Beta-almost Ricci solitons on Sasakian 3-manifolds
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Pradip Majhi
mpradipmajhi@gmail.com
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Debabrata Kar
debabratakar6@gmail.com
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DOI:
https://doi.org/10.4067/S0719-06462019000300063Abstract
In this paper we characterize the Sasakian 3-manifolds admitting β-almost Ricci solitons whose potential vector field is a contact vector field. Among others we prove that a β-almost Ricci soliton whose potential vector field is a contact vector field on a Sasakian 3-manifold is shrinking, Einstein and non-trivial. Moreover, we prove that this type of manifolds are isometric to a sphere of radius √7.
Keywords
[1] Barros, A. and Ribeiro, Jr., Some characterizations for compact almost Ricci solitons, Proc. Amer. Math. Soc., 140(3) (2012), 1033-1040.
[2] Blair, D. E., Lecture notes in Mathematics, 509, Springer-Verlag Berlin (1976).
[3] Blair, D. E., Riemannian Geometry of contact and symplectic manifolds, Birkhauser, Boston, 2002.
[4] Blair, D. E., Koufogiorgos, T. and Sharma, R., A classification of 3-dimensional contact metric manifolds with Qφ = φQ, Kodai Math. J., 13 (1990), 391-401.
[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.
[6] Chave, T. and Valent, G., Quasi-Einstein metrics and their renoirmalizability properties, Helv. Phys. Acta., 69 (1996) 344-347.
[7] 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.
[8] Deshmukh, S., Jacobi-type vector fields on Ricci solitons, Bull. Math. Soc. Sci. Math. Roumanie, 55(103)1 (2012), 41-50.
[9] Deshmukh, S., Alodan, H. and Al-Sodais, H., A Note on Ricci Soliton, Balkan J. Geom. Appl. 16 (1) (2011), 48-55.
[10] De, U. C. and Biswas, S., A note on ξ-conformally flat contact manifolds, Bull. Malays. Math. Soc., 29 (2006), 51-57.
[11] De, U. C. and Mondal, A. K., Three dimensional Quasi-Sasakian manifolds and Ricci solitons, SUT J. Math., 48(1) (2012), 71-81.
[12] De, U. C. and Mondal, A. K., The structure of some classes of 3-dimensional normal almost contact metric manifolds, Bull. Malays. Math. Soc., 36(2) (2013), 501-509.
[13] De, Avik and Jun, J. B., On N(k)-contact metric manifolds satisfying certain curvature conditions, Kyungpook Math. J., 51(4) (2011), 457-468.
[14] Friedan, D., Nonlinear models in 2 + Ç« dimensions, Ann. Phys., 163 (1985), 318-419.
[15] Ghosh, A. and Sharma, R., Sasakian metric as a Ricci soliton and related results, J. Geom. Phys., 75 (2014), 1-6.
[16] Ghosh, A. and Patra, D. S., The k-almost Ricci solitons and contact geometry, J. Korean. Math., 55(1) (2018), 161-174.
[17] 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.
[18] Hamilton, R. S., Three manifolds with positive Ricci curvature, J. Differential Geom., 17 (1982), 255-306.
[19] Ivey, T., Ricci solitons on compact 3-manifolds, Diff. Geom. Appl., 3 (1993),301-307.
[20] Kim, B.H., Fibered Riemannian spaces with quasi-Sasakian structures, Hiroshima Math. J., 20 (1990), 477-513.
[21] Majhi, P., and De, U. C., Classifications of N(k)-contact metric manifolds satisfying certain curvature conditions, Acta Math. Univ. Commenianae, LXXXIV(1) (2015), 167-178.
[22] Ozgur, C., Contact metric manifolds with cyclic-parallel Ricci tensor, Diff. Geom. Dynamical systems, 4 (2002), 21-25.
[23] Ozgur, C. and Sular, S., On N(k)-contact metric manifolds satisfying certain conditions, SUT J. Math., 44(1) (2008), 89-99.
[24] 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.
[25] Taleshian, A. and Hosseinzabeh, A. A., Investigation of Some Conditions on N(k)-Quasi Einstein Manifolds, Bull. Malays. Math. Soc. 34(3)(2011), 455-464.
[26] Gomes, J. N., Wang, Q. and Xia, C., On the h-almost Ricci soliton, J. Geom. Phys., 114 (2017), 216-222.
