Certain results for η-Ricci Solitons and Yamabe Solitons on quasi-Sasakian 3-Manifolds
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Sunil Kumar Yadav
prof_sky16@yahoo.com
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Abhishek Kushwaha
ab11sk1991@gmail.com
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Dhruwa Narain
dhruwanarain.dubey@gmail.com
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DOI:
https://doi.org/10.4067/S0719-06462019000200077Abstract
We classify quasi-Sasakian 3-manifold with proper η-Ricci soliton and investigate its geometrical properties. Certain results of Yamabe soliton on such manifold are also presented. Finally, we construct an example of non-existence of proper η-Ricci soliton on 3-dimensional quasi-Sasakian manifold to illustrate the results obtained in previous section of the paper.
Keywords
[1] Agricola, I. and Friedrich, T., Killing spinors in supergravity with 4-fluxes ,Class. Quant. Grav. 20(2003),4707-4717.
[2] Agricola, I., Friedrich, T., Nagy, P.A. and Puhle, C., On the Ricci tensor in the common sector of type II,string theory, Class. Quan. Grav.,22(2005), 2569-2577.
[3] Acharya, B. S., Figueroa A-O‘Farrell, Hull, M. C., and Spence, B. J., Branes at canonical singularities and holography, Adv. Theor. Math. Phys.2 (1999), 1249-1286.
[4] Blair, D. E.,The theory of quasi-Sasakian structure, Journal of Differential Geom. 1(1967), 331-345.
[5] Blair, D.E., Riemannian geometry of contact and symplectic maniifolds, Birkhauser, Boston.(2010).
[6] Barabosa, E. and Ribeiro, E. Jr., On conformal solution of the Yamabe flow, Arch., Math.,101, (2013), 73-89.
[7] Blaga, A. M., η-Ricci solitons on Lorentzian para-Sasakian manifolds, Filomat. 30, 2 (2016), 489-496.
[8] Blaga, A. M., η-Ricci solitons on para-Kenmotsu manifolds, Balkan J. Geom. Appl. 20 (2015), 1-13.
[9] 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.
[10] Cho, J. T. and Kimura, M., Ricci soliltons and real hypersurfaces in a complex space form, Tohoku Math. J.(61) (2009), no. 2, 205-212.
[11] Chaki, M. C. and Kawaguchi, T., On almost pseudo Ricci symmetric manifolds, Tensor N.S.(68)(2007), 10-14.
[12] Deshmukh, S., Alodan, H. and Al-Sodais, H., A note on Ricci soliton, Balkan J. Geom. Appl.1,16 (2011), 48-55.
[13] De, U. C. and Sarkar, A., On φ-Ricci symmetric Sasakian manifolds, Proc. Jangjeon Math. Soc.11 (2008), 47-52.
[14] Friedrich, T., and Ivanov, S.,Almost contact manifolds connection with torsion and parallel spinors, Journal Reine Angew. Math. 559, 217-236.
[15] Gray, A., Einstein-like manifolds which are not Einstein, Geom. Dedicata.7 (1978), 259-280.
[16] Gonzalez, J. C. and Chinea, D., Quasi-Sasakian homogeneous structures on the generalized Heisenberg group H(p, 1), Proc. Amer. Math. Soc.105 (189), 173-184.
[17] Hamilton, R. S., The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986), 237-262, Contemp. Math.71, Amer. Math. Soc,Providence, RI, 1988.
[18] Hui, S. K., Yadav, S. K. and Patra, A., Almost conformal Ricci solitons on f-Kenmotsu manifolds, Khayyam J. Math. 1,5 (2019),84-104.
[19] Hui, S. K., Yadav, S. K. and Chaubey, S. K., η-Ricci soliton on 3-dimensional f-Kenmotsu manifolds, Appl. Appl. Math., 13, 2 (2018), 933-951.
[20] Ivey, T., Ricci solitons on compact 3-manifolds, Diff. Geom. Appl.3 (1993), 301-307.
[21] Kanemaki, S., Quasi-Sasakian manifolds, Tohoku Math. J.29 (1977),227-233.
[22] Kim, B. H., Fibred Riemannian spaces with quasi-Sasakian structure, Hiroshima Math.J.20 (1990),477-513.
