Miao-Tam Equation and Ricci Solitons on Three-Dimensional Trans-Sasakian Generalized Sasakian Space-Forms

Uday Chand De; Tarak Mandal; Avijit Sarkar

doi:10.32323/ujma.1381621

EN

Miao-Tam Equation and Ricci Solitons on Three-Dimensional Trans-Sasakian Generalized Sasakian Space-Forms

Abstract

The aim of the present article is to characterize some properties of the Miao-Tam equation on three-dimensional generalized Sasakian space-forms with trans-Sasakian structures. It has been proved that in such space-forms if the Miao-Tam equation admits non-trivial solution, then the metric of the space form must be a gradient Ricci soliton. We have derived that a non-trivial solution of the Fischer-Marsden equation does not exist on the said space-forms. We have also investigated certain features of Ricci solitons and gradient Ricci solitons. At the end of the article, we construct an example to verify the obtained results.

Keywords

Generalized Sasakian space-forms, Gradient Ricci solitons, Miao-Tam equation, Sasakian space-forms, Trans-Sasakian manifolds

References

[1] X. Chen, On almost f -cosymplectic manifolds satisfying the Miao-Tam equation, J. Geom., 111 (2020), Article No 28.
[2] A. Barros, E. Ribeiro Jr., Critical point equation on four-dimensional compact manifolds, Math. Nachr., 287 (2014), 1618–1623.
[3] A. Besse, Einstein Manifolds, Springer-Verlag, New York, (2008).
[4] A. M. Blaga, On harmonicity and Miao-Tam critical metrics in a perfect fluid spacetime, Bol. Soc. Mat. Mexicana, 26 (2020), 1289–1299.
[5] A. Ghosh, D. S. Patra, Certain almost Kenmotsu metrics satisfying the Miao-Tam equation, arXiv:1701.04996v1[Math.DG], (2017).
[6] A. Ghosh, D. S. Patra, The critical point equation and contact geometry, arXiv:1711.05935v1[Math.DG], (2017).
[7] T. Mandal, Miao-Tam equation on almost coK¨ahler manifolds, Commun. Korean Math. Soc., 37 (2022), 881–891.
[8] T. Mandal, Miao-Tam equation on normal almost contact metric manifolds, Differ. Geom.-Dyn. Syst., 23 (2021), 135–143.
[9] D. S. Patra, A. Ghosh, Certain contact metrics satisfying the Miao-Tam critical condition, Ann. Polon. Math., 116 (2016), 263–271.
[10] A. Sarkar, G. G. Biswas, Critical point equation on K-paracontact manifolds, Balkan J. Geom. Appl., 5 (2020), 117–126.

[11] P. Miao, L.-F. Tam, On the volume functional of compact manifolds with boundary with constant scalar curvature, Calc. Var. PDE., 36(2009), 141–171.
[12] A. E. Fischer, J. Marsden, Manifolds of Riemannian metrics with prescribed scalar curvature, Bull. Am. Math. Soc., 80 (1974), 479–484.
[13] O. Kobayashi, A differential equation arising from scalar curvature function, J. Math. Soc. Jpn., 34 (1982), 665–675.
[14] R. S. Hamilton, The Ricci flow on surfaces, Contemp. Math., 71 (1988), 237–261.
[15] B.-Y. Chen, A survey on Ricci solitons on Riemannian submanifolds, Contemp. Math., 674 (2016), 27–39.
[16] A. Sarkar, G. G. Biswas, Ricci soliton on generalized Sasakian space forms with quasi-Sasakian metric, Afr. Mat., 31 (2020), 455–463.
[17] Y. Wang, Ricci solitons on 3-dimensional cosymplectic manifolds, Math. Slovaca., 67 (2017), 979–984.
[18] Y. Wang, Ricci solitons on almost coK¨ahler manifolds, Cand. Math. Bull., 62 (2019), 912–922.
[19] P. Alegre, D. E. Blair, A. Carriazo, Generalized Sasakian-space-forms, Israel J. Math., 141 (2004), 157–183.
[20] P. Alegre, A. Carriazo, Structures on generalized Sasakian space forms, Differential Geom. Appl., 26(6) (2008), 656–666.
[21] P. Alegre, D. E. Blair, A. Carriazo, Generalized Sasakian space-forms, Israel J. Math., 141 (2004), 157–183.
[22] P. Alegre, A. Carriazo, C. O¨ zgu¨r, S. Sular, New examples of Generalized Sasakian space-forms, Proc. Est. Acad. Sci., 60 (2011), 251–257.
[23] U. C. De , A. Sarkar, Some results on generalized Sasakian Space-forms, Thai. J. Math., 8 (2010), 1–10.
[24] U. K. Kim, Conformally flat generalized Sasakian space-forms and locally symmetric generalized Sasakian space-forms, Note Mat., 26 (2006), 55–67.
[25] A. Sarkar, M. Sen, Locally f-symmetric generalized generalized Sasakian space-forms, Ukr. Math. J., 65 (2014), 1588–1597.
[26] D. E. Blair, T. Koufogiorgos and B. J. Papantoniou, Contact metric manifolds satisfying a nullity condition, Israel J. Math., 19 (1995), 189–214.
[27] D. E. Blair, T. Koufogiorgos, R. Sharma, A classification of 3-dimensional contact metric manifolds with Qf = fQ, Kodai Math. J., 13(1990), 391–401.
[28] J. A. Oubina, New classes of almost contact metric structures, Publ. Math. Debreceen, 32 (1985), 187–193.
[29] U. C. De, M. M. Tripathi, Ricci tensor in 3-dimensional trans-Sasakian manifolds, Kyungpook Math. J., 43 (2003), 247–255.
[30] S. Deshmukh, M. M. Tripathi, A note on trans-Sasakian manifolds, Math. Slovaca, 63 (2013), 1361–1370.
[31] Y. Wang, W. Wang, A Remark on trans-Sasakian 3-manifolds, Rev. de la Union Mat. Argentina, 60 (2019), 257–264.
[32] S. Deshmukh, F. Al-Solamy, A note on compact trans-Sasakian manifolds, Mediterr. J. Math. 13 (2016), 2099–2104.
[33] S. Deshmukh, Trans-Sasakian manifolds homothetic to Sasakian manifolds, Mediterr. J. Math. 13 (2016), 2951–2958.
[34] D. E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, 203 (2010), Birkhauser, New York.
[35] S. Tanno, Some transformations on manifolds with almost contact and contact metric structure II, Tohoku Math. J., 15 (1963), 322–331.
[36] D. S. Patra and A. Ghosh, The Fischer-Marsden conjecture and contact geometry, Period. Math. Hungar., 76(2) (2017), 1–10.

