\( \eta \)-Ricci Solitons on 3-dimensional Trans-Sasakian Manifolds

  • Sampa Pahan Department of Mathematics, Mrinalini Datta Mahavidyapith Kolkata-700051, India.
Keywords: Trans-Sasakian manifold, η-Ricci solitons


In this paper, we study \( \eta \)-Ricci solitons on 3-dimensional trans-Sasakian manifolds. Firstly we give conditions for the existence of these geometric structures and then observe that they provide examples of \( \eta \)-Einstein manifolds. In the case of \( \phi \)-Ricci symmetric trans-Sasakian manifolds, the η-Ricci soliton condition turns them to Einstein manifolds. Afterward, we study the implications in this geometric context of the important tensorial conditions \( R \cdot S = 0\), \(S \cdot R = 0\), \(W_2\cdot S = 0\) and \(S \cdot W_2 = 0\).


C. S. Bagewadi, G. Ingalahalli, S. R. Ashoka, A stuy on Ricci solitons in Kenmotsu Manifolds, ISRN Geometry, (2013), Article ID 412593, 6 pages.

A. Bhattacharyya, T. Dutta, and S. Pahan, Ricci Soliton, Conformal Ricci Soliton And Torqued Vector Fields, Bulletin of the Transilvania University of Brasov Series III: Mathematics, Informatics, Physics,, Vol 10(59), No. 1 (2017), 39-52.

A. M. Blaga, Eta-Ricci solitons on para-Kenmotsu manifolds, Balkan Journal of Geometry and Its Applications, Vol.20, No.1, 2015, pp. 1-13.

C. Cālin, M. Crasmareanu, Eta-Ricci solitons on Hopf hypersurfaces in complex forms, Revue Roumaine de Math. Pures et app., 57 (1), (2012), 53-63.

B. Y. Chen, S. Deshmukh, Geometry of compact shrinking Ricci solitons, Balkan Journal of Geometry and Its Applications, Vol.19, No.1, 2014, pp. 13-21.

J.C. Cho, M. Kimura Ricci solitons and real hypersurfaces in a complex space form, Tohoku Math. J. 61 (2), (2009), 205-2012.

O. Chodosh, F. T.-H Fong, Rational symmetry of conical Kähler-Ricci solitons, Math. Ann., 364(2016), 777-792.

A. Futaki, H. Ono, G. Wang, Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds, J. Diff. Geom. 83 (3), (2009), 585-636.

S. Golab, On semi-symmetric and quarter-symmetric linear connection, Tensor. N. S., 29(1975), 249-254.

R. S. Hamilton, The formation of singularities in the Ricci flow, Surveys in Differential Geometry (Cambridge, MA, 1993), 2, 7-136, International Press, Combridge, MA, 1995.

R. S. Hamilton, The Ricci flow on surfaces, Mathematical and general relativity, Contemp. math, 71(1988), 237-261.

G. Ingalahalli, C. S. Bagewadi, Ricci solitons on α-Sasakian Manifolds, ISRN Geometry, (2012), Article ID 421384, 13 pages.

J. C. Marrero, Ihe local structure of trans-Sasakian manifolds, Ann. Mat. Pura. Appl., (4), 162(1992), 77-86.

J. Morgan, G. Tian, Ricci Flow and the Poincaré Conjecture, American Mathematical Society Clay Mathematics Institute, (2007).

J. A. Oubina, New classes of almost contact metric structures, pub. Math. Debrecen, 20 (1), (2015), 1-13.

G. P. Pokhariyal, R. S. Mishra, The curvature tensors and their relativistic significance, Yokohama Math. J., 18(1970), 105-106.

D. G. Prakasha, B. S. Hadimani, η-Ricci solitons on para-Sasakian manifolds, Journal of Geometry, (2016), DOI: 10.1007/s00022-016-0345-z, pp 1-10.

How to Cite
Pahan, S. (2020). \( \eta \)-Ricci Solitons on 3-dimensional Trans-Sasakian Manifolds. CUBO, A Mathematical Journal, 22(1), 23–37. https://doi.org/10.4067/S0719-06462020000100023