The Conharmonic Curvature Tensor on $N(\kappa)$-Paracontact Metric Manifold

Abstract

The object of the paper is to study $N(k)$-paracontact metric manifolds satisfying certain curvature conditions on conharmonic curvature tensor. Specially, we study the symmetric properties of conharmonic curvature tensor on $N(k)$-paracontact metric manifolds such as conharmonically $\varphi$-symmetric, 3-dimensional locally conharmonically $\varphi$-symmetric $N(k)$-paracontact metric manifolds and $\varphi$-conharmonically semisymmetric $N(k)$-paracontact metric manifolds and get some new results.

Keywords

• N(κ)-paracontact metric manifolds
• Conharmonic curvature tensor
• Einstein manifolds

References

1. [1] D. B. Abdussattar, \textit{On conharmonic transformations in general relativity}, Bull. Cal. Math. Soc. 41 (1996), 409–-416.
2. [2] C. $\ddot{O}$zg$\ddot{u}$r, On $\varphi$-conformally flat Lorentzian para-Sasakian manifolds, Radovi Mathemaicki 12 (2003), 96–-106.
3. [3] D. V. Alekseevski, C. Medori and A. Tomassini, Maximally homogeneous para-CR manifolds, Ann. Global Anal. Geom. 30 (2006), 1--27.
4. [4] D. V. Alekseevski, V. Cortes, A. S. Galaev and T. Leistner, Cones over pseudo-Riemannian manifolds and their holonomy, J. Reine Angew. Math. 635 (2009), 23--69.
5. [5] E. Boeckx, P. Buecken and L. Vanhecke, $\varphi$- symmetric contact metric spaces, Glasg. Math. J. 41 (1999), 409--416.
6. [6] G. Calvaruso and D. Perrone, Geometry of $H$-paracontact metric manifold}, arXiv:1307.7662v1.
7. [7] G. Calvaruso, Homogeneous paracontact metric three-manifolds, Illinois J. Math. 55 (2011), 697--718.
8. [8] U. C. De, On $\varphi$-symmetric Kenmotsu manifold, Int. Electron. J. Geom. 1 (2008), (1), 33--38.

9. [9] U. C. De, A. Yildiz and A. F. Yaliniz, Locally $\varphi$-symmetric almost contact metric manifolds of dimension 3, Appl. Math. Lett. 20 (2009), 723--727.
10. [10] U. C. De, A. Yildiz and A. F. Yaliniz, On three-dimensional $N(\kappa)$-paracontact metric manifolds and ricci solitons, Bull. Iranian Math. Soc. 43 (2017), No. 6, 1571–-1583.
11. [11] S. Erdem, On almost (para) contact (hyperbolic) metric manifolds and harmonicity of $(\varphi,\varphi ')$-holomorphic maps between them, Houston J. Math. 28 (2002), 21--45.
12. [12] S. Ghosh, U. C. De and A. Taleshian, Conharmonic curvature tensor on $N(\kappa)$-contact metric manifolds, ISRN Geom. DOI : 10.5402/2011/423798.
13. [13] Y. Ishii, On conharmonic transformations, Tensor N.S. 7 (1957), 73–-80.
14. [14] S. Ivanov, D.Vassilev and S. Zamkovoy, Conformal paracontact curvature and the local flatness theorem, Geom. Dedicata 144 (2012), 115--129.
15. [15] S. Kaneyuki and F. L. Williams, Almost paracontact and parahodge structures on manifolds, Nagoya Math. J. 99 (1985), 173--187.
16. [16] K. Mandal and D. Mandal, Certain results on N(k)-paracontact metric manifolds, Note Mat. 38 (2018) no. 2, 21–33.
17. [17] M. Manev and M. Staikova, On almost paracontact Riemannian manifolds of type $(n,n)$, J. Geom. 72 (2001), 108--114.
18. [18] B. C. Montano, I. K. Erken and C. Murathan, Nullity conditions in paracontact geometry, Differential Geom. Appl. 30(2010), 79--100.
19. [19] C. Murathan and I. Kupeli Erken, The harmonicity of the Reeb vector field on paracontact metric 3-manifolds, arXiv:1305.1511v2.
20. [20] D. M. Naik and V. Venkatesha, $\eta$-Ricci solitons and almost $\eta$-Ricci solitons on para-Sasakian manifolds}, International Journal of Geometric Methods in Modern Physics Vol. 16, No. 9, (2019) 1950134, 18 pages.
21. [21] D. G. Prakasha, A. T. Vanli, C. S. Bagewadi and D. A. Patil, Some classes of Kenmotsu manifolds with respect to semi-symmetric metric connection, Acta Math. Sinica (Eng. Series) DOI : 10.1007/s10114-013-0326-1.
22. [22] D. G. Prakasha and K. K. Mirji, On $(\kappa, \mu)$-paracontact metric manifolds, Gen. Math. Notes 25 (2014), No. 2, 68--77.
23. [23] D. G. Prakasha and K. K. Mirji, On $\varphi$-symmetric $N(\kappa)$-paracontact metric manifolds, J. Math. 2015, Article ID 728298, 6 pages.
24. [24] D. G. Prakasha, L. M. Fernandez and K. K. Mirji, The $\mathcal{M}$-projective curvature tensor field on generalized $(\kappa, \mu)$-paracontact metric manifolds, Georgian Math. J. 27, (1), 141--147.
25. [25] S. A. Siddiqui and Z. Ahsan, Conharmonic curvature tensor and the space - time of general relativity, Differ. Geom. Dyn. Syst. {\bf 12} (2010), 213-220.
26. [26] S. S. Shukla and M. K. Shukla, On $\varphi$-symmetric para-Sasakian manifolds, Internat. J. Math. Anal. 4(2010), No. 16, 761--769.
27. [27] Y. J. Suh and K. Mandal, Yamabe solitons on three-dimensional N(k)-paracontact metric manifolds, Bull. Iran. Math. Soc. (2018) 44: 183. https://doi.org/10.1007/s41980-018-0013-1
28. [28] T. Takahashi, Sasakian $\varphi$- symmetric spaces, Tohoku Math. J. 29 (1997), 91--113.
29. [29] A. Taleshian, D. G. Prakasha, K. Vikas and N. Asghari, On the Conharmonic Curvature Tensor of LP-Sasakian Manifolds, Palestine J. Math. {\bf 3} (2014), (1), 11--18.
30. [30] Venkatesha and D. M. Naik, Certain Results on $K$-Paracontact and Para Sasakian Manifolds, J. Geom. 108 (2017), 939–952.
31. [31] S. Zamkovoy, Canonical connections on paracontact manifolds}, Ann. Global Anal. Geom. 36(2009), 37--60.
32. [32] S. Zamkovoy and V. Tzanov, Non-existence of flat paracontact metric structures in dimension greater than or equal to five, Ann. Sofia Univ. Fac. Math and Inf. 100 (2010), 27--34.