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

Yıl 2024, Cilt: 7 Sayı: 1, 1 - 11, 18.03.2024

Uday Chand De Tarak Mandal Avijit Sarkar

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

Öz

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.

Anahtar Kelimeler

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

Kaynakça

  • [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.

Yıl 2024, Cilt: 7 Sayı: 1, 1 - 11, 18.03.2024

Uday Chand De Tarak Mandal Avijit Sarkar

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

Öz

Kaynakça

  • [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.
Toplam 36 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Kısmi Diferansiyel Denklemler, Temel Matematik (Diğer)
Bölüm Makaleler
Yazarlar

Uday Chand De 0000-0002-8990-4609

Tarak Mandal 0000-0003-4808-8454

Avijit Sarkar 0000-0002-7370-1698

Erken Görünüm Tarihi 16 Ocak 2024
Yayımlanma Tarihi 18 Mart 2024
Gönderilme Tarihi 26 Ekim 2023
Kabul Tarihi 24 Aralık 2023
Yayımlandığı Sayı Yıl 2024 Cilt: 7 Sayı: 1

Kaynak Göster

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 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. Mart 2024;7(1):1-11. doi:10.32323/ujma.1381621
Chicago De, Uday Chand, Tarak Mandal, ve Avijit Sarkar. “Miao-Tam Equation and Ricci Solitons on Three-Dimensional Trans-Sasakian Generalized Sasakian Space-Forms”. Universal Journal of Mathematics and Applications 7, sy. 1 (Mart 2024): 1-11. https://doi.org/10.32323/ujma.1381621.
EndNote De UC, Mandal T, Sarkar A (01 Mart 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 U. C. De, T. Mandal, ve A. Sarkar, “Miao-Tam Equation and Ricci Solitons on Three-Dimensional Trans-Sasakian Generalized Sasakian Space-Forms”, Univ. J. Math. Appl., c. 7, sy. 1, ss. 1–11, 2024, doi: 10.32323/ujma.1381621.
ISNAD De, Uday Chand vd. “Miao-Tam Equation and Ricci Solitons on Three-Dimensional Trans-Sasakian Generalized Sasakian Space-Forms”. Universal Journal of Mathematics and Applications 7/1 (Mart 2024), 1-11. https://doi.org/10.32323/ujma.1381621.
JAMA 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 vd. “Miao-Tam Equation and Ricci Solitons on Three-Dimensional Trans-Sasakian Generalized Sasakian Space-Forms”. Universal Journal of Mathematics and Applications, c. 7, sy. 1, 2024, ss. 1-11, doi:10.32323/ujma.1381621.
Vancouver 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.
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