η-Einstein Solitons in δ- Lorentzian Trans Sasakian Manifolds

Abstract

The object of the present research is to study the δ-Lorentzian Trans Sasakian manifolds addmitting the η-Einstein Solitons and gradient Einstein soliton. It is shown that a symmetric second order covariant tensor in a δ-Lorentzian Trans Sasakian manifold is a constant multiple of metric tensor. Also an example of η-Einstein soliton in 3-diemsional δ-Lorentzian Trans Sasakian manifold is provided in the region where δ-Lorentzian Trans Sasakian manifold expanding.

Keywords

• η-Einstein Solitons,δ-Lorentzian Trans Sasakian man- ifold

Kaynakça

1. 1] Bhati, S. M., On weakly Ricci ϕ-symmetric δ-Lorentzian trans Sasakian manifolds, Bull. Math. Anal. Appl., vol. 5, (1), (2013), 36-43.
2. [2] Blair, D. E. and Oubina, J. A., Conformal and related changes of metric on the product of two almost contact metric manifolds, Publ. Mat. 34 (1990), 199-207.
3. [3] Bejancu, A. and Duggal, K. L., Real hypersurfaces of indefnite Kaehler manifolds, Int. J. Math and Math Sci., 16(3) (1993), 545-556.
4. [4] Blaga, A. M., η-Ricci solitons on Lorentzian para-Sasakian manifolds, Filomat 30 (2016), no. 2, 489-496.
5. [5] Blaga, A. M.,η-Ricci solitons on para-Kenmotsu manifolds, Balkan J. Geom. Appl. 20 (2015), 1-13.
6. [6] Blaga, A. M., Perktas, S. Y., Acet, B. L. and Erdogan, F. E., η-Ricci solitons in (ϵ)-almost para contact metric manifolds, arXiv:1707.07528v2math. DG]25 jul. 2017.
7. [7] Bagewadi, C. S. and Ingalahalli, G., Ricci Solitons in Lorentzian α-Sasakian Manifolds, Acta Math. Acad. Paedagog. Nyhzi. (N.S.) 28(1) (2012), 59-68.
8. [8] Bagewadi, C. S. and Ingalahalli, G., Ricci solitons in (ϵ,δ)-Trans-Sasakain manifolds, Int. J. Anal.Apply., 2 (2017), 209-217.

