$\eta $-Ricci Bourguignon Soliton on Para-Sasakian Manifolds with SSNM-Connection

Abstract

The purpose of the present paper is to study $\eta $-Ricci-Bourguignon soliton on para-Sasakian manifold under some curvature conditions. We introduce here a new semi-symmetric non metric connection (briefly, SSNM-connection) on para-Sasakian manifold. We have obtained characterizations of para-Sasakian manifold based on both $\eta $% -Ricci-Bourguignon soliton and the $T_{\theta }$-curvature tensor with the SSNM-connection, where the $T_{\theta }$-curvature tensor is the generalization of conformal, concircular, conharmonic, projective, pseudo projective and $M$- projective curvature tensors. Moreover, we investigate $T_{\theta } $-Ricci symmetric para-Sasakian manifold admitting $\eta $% -Ricci-Bourguignon soliton with respect to SSNM-connection.

Keywords

• η-Ricci-Bourguignon soliton;
• η-Einstein manifold
• para-Sasakian manifold;
• semi-symmetric nonmetric connection
• τ-curvature tensor

References

1. [1] A. Adati and K. Matsumoto, On conformally recurrent and conformally symmetric P-Sasakian manifolds, TRU Math., Vol:13 (1997), 25-32.
2. [2] A. Adati and T. Miyazawa, On P-Sasakian manifolds satisfying certain conditions, Tensor, N.S., Vol:33 (1979), 173-178.
3. [3] G. Ayar, M. Yıldırım, Ricci solitons and gradient Ricci solitons on nearly Kenmotsu manifolds, Facta Universitatis, Series: Mathematics and Informatics, Vol:34 No.3 (2019), 503-510.
4. [4] G. Ayar, M. Yıldırım, h-Ricci solitons on nearly Kenmotsu manifolds, Asian-European Journal of Mathematics, ID-2040002, Vol:12 No.6 (2019)
5. [5] N. S. Agashe and M. R. Chafle, A semi-symmetric non-metric connection on a Riemannian manifold, Indian J. Pure Appl. Math., Vol:23 (1992), 399-409.
6. [6] K. Amur and S. S. Pujar, On submanifolds of a Riemannian manifold admitting a metric semi-symmetric connection, Tensor, N.S., Vol:32 (1978), 35-38.
7. [7] N. Bhunia, S. Pahan and A. Bhattacharyya, Application of t-curvature tensor in spacetimes, Jordan J. Math. Stat., Vol:15 No.3B (2022), 629-641.
8. [8] T. Q. Binh, On semi-symmetric connections, Periodic Math. Hungarica, Vol:21 (1990), 101-107.