[23] Kon, M., Invariant submanifolds in Sasakian manifolds, Mathematische Annalen.219 (1976), 277-290.
[24] Mandal, A. K. and De, A., Some theorems on 3-dimensional quasi-Sasakian manifolds, Tamsui Oxford Journal of Information and Mathematical Sciences.27 (2011), 411-427.
[25] Olszak, Z.,On three-dimensional conformally flat quasi-Sasakian manifolds, Period, Math. Hungar.2,33 (1996),105-113.
[26] Oubina, J. A.,New classes of almost contact metric structures, Publ. Math. Debreceen.32 (1985), 187-193.
[27] Olszak, Z., Normal almost contact metric manifolds of dimension three,Ann. Polon.Math. 47(1986),41-50.
[28] Pokhariyal, G. P., Yadav, S. K. and Chaubey, S. K., Ricci solitons on trans-Sasakian mani- folds, Differential Geometry-Dynamical Systems.20 (2018), 138-158.
[29] Singh, H., and Khan, Q., On special weakly symmetric Riemannian manifolds, Publ. Math. Debrecen.3,58 (2001), 523-536.
[30] Tanno, S.,Quasi-Sasakian structure of rank (2p + 1), Journal Differential Geom.5 (1971), 317- 324.
[31] Yadav, S. K., Ozturk, H., On (ǫ)-almost paracontact metric manifolds with conformal η-Ricci solitons, Differential Geometry-Dynamical Systems. 21 (2019), 202-215.
[32] Yadav, S. K., Chaubey, S. K. and Suthar, D. L., Some results of η-Ricci soliton on (LCS)- manifolds, Surveys in Mathematics and its Applications.13 (2018), 237-250.
[33] Yadav, S. K., Chaubey, S. K. and Suthar, D. L., Some geometric properties of η-Ricci solitons and gradient Ricci solitons on (LCS)n-manifolds, Cubo a Mathematical Journal, 2, 19 (2017), 33-48.
[34] Yano, K., Integral Formulas in Riemannian geometry, Marcel Dekker, New York, 1970.
[35] Ye, R., Global existence and convergence of Yamabe flow, Journal Differ., Geom. 1, 39 (1994),35-50.
[36] Yadav, S. K., Ricci solitons on para-Kahler manifolds, Extracta Mathematicae, Available online April 29, 2, 34 (2019).
[2] Agricola, I., Friedrich, T., Nagy, P.A. and Puhle, C., On the Ricci tensor in the common sector of type II,string theory, Class. Quan. Grav.,22(2005), 2569-2577.
[3] Acharya, B. S., Figueroa A-O‘Farrell, Hull, M. C., and Spence, B. J., Branes at canonical singularities and holography, Adv. Theor. Math. Phys.2 (1999), 1249-1286.
[4] Blair, D. E.,The theory of quasi-Sasakian structure, Journal of Differential Geom. 1(1967), 331-345.
[5] Blair, D.E., Riemannian geometry of contact and symplectic maniifolds, Birkhauser, Boston.(2010).
[6] Barabosa, E. and Ribeiro, E. Jr., On conformal solution of the Yamabe flow, Arch., Math.,101, (2013), 73-89.
[7] Blaga, A. M., η-Ricci solitons on Lorentzian para-Sasakian manifolds, Filomat. 30, 2 (2016), 489-496.
[8] Blaga, A. M., η-Ricci solitons on para-Kenmotsu manifolds, Balkan J. Geom. Appl. 20 (2015), 1-13.
[9] 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.
[10] Cho, J. T. and Kimura, M., Ricci soliltons and real hypersurfaces in a complex space form, Tohoku Math. J.(61) (2009), no. 2, 205-212.
[11] Chaki, M. C. and Kawaguchi, T., On almost pseudo Ricci symmetric manifolds, Tensor N.S.(68)(2007), 10-14.
[12] Deshmukh, S., Alodan, H. and Al-Sodais, H., A note on Ricci soliton, Balkan J. Geom. Appl.1,16 (2011), 48-55.
[13] De, U. C. and Sarkar, A., On φ-Ricci symmetric Sasakian manifolds, Proc. Jangjeon Math. Soc.11 (2008), 47-52.
