Geometry of $\ast$-$k$-Ricci-Yamabe soliton and gradient $\ast$-$k$-Ricci-Yamabe soliton on Kenmotsu manifolds

Abstract

The goal of the current paper is to characterize the $\ast$-$k$-Ricci-Yamabe soliton within the framework on Kenmotsu manifolds. Here, we have shown the nature of the soliton and found the scalar curvature when the manifold admits the $\ast$-$k$-Ricci-Yamabe soliton on the Kenmotsu manifold. Next, we have evolved the characterization of the vector field when the manifold satisfies the $\ast$-$k$-Ricci-Yamabe solitons. Also we have embellished some applications of vector field as torse-forming in terms of $\ast$-$k$-Ricci-Yamabe soliton on Kenmotsu manifold. Then, we studied the gradient $\ast$-$k$-Ricci-Yamabe soliton to yield the nature of the Riemannian curvature tensor. We have developed an example of a $\ast$-$k$-Ricci-Yamabe soliton on a 5-dimensional Kenmotsu manifold to prove our findings.

Keywords

• Ricci-Yamabe soliton
• *-k-Ricci-Yamabe soliton
• gradient *-k-Ricci-Yamabe soliton
• torse forming vector field
• conformal Killing vector field
• Kenmotsu manifold

References

1. [1] C.S. Bagewadi and V. S. Prasad, Note on Kenmotsu manifolds, Bull. Cal. Math. Soc. 91, 379-384, 1999.
2. [2] E. Barbosa and E. Ribeiro Jr., On conformal solutions of the Yamabe flow, Arch. Math. 101, 7989, 2013.
3. [3] H.D. Cao, X. Sun and Y. Zhang, On the structure of gradient Yamabe solitons, arXiv:1108.6316v2 [math.DG].
4. [4] G. Catino and L. Mazzieri, Gradient Einstein solitons, Nonlinear Anal. 132, 6694, 2016.
5. [5] D. Chakraborty, V.N. Mishra and S.K. Hui, Ricci solitons on three dimensional $\beta$-Kenmotsu manifolds with respect to Schouten-Van Kampen connection, Journal of Ultra Scientist of Physical Sciences Section A 30 (1), 86-91, 2018.
6. [6] B.Y. Chen, A simple characterization of generalized Robertson-Walker space-times, Gen. Relativity Gravitation 46 (12), Article No: 1833, 2014.
7. [7] B.Y. Chen, Classification of torqued vector fields and its applications to Ricci solitons, Kragujevac J. Math. 41 (2) , 239-250, 2017.
8. [8] X. Chen, Real hypersurfaces with *-Ricci solitons of non-flat complex space forms. Tokyo J. Math. 41, 433451, 2018.

