Generalized (k,µ)-Space forms and Ricci solitons

Abstract

In this paper, we study Ricci-semisymmetric and Ricci pseudo-symmetric generalized (k,µ)-space forms along with characterization of generalized (k,µ)-space forms satisfying the curvature conditions Q(g,S) = 0 and Q(S,R) = 0. Further, we study Ricci solitons in generalized (k,µ)-space forms and obtained some interesting results.

Keywords

• Ricci-semisymmetric,Ricci pseudosymmetric,Ricci solitons,shrinking,expanding,steady,generalized (kμ)-space forms

References

1. P. Alegre, D.E. Blair and A. Carriazo, Generalized Sasakian-space-forms, Israel J. Math. 141(2004), 157-183.
2. D.E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Math. Springer Verlag, New York, 509(1973), 1-16.
3. A. Carriazo and V. Martin-Molina, Generalized (k,μ)-space forms and D-homothetic deformations, Balkan Journal of Geometry and its Applications, 16, 1(2011), 37-47.
4. A. Carriazo, V. Martin-Molina and M.M. Tripathi, Generalized(k,μ)-space forms, Mediterr. J. Math. 10, 2(2013), 475-496.
5. U.C. De and K. Mandal, Certain results on generalized (κ,μ)-contact metric manifolds, Journal of Geometry, 108(2017), 611-621.
6. U.C. De, Y. Han and K. Mandal, On para-sasakian manifolds satisfying certain CurvatureConditions, Filomat 31(2017), 1941-1947.
7. M. Faghfouri and N. Ghaffarzadeh, On doubly warped product submanifolds of generalized(κ,μ)-space forms, Afrika Matematika 26, 7-8 (2015), 1443-1455.
8. R.S. Hamilton, The Ricci flow on surfaces, Contemporary Mathematics, 71 (1988), 237-261.

9. S.K. Hui and D. Chakraborty, Ricci almost solitons on Concircular Ricci pseudosymmetric β-Kenmotsu manifolds, Hacettepe Journal of Mathematics and Statistics, 47 (2018), 579-587.
10. J.B. Jun, A. Yildiz and U.C. De, On ϕ-recurrent (k,μ)-contact metric manifolds, Bulletin of the Korean Mathematical Society, 45(2008), 689-700.
11. Kiran Kumar D. L., Nagaraja H. G. and Venu K., D-homothetically deformed Kenmotsu metric as a Ricci soliton, Annales Mathematicae Silesianae, DOI: 10.2478/amsil-2018-0010, 2018.
12. D. Kowalczyk, On some subclass of semi-symmetric manifolds, Soochow J. Math., 27(2001), 445-461.
13. B. Laha, D-Conformal curvature tensor in Generalized(k,μ)-space forms, Mathematical Combinatorics, 2(2017), 43-51.
14. S. Makhal and U.C. De,On pseudo-symmetry curvature conditions of generalized(k,μ)-paracontact metric manifolds, Konuralp Journal of Mathematics, 5(2017), 239-247.
15. Nagaraja H. G. and Kiran Kumar D. L., Ricci solitons in Kenmotsu manifolds under generalized D-conformal deformation, Lobachevskii Journal of Mathematics, 40: 195-200, 2019.
16. Nagaraja H. G., Kiran Kumar D. L. and Prasad V. S., Ricci solitons on Kenmotsu manifolds under D-homothetic deformation, Khayyam J. Math., 4: 102-109, 2018.
17. Nagaraja H. G., Kiran Kumar D. L. and Prakasha D. G., Da-homothetic deformation and Ricci solitons in (k,μ)-contact metric manifolds, Konuralp journal of mathematics, 6 pages, 2019.
18. D.G. Prakasha, S.K. Hui and K. Mirji, On 3-Dimensional Contact Metric Generalized (κ,μ)-Space Forms, International Journal of Mathematics and Mathematical Sciences, (2014), 6 pages.
19. C.R. Premalatha and H.G. Nagaraja, Recurrent generalized (κ,μ)-space forms, Acta Universitatis Apulensis, 38(2014), 95-108.
20. A.A. Shaikh and C.K. Mondal, Some results in η-Ricci Soliton and gradient ρ-Einstein soliton in a complete Riemannian manifold, arXiv preprint arXiv:1808.04789 (2018).
21. Shanmukha B., Venkatesha and Vishnuvardhana S.V., Some results on generalized (k,μ)-space forms, New Trends in Mathematical Sciences, 6: 48-56, 2018.
22. R. Sharma, Certain results on K-contact and (k,μ)-contact manifolds. Journal of Geometry, 89, 1(2008), 138-147.
23. Shivaprasanna G.S., Some results on generalized (k,μ)-space forms, International Journal of Scientific Engineering and Applied Science, 2: 184-191, 2016.
24. L. Verstraelen, Comments on pseudo-symmetry in sense of R. Deszcz, Geometry and Topology of submanifolds, Geometry and Topology of submanifolds, World Sci. Publication. 6(1994), 199-209.