9. [9] Bagewadi, C. S., and Venkatesha, Some Curvature Tensors on a Trans-Sasakian Manifold, Turk. J. Math. 31 (2007), 111-121.
10. [10] Calin, C. and Crasmareanu, M., η-Ricci solitons on Hopf Hypersurfaces in complex space forms, Rev. Roumaine Math. Pures Appl. 57 (2012), no. 1, 55-63.
11. [11] Cho, J. T. and Kimura, M., Ricci solitons and Real hypersurfaces in a complex space form, Tohoku math.J., 61(2009), 205-212.
12. [12] Catino, G. and Mazzieri, L., Gradient Einstein solitons, Nonlinear Anal. 132 (2016), 6694.
13. [13] De, U. C. and Sarkar, A., On (ϵ)-Kenmotsu manifolds, Hadronic J. 32 (2009), 231-242.
14. [14] De, U. C. and Sarkar, A., On three-dimensional Trans-Sasakian Manifolds, Extracta Math. 23 (2008) 265277.
15. [15] De, U. C. and Krishnende De., On Lorentzian Trans-Sasakian manifolds, Commun. Fac. Sci. Univ. Ank. series A1, vol. 62 (2), (2013), 37-51.
16. [16] Eisenhart, L. P., Symmetric tensors of the second order whose first covariant derivatives are zero, Trans. Amer. Math. Soc., 25(2) (1923), 297-306.
17. [17] Gray, A. and Harvella, L. M., The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl., 123(4) (1980), 35-58.
18. [18] Gill, H. and Dube, K.K., Generalized CR- Submanifolds of a trans Lorentzian para Sasakian manifold, Proc. Nat. Acad. Sci. India Sec. A Phys. 2(2006), 119-124.
19. [19] Hamilton, R. S., The Ricci flow on surfaces, Mathematics and general relativity, (Santa Cruz. CA, 1986), Contemp. Math. 71, Amer. Math. Soc., (1988), 237-262.
20. [20] Ingalahalli, G. and Bagewadi, C. S., Ricci solitons in (ϵ)-Trans-Sasakain manifolds, J. Tensor Soc. 6 (1) (2012), 145-159.
21. [21] Ikawa, T. and Erdogan, M., Sasakian manifolds with Lorentzian metric, Kyungpook Math.J. 35(1996), 517-526.
22. [22] Kenmotsu, K., A class of almost contact Riemannian manifolds, Tohoku Math. J. 24(2) (1972), 93-103.
23. [23] Levy, H. Symmetric tensors of the second order whose covariant derivatives vanish, Ann. Math. 27(2) (1925), 91-98.
24. [24] Marrero, J. C., The local structure of Trans-Sasakian manifolds, Annali di Mat. Pura ed Appl. 162 (1992), 77-86.
25. [25] Matsumoto, K., On Lorentzian paracontact manifolds, Bull. Yamagata Univ. Nat. Science, 2(1989), 151-156.
26. [26] Mihai, I., Oiaga, A. and Rosca, R., Lorentzian Kenmotsu manifolds having two skew- sym- metric conformal vector ?elds, Bull. Math. Soc. Sci. Math. Roumania, 42(1999), 237-251.
27. [27] Nagaraja, H. G., Premalatha, C. R. and Somashekhara, G., On (ϵ,δ)-Trans-Sasakian Strucutre, Proc. Est. Acad. Sci. 61 (1) (2012), 20-28.
28. [28] Nagaraja, H.G. and C.R. Premalatha, C. R., Ricci solitons in Kenmotsu manifolds, J. Math. Anal. 3 (2) (2012), 18-24.
29. [29] Oubina, J. A., New classes of almost contact metric structures, Publ. Math. Debrecen 32 (1985), 187-193.
30. [30] Prakasha, D. G. and B. S. Hadimani, η-Ricci solitons on para-Sasakian manifolds, J. Geom., DOI 10.1007/s00022-016-0345-z.
31. [31] Pujar, S. S., and Khairnar, V. J., On Lorentzian trans-Sasakian manifold-I, Int.J.of Ultra Sciences of Physical Sciences, 23(1)(2011),53-66.
32. [32] Pujar, S. S., On ? Lorentzian ? Sasakian manifolds, to appear in Antactica J. of Mathematics 8(2012).
33. [33] Sharma, R., Certain results on K-contact and (k,µ)-contact manifolds, J. Geom., 89(1-2) (2008), 138-147.
34. [34] Shukla, S. S. and Singh, D. D., On (ϵ)-Trans-Sasakian manifolds, Int. J. Math. Anal. 49(4) (2010), 2401-2414.
35. [35] Siddiqi, M. D, Haseeb, A. and Ahmad, M., A Note On Generalized Ricci-Recurrent (ϵ,δ)- Trans-Sasakian Manifolds, Palestine J. Math., Vol. 4(1), 156-163 (2015).
36. [36] Tripathi, M. M., Ricci solitons in contact metric manifolds, arXiv:0801.4222 [math.DG].
37. [37] Turan, M., De, U. C. and Yildiz, A., Ricci solitons and gradient Ricci solitons on 3- dimensional trans-Sasakian manifolds, Filomat, 26(2) (2012), 363-370.
38. [38] Takahashi, T., Sasakian manifolds with Pseudo -Riemannian metric,Tohoku Math.J. 21 (1969),271-290.
39. [39] Thripathi, M. M., Kilic, E. and Perktas, S. Y., Indefinite almost metric manifolds, Int.J. of Math. and Mathematical Sciences, (2010) Article ID 846195,doi.10,1155/846195.
40. [40] Tanno, S., The automorphism groups of almost contact Riemannian manifolds,Tohoku Math.J. 21 (1969),21-38.
41. [41] Vinu, K. and Nagaraja, H. G., η-Ricci solitons in trans-Sasakian manifolds, Commun. Fac. sci. Univ. Ank. Series A1, 66 n0. 2 (2017), 218-224.
42. [42] Xufeng, X. and Xiaoli, C., Two theorems on (ϵ)-Sasakain manifolds, Int. J. Math. Math.Sci., 21(2) (1998), 249-254.
43. [43] Yaliniz, A.F., Yildiz, A. and Turan, M., On three-dimensional Lorentzian β- Kenmotsu man- ifolds, Kuwait J. Sci. Eng. 36 (2009), 51-62.
44. [44] Yildiz, A., Turan, M. and Murathan, C., A class of Lorentzian α- Sasakian manifolds, Kyung- pook Math. J. 49(2009), 789 -799.