Research Article
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Year 2018, Volume: 1 Issue: 1, 27 - 38, 18.05.2018

Mohd Siddiqi

Abstract

References

  • 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] 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] Bejancu, A. and Duggal, K. L., Real hypersurfaces of indefnite Kaehler manifolds, Int. J. Math and Math Sci., 16(3) (1993), 545-556.
  • [4] Blaga, A. M., η-Ricci solitons on Lorentzian para-Sasakian manifolds, Filomat 30 (2016), no. 2, 489-496.
  • [5] Blaga, A. M.,η-Ricci solitons on para-Kenmotsu manifolds, Balkan J. Geom. Appl. 20 (2015), 1-13.
  • [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] 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] Bagewadi, C. S. and Ingalahalli, G., Ricci solitons in (ϵ,δ)-Trans-Sasakain manifolds, Int. J. Anal.Apply., 2 (2017), 209-217.
  • [9] Bagewadi, C. S., and Venkatesha, Some Curvature Tensors on a Trans-Sasakian Manifold, Turk. J. Math. 31 (2007), 111-121.
  • [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] Cho, J. T. and Kimura, M., Ricci solitons and Real hypersurfaces in a complex space form, Tohoku math.J., 61(2009), 205-212.
  • [12] Catino, G. and Mazzieri, L., Gradient Einstein solitons, Nonlinear Anal. 132 (2016), 6694.
  • [13] De, U. C. and Sarkar, A., On (ϵ)-Kenmotsu manifolds, Hadronic J. 32 (2009), 231-242.
  • [14] De, U. C. and Sarkar, A., On three-dimensional Trans-Sasakian Manifolds, Extracta Math. 23 (2008) 265277.
  • [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] 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] 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] 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] 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] Ingalahalli, G. and Bagewadi, C. S., Ricci solitons in (ϵ)-Trans-Sasakain manifolds, J. Tensor Soc. 6 (1) (2012), 145-159.
  • [21] Ikawa, T. and Erdogan, M., Sasakian manifolds with Lorentzian metric, Kyungpook Math.J. 35(1996), 517-526.
  • [22] Kenmotsu, K., A class of almost contact Riemannian manifolds, Tohoku Math. J. 24(2) (1972), 93-103.
  • [23] Levy, H. Symmetric tensors of the second order whose covariant derivatives vanish, Ann. Math. 27(2) (1925), 91-98.
  • [24] Marrero, J. C., The local structure of Trans-Sasakian manifolds, Annali di Mat. Pura ed Appl. 162 (1992), 77-86.
  • [25] Matsumoto, K., On Lorentzian paracontact manifolds, Bull. Yamagata Univ. Nat. Science, 2(1989), 151-156.
  • [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] 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] Nagaraja, H.G. and C.R. Premalatha, C. R., Ricci solitons in Kenmotsu manifolds, J. Math. Anal. 3 (2) (2012), 18-24.
  • [29] Oubina, J. A., New classes of almost contact metric structures, Publ. Math. Debrecen 32 (1985), 187-193.
  • [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] 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] Pujar, S. S., On ? Lorentzian ? Sasakian manifolds, to appear in Antactica J. of Mathematics 8(2012).
  • [33] Sharma, R., Certain results on K-contact and (k,µ)-contact manifolds, J. Geom., 89(1-2) (2008), 138-147.
  • [34] Shukla, S. S. and Singh, D. D., On (ϵ)-Trans-Sasakian manifolds, Int. J. Math. Anal. 49(4) (2010), 2401-2414.
  • [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] Tripathi, M. M., Ricci solitons in contact metric manifolds, arXiv:0801.4222 [math.DG].
  • [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] Takahashi, T., Sasakian manifolds with Pseudo -Riemannian metric,Tohoku Math.J. 21 (1969),271-290.
  • [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] Tanno, S., The automorphism groups of almost contact Riemannian manifolds,Tohoku Math.J. 21 (1969),21-38.
  • [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] Xufeng, X. and Xiaoli, C., Two theorems on (ϵ)-Sasakain manifolds, Int. J. Math. Math.Sci., 21(2) (1998), 249-254.
  • [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] Yildiz, A., Turan, M. and Murathan, C., A class of Lorentzian α- Sasakian manifolds, Kyung- pook Math. J. 49(2009), 789 -799.

