$\star$-Ricci-Yamabe solitons on almost coKähler manifolds

Abstract

The aim of the present article is to analyze $\star$-Ricci--Yamabe solitons on almost coKähler manifolds and to characterize them when the potential vector field is pointwise collinear with the Reeb vector field. It is proved that a compact almost coKähler manifold admitting a $\star$-Ricci--Yamabe soliton under certain restriction on $\star$-scalar curvature is coKähler and $\star$-Ricci flat; in addition, that the soliton is steady. $(\kappa, \mu)$-almost coKähler manifolds admitting such solitons are also considered and finally, the obtained results are completed by non-trivial examples.

Keywords

• almost coKähler manifold
• $(\kappa$
• $\mu)$-nullity distribution
• $\star$-Ricci curvature
• Ricci soliton
• Yamabe soliton

References

1. [1] J.E. Andersen, Geometric quantization of symplectic manifolds with respect to reducible non-negative polarization, Commun. Math. Phys., 183, 401–421, 1997.
2. [2] R.J. Berman, Relative Kähler Ricci flow and their quantization, Anal. PDE, 6, 131– 180, 2013.
3. [3] A.M. Blaga, Geometric solitons in a D-homothetically deformed Kenmotsu manifold, Filomat, 36, 175–186, 2022.
4. [4] A.M. Blaga and H.M. Taştan, Some results on almost $\eta$-Ricci–Bourguignon solitons, J. Geom. Phys. 168, 104316, 2021.
5. [5] D.E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, 203, Birkhäuser, New York, 2010.
6. [6] D.E. Blair, The theory of quasi-Sasakian structures, J. Differential Geom. 1, 331–345, 1967.
7. [7] D.E. Blair, T. Koufogiorgos, and B.J. Papantoniou, Contact metric manifolds satisfying a nullity condition, Israel J. Math. 91, 189–214, 1995.
8. [8] X. Chen, Almost quasi-Yamabe solitons on almost cosymplectic manifolds, Int. J. Geom. Methods Mod. Phys. 17, 2050070, 2020.

9. [9] X. Chen, Cotton solitons on almost coKähler 3-manifolds, Quaest. Math. 44, 1055– 1075, 2021.
10. [10] X. Chen, The k-almost Yamabe solitons and coKähler manifolds, Int. J. Geom. Methods Mod. Phys. 18, 2150179, 2021.
11. [11] X. Chen, Three-dimensional contact metric manifolds with cotton solitons, Hiroshima J. Math. 51, 275–299, 2021.
12. [12] X. Dai, Y. Zhao, and U.C. De, $\star$-Ricci curvature on $(\kappa,\mu)$-almost Kenmotsu manifolds, Open Math. 17, 874–882, 2019.
13. [13] U.C. De, S.K. Chaubey, and Y.J. Suh, A note on almost coKähler manifolds, Int. J. Geom. Methods Mod. Phys. 17 (10), 2050153, 2020.
14. [14] U.C. De, S.K. Chaubey, and Y.J. Suh, Gradient Yamabe and gradient m-quasi- Einstein metrics on three-dimensional cosymplectic manifolds, Mediterr. J. Math. 18, Art. No. 80, 2021.
15. [15] U.C. De and A. Sardar, Classification of $(\kappa,\mu)$-almost coKähler manifolds with vanishing Bach tensor and divergence free Cotton tensor, Commun. Korean Math. Soc. 35, 1245–1254, 2020.
16. [16] U.C. De and Y.J. Suh, Yamabe and quasi-Yamabe solitons in para contact manifolds, Int. J. Geom. Methods Mod. Phys. 18, 2150196, 2021.
17. [17] U.C. De, Y.J. Suh, and S.K. Chaubey, Conformal vector fields on almost coKähler manifolds, Math. Slovaca 71, 1545–4552, 2021.
18. [18] A.E. Fischer, An introduction to conformal Ricci flow, Class. Quantum Grav. 21, 171–218, 2004.
19. [19] S. Güler and M. Crasmareanu, Ricci–Yamabe maps for Riemannian flow and their volume variation and volume entropy, Turk. J. Math. 43, 2631–2641, 2019.
20. [20] T. Hamada, Real hypersurfaces of complex space forms in terms of Ricci $\star$-tensor, Tokyo J. Math. 25, 473–483, 2002.
21. [21] R.S. Hamilton, Lectures on geometric flows, 1989, (Unpublished).
22. [22] R.S. Hamilton, The Ricci flow on surfaces, Contemp. Math. 71, 237–261, 1988.
23. [23] G. Kaimakamis and K. Panagiotidou, $\star$-Ricci solitons of real hypersurfaces in non-flat complex space forms, J. Geom. Phys. 86, 408–413, 2014.
24. [24] T. Mandal, Ricci–Yamabe solitons on $(\kappa,\mu)$-almost coKähler manifolds, Afr. Mat. 33, Art. No. 38, 10pp, 2022.
25. [25] C. Özgür, On Ricci solitons with a semisymmetric metric connection, Filomat, 35, 3635–3641, 2021.
26. [26] A. Sarkar and G.G. Biswas, $\star$-Ricci solitons on three dimensional trans-Sasakian manifolds, The Mathematics Student, 88, 153–164, 2019.
27. [27] A. Sarkar and G.G. Biswas, Ricci solitons on generalized Sasakian space forms with quasi-Sasakian metric, Afr. Mat. 31, 455–463, 2020.
28. [28] A. Sardar, M.N.I. Khan, and U.C. De, $\eta$-$\star$-Ricci solitons and almost coKähler manifolds, Mathematics 9(24), 3200, 2021.
29. [29] R. Sharma, A 3-dimensional Sasakian metric as a Yamabe soliton, Int. J. Geom. Methods Mod. Phys, 9(4), 1220003, 2012.
30. [30] Y.J. Suh and U.C. De, Yamabe solitons and Ricci Yamabe solitons on almost coKähler manifolds, Canad. Math. Bull. 62, 653–661, 2019.
31. [31] S. Tachibana, On almost-analytic vectors in almost Kählerian manifolds, Tohoku Math. J. 11, 247–265, 1959.
32. [32] S. Tanno, Some transformations on manifolds with almost contact and contact metric structures II, Tohoku Math. J. 15, 322–331, 1963.
33. [33] Y. Wang, A generalization of the Goldberg conjecture for coKähler manifolds, Mediterr. J. Math. 13, 2679–2690, 2016.
34. [34] Y. Wang, Yamabe solitons on three-dimensional Kenmotsu manifolds, Bull. Belg. Math. Soc. Simon Stevin 23, 345–355, 2016.
35. [35] Y. Wang, Ricci solitons on 3-dimensional cosymplectic manifolds, Math. Slovaca, 67, 979–984, 2017.
36. [36] Y. Wang, Ricci solitons on almost coKähler manifolds, Canad. Math. Bull. 62, 912– 922, 2019.
37. [37] Y. Wang, Contact 3-manifolds and $\star$-Ricci solitons, Kodai Math. J. 43, 256–267, 2020.
38. [38] Y. Wang, Almost Kenmotsu $(\kappa, \mu)$-manifolds with Yamabe solitons, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 115, Art. No. 14, 2021.
39. [39] W. Wang, Almost cosymplectic $(\kappa, \mu)$-metrics as $\eta$-Ricci solitons, J. Nonlinear Math. Phys. 29, 58–72, 2022.
40. [40] E. Woolgar, Some applications of Ricci flow in Physics, Canadian J. Phys. 86, 2008.
41. [41] K. Yano, Integral formulas in Riemannian geometry, Marcel Dekker, New York, 1970.