9. [9] J. P. Bourguignon, Une stratification de l’espace des structures riemanniennes, Compos. Math., Vol:30 (1975), 1-41.
10. [10] J. P. Bourguignon, Ricci curvature and Einstein metrics, Global Differential Geometry and Global Analysis (Berlin, 1979) (Lecture Notes in Mathematics, 838), Springer, Berlin, (1981), 42-63.
11. [11] G. Catino, L. Cremaschi, L, Cremaschi, Z. Djadli, C. Mantegazza and L. Mazzieri. The Ricci-Bourguignon flow, Pacific J. Math., Vol: 287(2) (2017), 337-370.
12. [12] G. Catino and L. Mazzieri, Gradient Einstein solitons, Nonlinear Anal., Vol:132 (2016), 66-94.
13. [13] S. K. Chaubey and R. H. Ojha, On semi-symmetric non-metric connection on a Riemannian manifold, Filomat, Vol:26 (2012), 63-69.
14. [14] S. K. Chaybey and A. Yieldiz, Riemannian manifold admitting a new type of semi-symmetric non-metric connection, Turk. J. Math., Vol:43 (2019), 1887-1904.
15. [15] U. C. De, On a type of semi-symmetric metric connection on a Riemannian manifold, Indian J. Pure Appl. Math., Vol:21 (1990), 334-338.
16. [16] U. C. De and D. Kamilya, Hypersurfaces of a Riemannian manifold with semi-symmetric non-metric connection, J. Indian Inst. Sci., Vol:75 (1995), 707-710.
17. [17] U. C. De, Y. L. Han and P. B. Zhao, A special type of semi-symmetric non-metric connection on a Riemannian manifold, Facta Univ. (NIS), Vol:31 No.2 (2006), 529-541.
18. [18] H. A. Hayden, Subspaces of space with torsion, Proc. London Math. Soc., Vol:34 (1932), 27-50.
19. [19] R. S. Hamilton, The Ricci flow on surfaces, Math. and General Relativity, American Math. Soc. Contemp. Math., Vol:7 No.1 (1988), 232-262.
20. [20] K. Mandal and U. C. De, Quarter symmetric metric connection in a P-Sasakian manifold, Annals of West University of Timisoara-Mathematics and Comp. Sci., Vol:53 No.1 (2015), 137-150.
21. [21] K. Matsumoto, S. Ianus and I. Mihai, On P-Sasakian manifolds which admit certain tensor-fields, Publicationes Mathematicae-Debrecen, Vol:33 (1986), 199-204.
22. [22] H. G. Nagaraja and C. R. Premalatha, Ricci solitons in Kenmotsu manifolds, J. of Mathematical Analysis, Vol. 3(2) (2012), 18-24.
23. [23] C. Ozgur, On a class of para-Sasakian manifolds, Turkish Journal of Mathematics, Vol:29 No.3 (2005), 249-258.
24. [24] C. Ozgur and M. M. Tripathi, On P-Sasakian manifolds satisfying certain conditions on concircular curvature tensor, Turkish Journal of Mathematics, Vol:31 No.3 (2007), 171-179.
25. [25] C. Ozgur and S. Sular, Wraped product manifolds with semi-symmetric metric connections, Taiwan J. Math., Vol:15 (2011), 1701-1719.
26. [26] B. Prasad and S. C. Singh, Some properties of semi-symmetric non-metric connection in a Riemannian manifold, Jour. Pure Math., Vol:23 (2006), 121-134.
27. [27] B. Prasad and R. K. Verma, On a type of semi-symmetric non-metric connection on a Riemannian manifold, Bull. Call. Math. Soc., Vol:96 No.6 (2004), 483-488.
28. [28] B. Prasad, On pseudo projective curvature tensor on a Riemannian manifold, Bull. Cal. Math. Soc., Vol:94 No.3 (2002), 163-166.
29. [29] G. P. Pokhariyal and R. S. Mishra, Curvature tensors and their relativistic significance II, Yokohama Math. J., Vol:19 No.2 (1971), 97-103.
30. [30] V. V. Reddy, R. Sharma and S. Sivaramkrishan, Space times through Hawking-Ellis construction with a back ground Riemannian metric, Class Quant. Grav., Vol:24 No.13 (2007), 3339-3346.
31. [31] S. Sasaki and Y. Hatakeyama, On differentiable manifolds with certain structures which are closely related to almost contact structures II, Tohoku Mathematical Journal, Vol:13 (1961), 281-294.
32. [32] I. Sato and K. Matsumoto, On P-Sasakian manifolds satisfying certain conditions, Tensor N. S., Vol:33 (1979), 173-178.
33. [33] R. N. Singh and M. K. Pandey, On a type of semi-symmetric non-metric connection on a Riemannian manifold, Bull. Call. Math. Soc., Vol:96 No.6 (2008), 179-184.
34. [34] R. Sharma, Certain results on K-contact and (k;m)-contact manifolds, J. of Geometry, Vol:89 (2008), 138-147.
35. [35] S. S. Shukla, M. K. Shukla, On f-symmetric Para-Sasakian manifolds, Int. J. Math. Analysis, Vol:16 No.4 (2010), 761-769.
36. [36] M. Traore, H. M. Tastan and S. Gerdan Aydn, On almost h-Ricci-Bourguignon solitons, Miskolc. Math. Notes, ID-493508, Vol:25 No.1 (2024).
37. [37] M. Traore, H. M. Tastan, On sequential warped product h-Ricci-Bourguignon solitons, Filomat, ID-67856797, Vol:38 No.19, (2024).
38. [38] M. Traore,, H. M. Tastan and S. Gerdan Aydn, Some characterizations on Gradient Almost h-Ricci-Bourguignon Solitons, Boletim da Sociedade Paranaense de Matemtica, Vol:43 (2025), 1-12.
39. [39] M. M. Tripathi and P. Gupta, t-curvature tensor on a semi-Riemannian manifold, J. Adv. Math. Stud., Vol:4 No.1 (2011), 117-129.
40. [40] M. M. Tripathi, Ricci solitons in contact metric manifold, ArXiv: 0801. 4222 vl [math. D. G.], (2008).
41. [41] K. Yano, On semi-symmetric metric connections, Rev. Roumaine Math. Pure Appl., Vol. 15 (1970), 1579-1586.
42. [42] K. Yano, Concircular Geometry I. Concircular transformations, Math. Institute, Tokyo Imperial Univ. Proc. Vol:16 (1940), 195-200.
43. [43] K. Yano and S. Bochner, Curvature and Betti numbers, Annals of Mathematics Studies 32, Princeton University Press, 1953.
44. [44] M. Yıldırım, Semi-symmetric non-metric connections on statistical manifolds, Journal of Geometry and Physics, Vol:176 (2022),
45. [45] M. Yıldırım and G. Ayar, Ricci solitons and gradient Ricci solitons on nearly cosymplectic manifolds, Journal of Universal Mathematics, Vol:4 No.2 (2021), 201-208
46. [46] Y. Ishii, On conharmonic transformations, Tensor N. S., Vol. 7 (1957), 73-80.
47. [47] S. Zamkovoy, Canonical connection on paracontact manifolds, Ann. global Anal. Geom., Vol:36 (2009), 37-60.