Ricci almost solitons on Concircular Ricci pseudosymmetric $\beta$-Kenmotsu manifolds

Abstract

The object of the present paper is to study concircular Ricci pseudosymmetric $\beta$-Kenmotsu manifolds whose metric is Ricci almost soliton. We found the conditions when Ricci almost soliton on concircular Ricci pseudosymmetric $\beta$-Kenmotsu manifold to be shrinking, steady and expanding respectively. We also construct an example of concircular Ricci pseudosymmetric $\beta$-Kenmotsu manifold whose metric is Ricci almost soliton.

Keywords

• Ricci almost soliton,$\beta$-Kenmotsu manifold,Ricci pseudosymmetric manifold,concircular curvature tensor

References

1. Ashoka, S. R., Bagewadi, C. S. and Ingalahalli, G., Certain results on Ricci solitons in - Sasakian manifolds, Hindawi Publ. Corporation, Geometry, Vol. 2013, Article ID 573925, 4 pages.
2. Ashoka, S. R., Bagewadi, C. S. and Ingalahalli, G., A geometry on Ricci solitons in (LCS)n- manifolds, Di. Geom.-Dynamical Systems, 16 (2014), 5062.
3. Bagewadi, C. S. and Ingalahalli, G., Ricci solitons in Lorentzian -Sasakian manifolds, Acta Math. Acad. Paeda. Nyire., 28 (2012), 5968.
4. Bejan, C. L. and Crasmareanu, M., Ricci solitons in manifolds with quasi constant curva- ture, Publ. Math. Debrecen, 78 (2011), 235243.
5. Blaga, A. M., $\eta$-Ricci solitons on para-kenmotsu manifolds, Balkan J. Geom. Appl., 20 (2015), 113.
6. Blair, D. E., Contact manifolds in Riemannian geometry, Lecture Notes in Math. 509, Springer-Verlag, 1976.
7. Calin, C. and Crasmareanu, M., From the Eisenhart problem to Ricci solitons in f- Kenmotsu manifolds, Bull. Malays. Math. Sci. Soc. (2), 33 (2010), 361368.
8. Chandra, S., Hui, S. K. and Shaikh, A. A., Second order parallel tensors and Ricci solitons on (LCS)n-manifolds, Commun. Korean Math. Soc., 30 (2015), 123130.

9. Chen, B. Y. and Deshmukh, S., Geometry of compact shrinking Ricci solitons, Balkan J. Geom. Appl., 19 (2014), 1321.
10. Deshmukh, S., Al-Sodais, H. and Alodan, H., A note on Ricci solitons, Balkan J. Geom. Appl.,16 (2011), 4855.
11. Deszcz, R., On Ricci-pseudosymmetric warped products, Demonstratio Math., 22 (1989), 10531065.
12. Deszcz R., On pseudosymmetric spaces, Bull. Soc. Math. Belg. Ser. A, 44(1) (1992), 134.
13. Hamilton, R. S., Three-manifolds with positive Ricci curvature, J. Di. Geom., 17 (1982), 255306.
14. Hamilton, R. S., The Ricci ow on surfaces, Mathematics and general relativity, Contemp. Math., 71, American Math. Soc., 1988, 237262.
15. Hui, S. K. and Chakraborty, D., Some types of Ricci solitons on (LCS)n-manifolds, J. Math. Sci. Advances and Applications, 37 (2016), 117.
16. Hui, S. K. and Chakraborty, D., Generalized Sasakian-space-forms and Ricci almost solitons with a conformal killing vector eld, New Trends in Math. Sci., 4 (2016), 263269.
17. Hui, S. K. and Lemence, R. S., Ricci pseudosymmetric generalized quasi-Einstein manifolds, SUT J. Math., 51 (2015), 6785.
18. Hui, S. K., Lemence, R. S. and Chakraborty, D., Ricci solitons on three dimensional gen- eralized Sasakian-space-forms, Tensor Society, N. S., 76 (2015), 75-83.
19. Ingalahalli, G. and Bagewadi, C. S., Ricci solitons in $\alpha$-Sasakian manifolds,Hindawi Pub- lishing Corporation, ISRN Geometry, Vol. 2012, Article ID 421384, 13 pages.
20. Jahanara, B., Haesen, S., Senturk, Z. and Verstraelen, L., On the parallel transport of the Ricci curvatures, J. Geom. Phys., 57 (2007), 17711777.
21. Kenmotsu, K., A class of almost contact Riemannian manifolds, Tohoku Math. J., 24 (1972), 93103.
22. Nagaraja, H. G. and Premalatha, C. R., Ricci solitons in Kenmotsu manifolds, J. Math. Analysis, 3 (2012), 1824.
23. Oubi~na, J. A., New classes of almost contact metric structures, Publ. Math. Debrecen, 32 (1985), 187193.
24. Perelman, G., The entropy formula for the Ricci ow and its geometric applications, http://arXiv.org/abs/math/0211159, 2002, 139.
25. Perelman, G., Ricci ow with surgery on three manifolds, http://arXiv.org/abs/math/0303109, 2003, 122.
26. Pigola, S., Rigoli, M., Rimoldi, M. and Setti, A. G., Ricci almost solitons, Ann. Sc. Norm. Super. Pisa Cl. Sci, 10 (2011), 757799.
27. Shaikh, A. A. and Hui, S. K., On some classes of generalized quasi-Einstein manifolds, Commun. Korean Math. Soc., 24(3) (2009), 415424.
28. Shaikh, A. A. and Hui, S. K., On locally $\phi$-symmetric $\beta$-kenmotsu manifolds, Extracta Mathematicae, 24(3) (2009), 301316.
29. Shaikh, A. A. and Hui, S. K., On extended generalized $\phi$-recurrent $\beta$-Kenmotsu Manifolds, Publ. de l'Institut Math. (Beograd), 89(103) (2011), 7788.
30. Sharma, R., Certain results on k-contact and (k,\mu)-contact manifolds, J. of Geom., 89 (2008), 138147.
31. Tripathi, M. M., Ricci solitons in contact metric manifolds, arxiv:0801.4221 [Math.DG] (2008).
32. Yano, K., Concircular geometry, I, Proc. Imp. Acad. Tokyo, 16 (1940), 195200.