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

Krishnendu De, Uday Chand De


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.


3-dimensional para-Sasakian manifold, Almost Ricci soliton, Gradient almost Ricci soliton.

Subject classification

53C21, 53C25, 53C50.


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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)

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

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

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

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

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

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

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

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)

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)

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)

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)

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

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

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)

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

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

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)

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)


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