9. [9] X.M. Chen, The k-almost Yamabe solitons and contact metric manifolds, Rocky Mountain J. Math. 51, 125137, 2021.
10. [10] J.T. Cho and M. Kimura, Ricci solitons and real hypersurfaces in a complex space form, Tohoku Math. J. 61 (2), 205-212, 2009.
11. [11] D. Dey, *-Ricci-Yamabe Soliton and contact geometry, arXiv:2109.04220v1 [math.DG].
12. [12] S. Dey, Conformal Ricci soliton and almost conformal Ricci soliton in para- contact geometry, Int. J. Geom. Methods Mod. Phys., to appear, 2022, doi: 10.1142/S021988782350041X.
13. [13] S. Dey and S. Roy, Characterization of general relativistic spacetime equipped with $\eta$-Ricci-Bourguignon soliton, J. Geom. Phys. 178, Article No: 104578, 2022.
14. [14] S. Dey and S. Roy, *-$\eta$-Ricci Soliton within the framework of Sasakian manifold, Journal of Dynamical Systems & Geometric Theories 18 (2), 163-181, 2020.
15. [15] S. Dey, S. Sarkar and A. Bhattacharyya, *-$\eta$-Ricci soliton and contact geometry, Ric. Mat., to appear, 2021, doi: 10.1007/s11587-021-00667-0.
16. [16] S. Dey and S. Uddin, Conformal $\eta$-Ricci almost solitons on Kenmotsu manifolds, Int. J. Geom. Methods Mod. Phys. 19 (08), Article No: 2250121, 2022.
17. [17] D. Ganguly, S. Dey, A. Ali and A. Bhattacharyya, Conformal Ricci soliton and Quasi- Yamabe soliton on generalized Sasakian space form, J. Geom. Phys. 169, Article No: 104339, 2021.
18. [18] D. Ganguly, S. Dey and A. Bhattacharyya, On trans-Sasakian 3-manifolds as $\eta$- Einstein solitons,Carpathian Math. Publ. 13 (2), 460-474, 2021.
19. [19] A. Ghosh, Yamabe soliton and Quasi Yamabe soliton on Kenmotsu manifold, Math. Slovaca 70 (1), 151-160, 2020.
20. [20] S. Güler and M. Crasmareanu, Ricci-Yamabe maps for Riemannian flows and their volume variation and volume entropy, Turk. J. Math. 43, 2361-2641, 2019.
21. [21] T. Hamada, Real hypersurfaces of complex space forms in terms of Ricci *-tensor, Tokyo J. Math. 25, 473-483, 2002.
22. [22] R.S. Hamilton, Three Manifold with positive Ricci curvature, J. Differential Geom. 17 (2), 255-306, 1982.
23. [23] R.S. Hamilton, The Ricci flow on surfaces, Contemp. Math. 71, 237-261, 1988.
24. [24] S.K. Hui and D. Chakraborty, Ricci almost solitons on concircular Ricci pseudosym- metric $\beta$-Kenmotsu manifolds, Hacet. J. Math. Stat. 47 (3), 579-587, 2018.
25. [25] S.K. Hui, S. Uddin and D. Chakraborty, Generalized Sasakian-space forms whose metric is $\eta$-Ricci almost solitons, Differ. Geom. Dyn. Syst. 19, 45-55, 2017.
26. [26] S.K. Hui, S.K. Yadav and S.K. Chaubey, $\eta$-Ricci soliton on 3-dimensional f- Kenmotsu manifolds, Appl. Appl. Math. 13 (2), 933-951, 2018.
27. [27] G. Kaimakamis and K. Panagiotidou, *-Ricci Solitons of real hypersurface in non-flat complex space forms, J. Geom. Phys. 76, 408-413, 2014.
28. [28] K. Kenmotsu, A class of almost contact Riemannian manifolds, The Tˆohoku Mathematical Journal 24, 93-103, 1972.
29. [29] Y.L. Li, S. Dey, S. Pahan and A. Ali, Geometry of conformal $\eta$-Ricci solitons and conformal $\eta$-Ricci almost solitons on paracontact geometry, Open Math. 20 (1), 574589, 2022.
30. [30] Y.L. Li, D. Ganguly, S. Dey and A. Bhattacharyya, Conformal $\eta$-Ricci solitons within the framework of indefinite Kenmotsu manifolds, AIMS Math. 7, 54085430, 2022.
31. [31] Y.L. Li, S. Mondal, S. Dey, A. Bhattacharyya and A. Ali, A Study of Conformal $\eta$- Einstein Solitons on Trans-Sasakian 3-Manifold, J. Nonlinear Math. Phys., to appear, 2022, doi:/10.1007/s44198-022-00088-z.
32. [32] S. Roy and A. Bhattacharyya, Conformal Ricci solitons on 3-dimensional trans- Sasakian manifold, Jordan J. Math. Stat. 13 (1), 89-109, 2020.