[27] Wang, Y. and Liu, X., Ricci solitons on three-dimensional η-Einstein almost Kenmotsu manifolds, Taiwanese Journal of Mathematics, 19(1) (2015), 91-100.
[28] Wang, Y., Ricci solitons on 3-dimensional cosympletic manifolds, Math. Slovaca, 67(4) (2017), 979-984.
[2] Blair, D. E., Lecture notes in Mathematics, 509, Springer-Verlag Berlin (1976).
[3] Blair, D. E., Riemannian Geometry of contact and symplectic manifolds, Birkhauser, Boston, 2002.
[4] Blair, D. E., Koufogiorgos, T. and Sharma, R., A classification of 3-dimensional contact metric manifolds with Qφ = φQ, Kodai Math. J., 13 (1990), 391-401.
[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.
[6] Chave, T. and Valent, G., Quasi-Einstein metrics and their renoirmalizability properties, Helv. Phys. Acta., 69 (1996) 344-347.
[7] 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.
[8] Deshmukh, S., Jacobi-type vector fields on Ricci solitons, Bull. Math. Soc. Sci. Math. Roumanie, 55(103)1 (2012), 41-50.
[9] Deshmukh, S., Alodan, H. and Al-Sodais, H., A Note on Ricci Soliton, Balkan J. Geom. Appl. 16 (1) (2011), 48-55.
[10] De, U. C. and Biswas, S., A note on ξ-conformally flat contact manifolds, Bull. Malays. Math. Soc., 29 (2006), 51-57.
[11] De, U. C. and Mondal, A. K., Three dimensional Quasi-Sasakian manifolds and Ricci solitons, SUT J. Math., 48(1) (2012), 71-81.
[12] De, U. C. and Mondal, A. K., The structure of some classes of 3-dimensional normal almost contact metric manifolds, Bull. Malays. Math. Soc., 36(2) (2013), 501-509.
[13] De, Avik and Jun, J. B., On N(k)-contact metric manifolds satisfying certain curvature conditions, Kyungpook Math. J., 51(4) (2011), 457-468.
[14] Friedan, D., Nonlinear models in 2 + Ç« dimensions, Ann. Phys., 163 (1985), 318-419.
[15] Ghosh, A. and Sharma, R., Sasakian metric as a Ricci soliton and related results, J. Geom. Phys., 75 (2014), 1-6.
[16] Ghosh, A. and Patra, D. S., The k-almost Ricci solitons and contact geometry, J. Korean. Math., 55(1) (2018), 161-174.
[17] 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.
[18] Hamilton, R. S., Three manifolds with positive Ricci curvature, J. Differential Geom., 17 (1982), 255-306.
[19] Ivey, T., Ricci solitons on compact 3-manifolds, Diff. Geom. Appl., 3 (1993),301-307.
[20] Kim, B.H., Fibered Riemannian spaces with quasi-Sasakian structures, Hiroshima Math. J., 20 (1990), 477-513.
[21] Majhi, P., and De, U. C., Classifications of N(k)-contact metric manifolds satisfying certain curvature conditions, Acta Math. Univ. Commenianae, LXXXIV(1) (2015), 167-178.
[22] Ozgur, C., Contact metric manifolds with cyclic-parallel Ricci tensor, Diff. Geom. Dynamical systems, 4 (2002), 21-25.
[23] Ozgur, C. and Sular, S., On N(k)-contact metric manifolds satisfying certain conditions, SUT J. Math., 44(1) (2008), 89-99.
[24] 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.
[25] Taleshian, A. and Hosseinzabeh, A. A., Investigation of Some Conditions on N(k)-Quasi Einstein Manifolds, Bull. Malays. Math. Soc. 34(3)(2011), 455-464.
[26] Gomes, J. N., Wang, Q. and Xia, C., On the h-almost Ricci soliton, J. Geom. Phys., 114 (2017), 216-222.
[27] Wang, Y. and Liu, X., Ricci solitons on three-dimensional η-Einstein almost Kenmotsu manifolds, Taiwanese Journal of Mathematics, 19(1) (2015), 91-100.
[28] Wang, Y., Ricci solitons on 3-dimensional cosympletic manifolds, Math. Slovaca, 67(4) (2017), 979-984.
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Published
2020-01-20
How to Cite
[1]
P. Majhi and D. Kar, “Beta-almost Ricci solitons on Sasakian 3-manifolds”, CUBO, vol. 21, no. 3, pp. 63–74, Jan. 2020.
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