Cotton tensor on Sasakian 3-manifolds admitting eta-Ricci solitons

Abstract

The object of the present paper is to characterize Cotton tensor on a 3-dimensional Sasakian manifold admitting $\eta$-Ricci solitons. After introduction, we study 3-dimensional Sasakian manifolds and introduce a new notion, namely, Cotton pseudo-symmetric manifolds. Next we deal with the study Cotton tensor on a Sasakian 3-manifold admitting $\eta$-Ricci solitons. Among others we prove that such a manifold is a manifold of constant scalar curvature and Einstein manifold with some appropriate conditions. Also, we classify the nature of the soliton metric. Finally, we give an important remark.

Keywords

• Einstein manifold
• Ricci soliton
• $\eta$-Ricci soliton
• Cotton tensor
• Sasakian manifold

References

1. Alegre, P. and Carriazo, A., Semi-Riemannian generalized Sasakian space forms, Bulletin of the Malaysian Mathematical Sciences Society, 41 (2018), 1-14. DOI:10.1007/ s40840-015-0215-0.
2. Ayar, G. and Yildirim, M., η-Ricci solitons on nearly Kenmotsu manifolds, Asian-European Journal of Mathematics, 12(6) (2019), 2040002 (8pp.). DOI:10.1142/S1793557120400021
3. Blaga, A. M., η-Ricci solitons on Lorentzian para-Sasakian manifolds, Filomat, 30(2) (2016), 489-496.
4. Blaga, A. M., η-Ricci solitons on para-Kenmotsu manifolds, Balkan J. Geom. Appl., 20 (2015), 1-13.
5. Blaga, A. M., Torse-forming η-Ricci solitons in almost paracontact η-Einstein geometry, Filomat, 31(2) (2017), 499-504.
6. Boeckx, E., Kowalski, O. and Vanhecke, L., Riemannian Manifolds of Conullity Two, Singapore World Sci. Publishing, 1996.
7. Blair, D. E., Lecture Notes in Mathematics, 509, Springer-Verlag Berlin, 1976.
8. Blair, D. E., Riemannian Geometry of Contact and Symplectic Manifolds, Birkhäuser, Boston, 2010.

9. 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.
10. Cartan, E., Sur une classe remarqable d'espaces de Riemann, Bull. Soc. Math. France., 54 (1926), 214-264.
11. Cho, J. T. and Kimura, M., Ricci solitons and real hypersurfaces in a complex space form, Tohoku Math. J., 61(2) (2009), 205-212.
12. 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.
13. Chave, T. and Valent, G., Quasi-Einstein metrics and their renoirmalizability properties, Helv. Phys. Acta., 69 (1996), 344-347.
14. 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.
15. Deshmukh, S., Jacobi-type vector fields on Ricci solitons, Bull. Math. Soc. Sci. Math. Roumanie, 55(103) 1 (2012), 41-50.
16. Deshmukh, S., Alodan, H. and Al-Sodais, H., A Note on Ricci Soliton, Balkan J. Geom. Appl., 16(1) (2011), 48-55.
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. Di¤ erential Geom., 17 (1982), 255-306.
19. Ivey, T., Ricci solitons on compact 3-manifolds, Diff. Geom. Appl., 3 (1993), 301-307.
20. Kar, D., Majhi, P. and De, U. C., η-Ricci solitons on 3-dimensional N(k)-contact metric manifolds, Acta Universitatis Apulensis, 54 (2018), 71-88.
21. Kim, B.H., Fibered Riemannian spaces with quasi-Sasakian structures, Hiroshima Math. J., 20 (1990), 477-513.
22. Kowalski, O., An explicit classification of 3-dimensional Riemannian spaces satisfying R(X; Y ):R = 0, Czechoslovak Math. J., 46(121) (1996) (3), 427-474.
23. Majhi, P., De, U. C. and Kar, D., η-Ricci Solitons on Sasakian 3-Manifolds, Anal. de Vest Timisoara, LV(2) (2017), 143-156.
24. Prakasha, D. G. and Hadimani, B. S., η-Ricci solitons on para-Sasakian manifolds, J. Geometry, 108 (2017), 383-392.
25. Prakasha, D. G., Zengin, F. O. and Chavan, V., On M-projectively semisymmetric Lorentzian α-Sasakian manifolds, Afrika Matematika, 28 (2017), 899-908. DOI:10.1007/s13370-017-0493-9.
26. Szabo , Z. I., Structure theorems on Riemannian spaces satisfying R(X; Y ):R = 0, the local version, J. Diff. Geom., 17 (1982), 531-582.
27. Turan, M., Yetim, C. and Chaubey, S. K., On quasi-Sasakian 3-manifolds admitting η-Ricci solitons, Filomat, 33(15) (2019), 4923-4930. https://doi.org/10.2298/FIL1915923T
28. Verstraelen, L., Comments on Pseudosymmetry in the Sense of Ryszard Deszcz, In: Geometry and Topology of Submanifolds, VI. River Edge, NJ: World Sci. Publishing, 1994, 199-209.
29. Wang, Y. and Liu, X., Ricci solitons on three-dimensional η-Einstein almost Kenmotsu manifolds, Taiwanese Journal of Mathematics, 19(1) (2015), 91-100.