33. [33] S. Roy, S. Dey and A. Bhattacharyya, Yamabe Solitons on $(LCS)_{n}$-manifolds, J. Dyn. Syst. Geom. Theor. 18 (2), 261-279, 2020.
34. [34] S. Roy, S. Dey and A. Bhattacharyya, Conformal Yamabe soliton and *-Yamabe soliton with torse forming potential vector field, Mat. Vesnik 73 (4), 282-292, 2021.
35. [35] S. Roy, S. Dey and A. Bhattacharyya, Conformal Einstein soliton within the framework of para-Kähler manifold, Differ. Geom. Dyn. Syst. 23, 235-243, 2021.
36. [36] S. Roy, S. Dey and A. Bhattacharyya, A Kenmotsu metric as a conformal $\eta$-Einstein soliton, Carpathian Math. Publ. 13 (1), 110-118, 2021.
37. [37] S. Roy, S. Dey and A. Bhattacharyya, Some results on $\eta$-Yamabe Solitons in 3- dimensional trans-Sasakian manifold, Carpathian Math. Publ. 14 (1), 158170, 2022.
38. [38] S. Roy, S. Dey, A. Bhattacharyya and S.K. Hui *-Conformal $\eta$-Ricci Soliton on Sasakian manifold, Asian-Eur. J. Math. 15 (2), Article No: 2250035, 2022.
39. [39] S. Sarkar, S. Dey, A.H. Alkhaldi and A. Bhattacharyya, Geometry of para-Sasakian metric as an almost conformal $\eta$-Ricci soliton, J. Geom. Phys. 181, Article No: 104651, 2022.
40. [40] S. Sarkar, S. Dey and A. Bhattacharyya, A study of conformal almost Ricci soli- ton on Kenmotsu manifolds, Int. J. Geom. Methods Mod. Phys., to appear, 2022, doi:10.1142/S0219887823300015.
41. [41] S. Sarkar, S. Dey and X. Chen, Certain results of conformal and *-conformal Ricci soliton on para-cosymplectic and para-Kenmotsu manifolds, Filomat 35 (15), 50015015, 2021.
42. [42] J.A. Schouten, Ricci Calculus, Springer-Verlag, Berlin, 1954.
43. [43] R. Sharma, Certain results on K-contact and $(\kappa,\mu)$-contact manifolds, J. Geom. 89, 138-147, 2008.
44. [44] M.D. Siddiqi and M.A. Akyol, $\eta$-Ricci-Yamabe Soliton on Riemannian submersions from Riemannian manifolds, arXiv:2004.14124v1 [math.DG].
45. [45] A. Singh and S. Kishor, Some types of $\eta$-Ricci solitons on Lorentzian para-Sasakian manifolds,Facta Univ. Ser. Math. Inform. 33 (2), 217-230, 2018.
46. [46] S. Tachibana, On almost-analytic vectors in almost K$\ddot{a}$hlerian manifolds, Tohoku Math. J. 11 (2), 247-265, 1959.
47. [47] P. Topping, Lecture on the Ricci Flow, Cambridge University Press, 2006.
48. [48] V. Venkatesha, D.M. Naik and H.A. Kumara, *-Ricci solitons and gradient almost *-Ricci solitons on Kenmotsu manifolds, arXiv:1901.05222 [math.DG].
49. [49] Q. Wang, J.N. Gomes and C. Xia, On the h-almost Ricci soliton, J. Geom. Phys. 114, 216-222, 2017.
50. [50] Y. Wang, Contact 3-manifolds and *-Ricci soliton, Kodai Math. J. 43 (2), 256267, 2020.
51. [51] K. Yano, Concircular geometry I. Concircular transformations, Proc. Imp. Acad. Tokyo 16, 195-200, 1940.
52. [52] K. Yano, On the torse-forming directions in Riemannian spaces, Proc. Imp. Acad. Tokyo 20, 340345, 1944.
53. [53] K. Yano and B.Y. Chen, On the concurrent vector fields of immersed manifolds, Kodai Math. Sem. Rep. 23, 343-350, 1971.
54. [54] I.H. Yoldaş, On Kenmotsu manifolds admitting $\eta$-Ricci-Yamabe solitons, Int. J. Geom. Methods Mod. Phys. 18 (12), Article No: 2150189, 2021.
55. [55] H.İ. Yoldaş, Ş.E. Meriç and E. Yaşar, Some special vector fields on a cosymplectic manifold admitting a Ricci soliton, Miskolc Math. Notes 22 (2), 10391050, 2021.
56. [56] P. Zhang, Y.L. Li, S. Roy and A. Bhattacharyya Geometry of $\alpha$-cosymplectic metric as *-conformal $\eta$-RicciYamabe solitons admitting quarter-symmetric metric connection, Symmetry 13 (11), Article No: 2189, 2021.