Korean J. Math. Vol. 27 No. 3 (2019) pp.691-705
DOI: https://doi.org/10.11568/kjm.2019.27.3.691

Beta-almost Ricci solitons on almost CoKahler manifolds

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Debabrata Kar
Pradip Majhi


In the present paper is to classify Beta-almost ($\beta$-almost) Ricci solitons and $\beta$-almost gradient Ricci solitons on almost CoK\"ahler manifolds with $\xi$ belongs to $(k,\mu)$-nullity distribution. In this paper, we prove that such manifolds with $V$ is contact vector field and $Q\phi = \phi Q$ is $\eta$-Einstein and it is steady when the potential vector field is pointwise collinear to the reeb vectoer field. Moreover, we prove that a $(k,\mu)$-almost CoK\"ahler manifolds admitting $\beta$-almost gradient Ricci solitons is isometric to a sphere.

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[1] Barros, A. and Ribeiro, Jr., Some characterizations for compact almost Ricci solitons, Proc. Amer. Math. Soc. 140 (3) (2012), 1033–1040. Google Scholar

[2] Blair, D. E., Contact manifold in Riemannian Geometry, Lecture Notes on Mathematics, Springer, Berlin, 509 (1976). Google Scholar

[3] Blair, D. E., Riemannian Geometry on contact and sympletic manifolds, Progr. Math., Birkh ̈auser, Boston, 203 (2010). Google Scholar

[4] Blair, D. E., Koufogiorgos, T. and Papantoniou, B. J., Contact metric manifolds satisfying a nullity condition, Israel J. of Math. 91 (1995), 189–214. Google Scholar

[5] Calin, C. and Crasmareanu, M., From the Eisenhart problem to Ricci solitons in f-Kenmotsu manifolds, Bull. Malays. Math. Soc. 33 (3) (2010), 361–368. Google Scholar

[6] Chinea, D., L eon, M. de and Marrero, J. C., Topology of cosympletic manifolds, J. Math. Pures Appl. 72 (6) (1993), 567-591. Google Scholar

[7] Chave, T. and Valent, G., Quasi-Einstein metrics and their renoirmalizability properties, Helv. Phys. Acta. 69 (1996) 344–347. Google Scholar

[8] Chave, T. and Valent, G., On a class of compact and non-compact quasi-Einstein metrics and their renoirmalizability properties, Nuclear Phys. B. 478 (1996), 758–778. Google Scholar

[9] Deshmukh, S., Jacobi-type vector fields on Ricci solitons, Bull. Math. Soc. Sci. Math. Roumanie 55 (103)1 (2012), 41–50. Google Scholar

[10] Deshmukh, S., Alodan, H. and Al-Sodais, H., A Note on Ricci Soliton, Balkan J. Geom. Appl. 16 (1) (2011), 48–55. Google Scholar

[11] Dacko, P. and Olszak, Z., On almost cosympletic (k, μ, ν)-space, Banach Center Publ. 69 (2005), 211–220. Google Scholar

[12] Friedan, D., Nonlinear models in 2 + ε dimensions, Ann. Phys. 163 (1985), 318–419. Google Scholar

[13] Ghosh, A. and Sharma, R., Sasakian metric as a Ricci soliton and related results, J. Geom. Phys. 75 (2014), 1–6. Google Scholar

[14] Ghosh, A. and Patra, D. S., The k-almost Ricci solitons and contact geometry, J. Korean Math. Soc. 55 (1) (2018), 161–174. Google Scholar

[15] Hamilton, R. S., The Ricci flow on surfaces, Mathematics and general relativity, (Santa Cruz, CA, 1986), 237–262. Contemp. Math., 71, American Math. Soc., 1988. Google Scholar

[16] Hamilton, R. S., Three manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), 255–306. Google Scholar

[17] Ivey, T., Ricci solitons on compact 3-manifolds, Diff. Geom. Appl. 3 (1993), 301–307. Google Scholar

[18] Kim, B.H., Fibered Riemannian spaces with quasi-Sasakian structures, Hiroshima Math. J. 20 (1990), 477–513. Google Scholar

[19] Marrero, J. C. and Pad ron, E., New examples of compact cosympletic solvmanifolds, Arch. Math. (Brno) 34 (3) (1998), 337-345. Google Scholar

[20] Patra, D. S. and Ghosh, A., Certain contact metrics satisfying Miao-Tam critical condition, Ann. Polon. Math. 116 (3) (2016), 263–271. Google Scholar

[21] Pigola, S., Rigoli, M., Rimoldi, M. and Setti, A., Ricci almost solitons, Ann. Sc. Norm. Super. Pisa Cl. Sci (5) 10 (4) (2011), 757–799. Google Scholar

[22] Sharma, R., Certain results on K-contact and (k, μ)-contact manifolds, J. Geom. 89 (1-2) (2008), 138–147. Google Scholar

[23] Wang, Q., Gomes, J. N. and Xia, C., On the h-almost Ricci soliton, J. Geom. Phys. 114 (2017), 216–222. Google Scholar

[24] Wang, Y. and Liu, X., Ricci solitons on three-dimensional η-Einstein almost Kenmotsu manifolds, Taiwanese Journal of Mathematics 19 (1) (2015), 91–100. Google Scholar

[25] Wang, Y., Ricci solitons on 3-dimensional cosympletic manifolds, Math. Slovaca 67 (4) (2017), 979–984. Google Scholar