Korean J. Math. Vol. 28 No. 4 (2020) pp.739-751
DOI: https://doi.org/10.11568/kjm.2020.28.4.739

A note on almost Ricci soliton and gradient almost Ricci soliton on para-Sasakian manifolds

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Published: Dec 30, 2020
Keywords:
3-dimensional para-Sasakian manifold, Almost Ricci soliton, Gradient almost Ricci soliton.
Mathematics Subject Classification:
53C21, 53C25, 53C50.

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Krishnendu De
Kabi Sukanta Mahavidyalaya, Bhadreswar, P.O.-Angus, Hooghly Pin 712221, West Bengal, India.
Uday Chand De
Department of Pure Mathematics, University of Calcutta 35, Ballygunge Circular Road, Kol- 700019, West Bengal, India.

Abstract

The object of the offering exposition is to study almost Ricci soliton and gradient almost Ricci soliton in 3-dimensional para-Sasakian manifolds. At first, it is shown that if $(g, V,\lambda)$ be an almost Ricci soliton on a 3-dimensional para-Sasakian manifold $M$, then it reduces to a Ricci soliton and the soliton is expanding for $\lambda$=-2. Besides these, in this section, we prove that if $V$ is pointwise collinear with $\xi$, then $V$ is a constant multiple of $\xi$ and the manifold is of constant sectional curvature $-1$. Moreover, it is proved that if a 3-dimensional para-Sasakian manifold admits gradient almost Ricci soliton under certain conditions then either the manifold is of constant sectional curvature $-1$ or it reduces to a gradient Ricci soliton. Finally, we consider an example to justify some results of our paper.


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References

[1] Barros, A., Batista, R. and Ribeiro Jr., E., Compact almost Ricci solitons with constant scalar curvature are gradient, Monatsh. Math., DOI 10.1007/s00605- 013-0581-3. Google Scholar

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

[3] Blaga, A.M., Some Geometrical Aspects of Einstein, Ricci and Yamabe solitons, J. Geom. symmetry Phys. 52 (2019), 17–26. Google Scholar

[4] Blaga, A.M., η-Ricci solitons on para-Kenmotsu manifolds, Balkan J. Geom. Appl. 20 (2015), 1–13. Google Scholar

[5] Cappelletti-Montano, B., Erken, I.K. and Murathan, C.,Nullity conditions in paracontact geometry, Differ. Geom. Appl. 30 (2012), 665–693. Google Scholar

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

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

[8] Duggal, K. L., Almost Ricci Solitons and Physical Applications, Int. El. J. Geom., 2 (2017), 1–10. Google Scholar

[9] Duggal, K. L., A New Class of Almost Ricci Solitons and Their Physical Interpretation, Hindawi Pub. Cor. Int. S. Res. Not., Volume 2016, Art. ID 4903520, 6 pages. Google Scholar

[10] 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

[11] Erken, I.K. and Murathan, C., A complete study of three-dimensional paracontact (κ, μ, ν)-spaces, Int. J.Geom. Methods Mod. Phys. (2017). https://doi.org/10.1142/S0219887817501067. Google Scholar

[12] Erken, I.K., Yamabe solitons on three-dimensional normal almost paracontact metric manifolds, Periodica Mathematica Hungarica https://doi.org/10.1007/s10998-019-00303-3. Google Scholar

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

[14] Kaneyuki, S. and Williams, F.L., Almost paracontact and parahodge structures on manifolds, Nagoya Math. J. 99 (1985), 173–187. Google Scholar

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

[16] Sat ̄o, I., On a structure similar to the almost contact structure, Tensor (N.S.) 30(1976), 219–224. Google Scholar

[17] Sharma, R., Almost Ricci solitons and K-contact geometry, Monatsh Math. 175 (2014), 621–628. Google Scholar

[18] Turan, M., De, U. C. and Yildiz, A., Ricci solitons and gradient Ricci solitons in three-dimensional trans-Sahakian manifolds, Filomat 26 (2012), 363–370. [19] Yano, K., Integral Formulas in Riemannian Geometry, Marcel Dekker, New York, 1970. Google Scholar

[19] Yildiz, A., De, U. C. and Turan, M., On 3-dimensional f-Kenmotsu manifolds and Ricci solitons, Ukrainian Math.J. 65 (2013), 684–693. Google Scholar