η-Einstein Solitons in δ- Lorentzian Trans Sasakian Manifolds

Year 2018, Volume: 1 Issue: 1, 27 - 38, 18.05.2018

Mohd Siddiqi

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

References

  • 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] 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] Bejancu, A. and Duggal, K. L., Real hypersurfaces of indefnite Kaehler manifolds, Int. J. Math and Math Sci., 16(3) (1993), 545-556.
  • [4] Blaga, A. M., η-Ricci solitons on Lorentzian para-Sasakian manifolds, Filomat 30 (2016), no. 2, 489-496.
  • [5] Blaga, A. M.,η-Ricci solitons on para-Kenmotsu manifolds, Balkan J. Geom. Appl. 20 (2015), 1-13.
  • [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] 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] Bagewadi, C. S. and Ingalahalli, G., Ricci solitons in (ϵ,δ)-Trans-Sasakain manifolds, Int. J. Anal.Apply., 2 (2017), 209-217.
  • [9] Bagewadi, C. S., and Venkatesha, Some Curvature Tensors on a Trans-Sasakian Manifold, Turk. J. Math. 31 (2007), 111-121.
  • [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] Cho, J. T. and Kimura, M., Ricci solitons and Real hypersurfaces in a complex space form, Tohoku math.J., 61(2009), 205-212.
  • [12] Catino, G. and Mazzieri, L., Gradient Einstein solitons, Nonlinear Anal. 132 (2016), 6694.
  • [13] De, U. C. and Sarkar, A., On (ϵ)-Kenmotsu manifolds, Hadronic J. 32 (2009), 231-242.
  • [14] De, U. C. and Sarkar, A., On three-dimensional Trans-Sasakian Manifolds, Extracta Math. 23 (2008) 265277.
  • [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] 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] 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] 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] 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] Ingalahalli, G. and Bagewadi, C. S., Ricci solitons in (ϵ)-Trans-Sasakain manifolds, J. Tensor Soc. 6 (1) (2012), 145-159.
  • [21] Ikawa, T. and Erdogan, M., Sasakian manifolds with Lorentzian metric, Kyungpook Math.J. 35(1996), 517-526.
  • [22] Kenmotsu, K., A class of almost contact Riemannian manifolds, Tohoku Math. J. 24(2) (1972), 93-103.
  • [23] Levy, H. Symmetric tensors of the second order whose covariant derivatives vanish, Ann. Math. 27(2) (1925), 91-98.
  • [24] Marrero, J. C., The local structure of Trans-Sasakian manifolds, Annali di Mat. Pura ed Appl. 162 (1992), 77-86.
  • [25] Matsumoto, K., On Lorentzian paracontact manifolds, Bull. Yamagata Univ. Nat. Science, 2(1989), 151-156.
  • [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] 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] Nagaraja, H.G. and C.R. Premalatha, C. R., Ricci solitons in Kenmotsu manifolds, J. Math. Anal. 3 (2) (2012), 18-24.
  • [29] Oubina, J. A., New classes of almost contact metric structures, Publ. Math. Debrecen 32 (1985), 187-193.
  • [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] 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] Pujar, S. S., On ? Lorentzian ? Sasakian manifolds, to appear in Antactica J. of Mathematics 8(2012).
  • [33] Sharma, R., Certain results on K-contact and (k,µ)-contact manifolds, J. Geom., 89(1-2) (2008), 138-147.
  • [34] Shukla, S. S. and Singh, D. D., On (ϵ)-Trans-Sasakian manifolds, Int. J. Math. Anal. 49(4) (2010), 2401-2414.
  • [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] Tripathi, M. M., Ricci solitons in contact metric manifolds, arXiv:0801.4222 [math.DG].
  • [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] Takahashi, T., Sasakian manifolds with Pseudo -Riemannian metric,Tohoku Math.J. 21 (1969),271-290.
  • [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] Tanno, S., The automorphism groups of almost contact Riemannian manifolds,Tohoku Math.J. 21 (1969),21-38.
  • [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] Xufeng, X. and Xiaoli, C., Two theorems on (ϵ)-Sasakain manifolds, Int. J. Math. Math.Sci., 21(2) (1998), 249-254.
  • [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] Yildiz, A., Turan, M. and Murathan, C., A class of Lorentzian α- Sasakian manifolds, Kyung- pook Math. J. 49(2009), 789 -799.
There are 44 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Mohd Siddiqi

Publication Date May 18, 2018
Published in Issue Year 2018 Volume: 1 Issue: 1

Cite

APA Siddiqi, M. (2018). η-Einstein Solitons in δ- Lorentzian Trans Sasakian Manifolds. Mathematical Advances in Pure and Applied Sciences, 1(1), 27-38.
AMA Siddiqi M. η-Einstein Solitons in δ- Lorentzian Trans Sasakian Manifolds. MAPAS. May 2018;1(1):27-38.
Chicago Siddiqi, Mohd. “η-Einstein Solitons in δ- Lorentzian Trans Sasakian Manifolds”. Mathematical Advances in Pure and Applied Sciences 1, no. 1 (May 2018): 27-38.
EndNote Siddiqi M (May 1, 2018) η-Einstein Solitons in δ- Lorentzian Trans Sasakian Manifolds. Mathematical Advances in Pure and Applied Sciences 1 1 27–38.
IEEE M. Siddiqi, “η-Einstein Solitons in δ- Lorentzian Trans Sasakian Manifolds”, MAPAS, vol. 1, no. 1, pp. 27–38, 2018.
ISNAD Siddiqi, Mohd. “η-Einstein Solitons in δ- Lorentzian Trans Sasakian Manifolds”. Mathematical Advances in Pure and Applied Sciences 1/1 (May2018), 27-38.
JAMA Siddiqi M. η-Einstein Solitons in δ- Lorentzian Trans Sasakian Manifolds. MAPAS. 2018;1:27–38.
MLA Siddiqi, Mohd. “η-Einstein Solitons in δ- Lorentzian Trans Sasakian Manifolds”. Mathematical Advances in Pure and Applied Sciences, vol. 1, no. 1, 2018, pp. 27-38.
Vancouver Siddiqi M. η-Einstein Solitons in δ- Lorentzian Trans Sasakian Manifolds. MAPAS. 2018;1(1):27-38.