Details

Primary Language

English

Subjects

Partial Differential Equations, Pure Mathematics (Other)

Journal Section

Research Article

Authors

Uday Chand De
0000-0002-8990-4609
India

Tarak Mandal
0000-0003-4808-8454
India

Avijit Sarkar ^*
0000-0002-7370-1698
India

Early Pub Date

January 16, 2024

Publication Date

March 18, 2024

Submission Date

October 26, 2023

Acceptance Date

December 24, 2023

Published in Issue

Year 2024 Volume: 7 Number: 1

DOI

https://doi.org/10.32323/ujma.1381621

IZ

https://izlik.org/JA72XN88LY

APA

De, U. C., Mandal, T., & Sarkar, A. (2024). Miao-Tam Equation and Ricci Solitons on Three-Dimensional Trans-Sasakian Generalized Sasakian Space-Forms. Universal Journal of Mathematics and Applications, 7(1), 1-11. https://doi.org/10.32323/ujma.1381621

AMA

1.De UC, Mandal T, Sarkar A. Miao-Tam Equation and Ricci Solitons on Three-Dimensional Trans-Sasakian Generalized Sasakian Space-Forms. Univ. J. Math. Appl. 2024;7(1):1-11. doi:10.32323/ujma.1381621

Chicago

De, Uday Chand, Tarak Mandal, and Avijit Sarkar. 2024. “Miao-Tam Equation and Ricci Solitons on Three-Dimensional Trans-Sasakian Generalized Sasakian Space-Forms”. Universal Journal of Mathematics and Applications 7 (1): 1-11. https://doi.org/10.32323/ujma.1381621.

EndNote

De UC, Mandal T, Sarkar A (March 1, 2024) Miao-Tam Equation and Ricci Solitons on Three-Dimensional Trans-Sasakian Generalized Sasakian Space-Forms. Universal Journal of Mathematics and Applications 7 1 1–11.

IEEE

[1]U. C. De, T. Mandal, and A. Sarkar, “Miao-Tam Equation and Ricci Solitons on Three-Dimensional Trans-Sasakian Generalized Sasakian Space-Forms”, Univ. J. Math. Appl., vol. 7, no. 1, pp. 1–11, Mar. 2024, doi: 10.32323/ujma.1381621.

ISNAD

De, Uday Chand - Mandal, Tarak - Sarkar, Avijit. “Miao-Tam Equation and Ricci Solitons on Three-Dimensional Trans-Sasakian Generalized Sasakian Space-Forms”. Universal Journal of Mathematics and Applications 7/1 (March 1, 2024): 1-11. https://doi.org/10.32323/ujma.1381621.

JAMA

1.De UC, Mandal T, Sarkar A. Miao-Tam Equation and Ricci Solitons on Three-Dimensional Trans-Sasakian Generalized Sasakian Space-Forms. Univ. J. Math. Appl. 2024;7:1–11.

MLA

De, Uday Chand, et al. “Miao-Tam Equation and Ricci Solitons on Three-Dimensional Trans-Sasakian Generalized Sasakian Space-Forms”. Universal Journal of Mathematics and Applications, vol. 7, no. 1, Mar. 2024, pp. 1-11, doi:10.32323/ujma.1381621.

Vancouver

1.Uday Chand De, Tarak Mandal, Avijit Sarkar. Miao-Tam Equation and Ricci Solitons on Three-Dimensional Trans-Sasakian Generalized Sasakian Space-Forms. Univ. J. Math. Appl. 2024 Mar. 1;7(1):1-11. doi:10.32323/ujma.1381621