor.



Keywords:

Conharmonic curvature tensor, Z-symmetric tensor, Codazzi tensor, Torse-forming vector field, Recurrent tensor.



References:

[1]   D. B. Abdussatter, On conharmonic transformations in general relativity, Bull. Calcutta Math. Soc. 41 (1966), 409–416.

[2]   S. Bergman, The Kernel Function and Conformal Mapping, American Mathematical Society, United States of America, 1950.

[3]   A. L. Besse, Einstein Manifolds, Springer, Berlin, 1987.

[4]   G. Caristi and M. Ferrara, On torse-forming vector valued 1-forms, Differ. Geom. Dyn. Syst. 5(1) (2003), 13–16.

[5]   M.C. Chaki, Some theorems on recurrent and Ricci-recurrent spaces, Rend. Semin. Mat. Univ. Padova 26 (1956), 168–176.

[6]   U. C. De, N. Guha and D. Kamilya, On generalized Ricci-recurrent manifolds, Tensor (N.S.) 56 (1995), 312–317.

[7]   U. C. De, C. A. Mantica and Y. J. Suh, On weakly cyclic Z-symmetric manifolds, Acta Math. Hungar. 146(1) (2015), 153–167. https://doi.org/10.1007/s10474-014-0462-9

[8]   U. C. De and P. Pal, On almost pseudo-Z-symmetric manifolds, Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 53(1) (2014), 25–43.

[9]   F. de Felice and C. J. S. Clarke, Relativity on Curved Manifolds, Cambridge University Press, Cambridge, 1990.

[10]   A. Derdzinski and C. L. Shen, Codazzi tensor fields, curvature and Pontryagin forms, Proc. Lond. Math. Soc. 47(1) (1983), 15–26. https://doi.org/10.1112/plms/s3-47.1.15

[11]   L. P. Eisenhart, Riemannian Geometry, Princeton University Press, Princeton, 1926.

[12]   S. Ghosh, U. C. De and A. Taleshian, Conharmonic curvature tensor on N(k)-contact metric manifolds, ISRN Geometry (2011), 1–11. https://doi.org/10.5402/2011/423798

[13]   İ. Hinterleitner and V. A. Kiosak, ϕ(Ric)-vector fields in Riemannian spaces, Arch. Math. 44(5) (2008), 385–390.

[14]   Y. Ishii, On conharmonic transformations, Tensor (N.S.) 7 (1957), 73–80.

[15]   C. A. Mantica and L. G. Molinari, Weakly Z-symmetric manifold, Acta Math. Hungar. 135 (2012), 80–96. https://doi.org/10.1007/s10474-011-0166-3

[16]   C. A. Mantica and Y. J. Suh, Pseudo Z symmetric riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9(1) (2012), Article ID 1250004, 21 pages. https://doi.org/10.1142/S0219887812500041

[17]   C. A. Mantica and Y. J. Suh, Pseudo-Z symmetric spacetimes, J. Math. Phys. 55(4) (2014), Article ID 042502. https://doi.org/10.1063/1.4871442

[18]   C. A. Mantica and Y. J. Suh, Recurrent Z forms on Riemannian and Kaehler manifolds, Int. J. Geom. Methods Mod. Phys. 9(7) (2012), Article ID 1250059, 26 pages. https://doi.org/10.1142/S0219887812500594

[19]   J. Mikesh and M. Chodorova, On concircular and torse-forming vector fields on compact manifolds, Acta Math. Acad. Paedagog. Nyházi. (N.S.) 26(2) (2010), 329–335.

[20]   J. Mikesh, V. E. Berezovski, B. Sandor and O. Chepurna, Differential Geometry of Special Mappings, Palacky University, Olomouc, 2015.

[21]   P. M. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, 1953.

[22]   Z. Nehari, Conformal Mapping, McGraw-Hill, New York, 1952.

[23]   N. Prakash, A note on Ricci-recurrent and recurrent spaces, Bull. Calcutta Math. Soc. 54 (1962), 1–7.

[24]   W. Roter, On a generalization of conformally symmetric metrics, Tensor (N.S.) 46 (1987), 278–286.

[25]   H. S. Ruse, Three-dimensional spaces of recurrent curvature, Proc. Lond. Math. Soc. 50 (1949), 438–446. https://doi.org/10.1112/plms/s2-50.6.438

[26]   A. A. Shaikh and S. K. Hui, On weakly conharmonically symmetric manifolds, Tensor (N.S.) 70(2) (2008), 119–134.

[27]   S. A. Siddiqui and Z. Ahsan, Conharmonic curvature tensor and the spacetime of general relativity, Differ. Geom. Dyn. Syst. 12 (2010), 213–220.

[28]   S. Yamaguchi and M. Matsumoto, On Ricci-recurrent spaces, Tensor (N.S.) 19 (1968), 64–68.

[29]   K. Yano, On the torse forming directions in Riemannian spaces, Proceedings of the Imperial Academy 20(6) (1944), 340–345. https://doi.org/10.3792/pia/1195572958

[30]   A. Yavuz Taşcı and F. Özen Zengin, Concircularly flat Z-symmetric manifolds, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) LXV(2) (2019), 241–250.

[31]   A. Yavuz Taşcı and F. Özen Zengin, Z-symmetric manifold admitting concircular Ricci symmetric tensor, Afr. Mat. 31 (2020), 1093–1104. https://doi.org/10.1007/s13370-020-00782-5