[14] Friedrich, T., and Ivanov, S.,Almost contact manifolds connection with torsion and parallel spinors, Journal Reine Angew. Math. 559, 217-236.
[15] Gray, A., Einstein-like manifolds which are not Einstein, Geom. Dedicata.7 (1978), 259-280.
[16] Gonzalez, J. C. and Chinea, D., Quasi-Sasakian homogeneous structures on the generalized Heisenberg group H(p, 1), Proc. Amer. Math. Soc.105 (189), 173-184.
[17] Hamilton, R. S., The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986), 237-262, Contemp. Math.71, Amer. Math. Soc,Providence, RI, 1988.
[18] Hui, S. K., Yadav, S. K. and Patra, A., Almost conformal Ricci solitons on f-Kenmotsu manifolds, Khayyam J. Math. 1,5 (2019),84-104.
[19] Hui, S. K., Yadav, S. K. and Chaubey, S. K., η-Ricci soliton on 3-dimensional f-Kenmotsu manifolds, Appl. Appl. Math., 13, 2 (2018), 933-951.
[20] Ivey, T., Ricci solitons on compact 3-manifolds, Diff. Geom. Appl.3 (1993), 301-307.
[21] Kanemaki, S., Quasi-Sasakian manifolds, Tohoku Math. J.29 (1977),227-233.
[22] Kim, B. H., Fibred Riemannian spaces with quasi-Sasakian structure, Hiroshima Math.J.20 (1990),477-513.
[23] Kon, M., Invariant submanifolds in Sasakian manifolds, Mathematische Annalen.219 (1976), 277-290.
[24] Mandal, A. K. and De, A., Some theorems on 3-dimensional quasi-Sasakian manifolds, Tamsui Oxford Journal of Information and Mathematical Sciences.27 (2011), 411-427.
[25] Olszak, Z.,On three-dimensional conformally flat quasi-Sasakian manifolds, Period, Math. Hungar.2,33 (1996),105-113.
[26] Oubina, J. A.,New classes of almost contact metric structures, Publ. Math. Debreceen.32 (1985), 187-193.
[27] Olszak, Z., Normal almost contact metric manifolds of dimension three,Ann. Polon.Math. 47(1986),41-50.
[28] Pokhariyal, G. P., Yadav, S. K. and Chaubey, S. K., Ricci solitons on trans-Sasakian mani- folds, Differential Geometry-Dynamical Systems.20 (2018), 138-158.
[29] Singh, H., and Khan, Q., On special weakly symmetric Riemannian manifolds, Publ. Math. Debrecen.3,58 (2001), 523-536.
[30] Tanno, S.,Quasi-Sasakian structure of rank (2p + 1), Journal Differential Geom.5 (1971), 317- 324.
[31] Yadav, S. K., Ozturk, H., On (ǫ)-almost paracontact metric manifolds with conformal η-Ricci solitons, Differential Geometry-Dynamical Systems. 21 (2019), 202-215.
[32] Yadav, S. K., Chaubey, S. K. and Suthar, D. L., Some results of η-Ricci soliton on (LCS)- manifolds, Surveys in Mathematics and its Applications.13 (2018), 237-250.
[33] Yadav, S. K., Chaubey, S. K. and Suthar, D. L., Some geometric properties of η-Ricci solitons and gradient Ricci solitons on (LCS)n-manifolds, Cubo a Mathematical Journal, 2, 19 (2017), 33-48.
[34] Yano, K., Integral Formulas in Riemannian geometry, Marcel Dekker, New York, 1970.
[35] Ye, R., Global existence and convergence of Yamabe flow, Journal Differ., Geom. 1, 39 (1994),35-50.
[36] Yadav, S. K., Ricci solitons on para-Kahler manifolds, Extracta Mathematicae, Available online April 29, 2, 34 (2019).
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Published
2019-08-10
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[1]
S. . Kumar Yadav, A. . Kushwaha, and D. . Narain, “Certain results for η-Ricci Solitons and Yamabe Solitons on quasi-Sasakian 3-Manifolds”, CUBO, vol. 21, no. 2, pp. 77–98, Aug. 2019.
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