Certain solitons on generalized $(\kappa, \mu)$ contact metric manifolds
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Abstract
The aim of the present paper is to study some solitons on three dimensional generalized $(\kappa, \mu)$-contact metric manifolds. We study gradient Yamabe solitons on three dimensional generalized $(\kappa, \mu)$-contact metric manifolds. It is proved that if the metric of a three dimensional generalized $(\kappa, \mu)$-contact metric manifold is gradient Einstein soliton then $\mu = \frac{2\kappa}{\kappa - 2}.$ It is shown that if the metric of a three dimensional generalized $(\kappa, \mu)$-contact metric manifold is closed m-quasi Einstein metric then $\kappa = \frac{\lambda}{m + 2}$ and $\mu = 0.$ We also study conformal gradient Ricci solitons on three dimensional generalized $(\kappa, \mu)$-contact metric manifolds.
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References
[1] A. Barros and E. Ribeiro Jr, Integral formulae on quasi-Einstein manifolds and its applications, Glasgow Math. J. 54 (2012), 213–223. Google Scholar
[2] N., Basu and A. Bhattacharyya, conformal Ricci solition in Kenmotsu manifold, Golb. J. Adv. Res. class. Mod. Geom. 4 (2015), 15–21. Google Scholar
[3] D. E. Blair, Contact Manifolds in Riemannian Geometry, Lecture Notes in Math. 509 (1976), 199–207. Google Scholar
[4] D. E. Blair, T. Koufogiorgos and B. J. Papantoniou, Contact metric manifolds satisying nullity condition, Israel J. Math. 91 (1995), 189–214. Google Scholar
[5] D. E. Blair, T. Koufogiorgos and R. Sharma, A classification of 3-dimensional contact tensor of a contact metric manifold with Qφ = φQ, Kodai Math. J. 13 (2007), 391–401. Google Scholar
[6] G. Catino, L. Mazzieri, Gradient Einstein soliton, arXiv:1201.6620v5 [math.DG] 29 Nov 2013. Google Scholar
[7] U. C. De and S. Samui, Quasi-conformal curvature tensor on generalized (κ, μ)- contact metric manifolds, Acta Univ. Apulensis Math. Inform. 40 (2014), 291– 303. Google Scholar
[8] A. Ghosh, Certain contact metric as Ricci almost solitons, Results Math. 65 (2014), 81–94. Google Scholar
[9] F. Gouli-Andreou and P. J. Xenos, A class of contact metric 3-manifolds with ξ ∈ (κ, μ) and κ μ are functions, Algebras, Groups and Geom. 17 (2000), 401– 407. Google Scholar
[10] R. S. Hamilton, Ricci flow on surfaces, Contemp. Math. 71 (1988), 237–261. Google Scholar
[11] S. K. Hui, Almost conformal Ricci solitons on f-Kenmotsu manifolds, Khayyam Journal of Mathematics 5 (2019), 89–104. Google Scholar
[12] T. Koufogiorgos and C. Tsichlias, On the existance of new class of contact metric manifolds, Cand. Math. Bull., Vol. 43 (2000), 440–447. Google Scholar
[13] J. B. Jun, U. C. De and G. Pathak, On Kenmotsu manifolds, J. Korean Math. Soc. 42 (2005), 435–445. Google Scholar
[14] M. Limoncu,Modification of the Ricci tensor and its applications, Arch. Math. 95 (2010), 191–199. Google Scholar
[15] P. Majhi, and G. Ghosh,Certain results on generalized (κ, μ)-contact manifolds, Bol. Soc. Parana. Mat. 37 (2019), 131–142. Google Scholar
[16] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv: 0211159 mathDG, (2002)(Preprint). Google Scholar
[17] S. Pigola, et al., Ricci almost solitons, Ann. Sc. Norm. Sup. Pisa. Cl. Sci. 10 (2011), 757–799. Google Scholar
[18] A. Sarkar and P. Bhakta, Ricci almost soliton on (κ, μ) space forms, Acta Universitatis Apulensis, 57 (2019), 75–85. Google Scholar
[19] A. Sarkar and P. Bhakta, On certain soliton and Ricci tensor of generalized (κ, μ) manifolds, J. Adv. Math. Stud. 12 (2019), 314–323. Google Scholar
[20] A. Sarkar, A. Sil and A. K. Paul, Ricci almost solitons on three diemensional quasi-Sasakian manifolds, Proc. Nat. Acad. Sci, Ind., Sec A. Ph. Sc. 89 (2019), 705–710. Google Scholar
[21] A. Sarkar and R. Mandal, On N (κ)-para contact 3-manifolds with Ricci solitons, Math. Students. 88 (2019), 137–145. Google Scholar
[22] A. Sarkar and G. G. Biswas, Ricci solitons on three-dimensional generalized Sasakian space forms with quasi-Sasakian metric, Africa Mat. 31 (2020), 455– 463. Google Scholar
[23] A. Sarkar, A. K. Paul and R. Mandal, On α-para Kenmotsu 3-manifolds with Ricci solitons, Balkan. J. Geom. Appl. 23 (2018), 100–112. Google Scholar
[24] A. Sarkar and G. G.Biswas, A Ricci soliton on three-dimensional trans-Sasakian manifolds, Mathematics students. 88 (2019), 153–164. Google Scholar
[25] R. Sharma, Almost Ricci solitons and K-contact geometry, Montash Math. 175 (2014), 621–628. Google Scholar
[26] T. Taniguchi, Charactrizations of real hypersurfaces of a complex hyperbolic space interms of holomorphic distribution, Tsukuba J. Math. 18 (1994), 469– 482. Google Scholar
[27] S. Tanno, Ricci curvature of contact Riemannian manifolds, Tohoku Math. J. 40 (1988), 441–448. Google Scholar
[28] A. Yildiz, U. C. De, and A. Cetinkaya, On some classes of 3-dimensional gen- eralized (κ, μ)-contact metric manifolds, Turkish J. Math. 39 (2015), 356–368. Google Scholar