대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제38권3호
  • /
  • Pages.865-880
  • /
  • 2023
  • /
  • 1225-1763(pISSN)
  • /
  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

*-CONFORMAL RICCI SOLITONS ON ALMOST COKÄHLER MANIFOLDS

  • Tarak Mandal (Department of Mathematics University of Kalyani) ;
  • Avijit Sarkar (Department of Mathematics University of Kalyani)
  • 투고 : 2022.05.22
  • 심사 : 2023.05.09
  • 발행 : 2023.07.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

The main intention of the current paper is to characterize certain properties of *-conformal Ricci solitons on non-coKähler (𝜅, 𝜇)-almost coKähler manifolds. At first, we find that there does not exist *-conformal Ricci soliton if the potential vector field is the Reeb vector field θ. We also prove that the non-coKähler (𝜅, 𝜇)-almost coKähler manifolds admit *-conformal Ricci solitons if the potential vector field is the infinitesimal contact transformation. It is also studied that there does not exist *-conformal gradient Ricci solitons on the said manifolds. An example has been constructed to verify the obtained results.

키워드

과제정보

The authors are thankful to the referee for his/her valuable suggestions towards the improvement of the paper.

참고문헌

  1. Y. S. Balkan, S. Uddin, and A. H. Alkhaldi, A class of φ-recurrent almost cosymplectic space, Honam Math. J. 40 (2018), no. 2, 293-304.
  2. N. Basu and A. Bhattacharyya, Conformal Ricci soliton in Kenmotsu manifold, Glob. J. Adv. Res. Class. Mod. Geom. 4 (2015), no. 1, 15-21.
  3. C.-L. Bejan and M. Crasmareanu, Ricci solitons in manifolds with quasi-constant curvature, Publ. Math. Debrecen 78 (2011), no. 1, 235-243. https://doi.org/10.5486/PMD.2011.4797
  4. D. E. Blair, The theory of quasi-Sasakian structures, J. Differential Geometry 1 (1967), 331-345. http://projecteuclid.org/euclid.jdg/1214428097
  5. D. E. Blair, Riemannian geometry of contact and symplectic manifolds, second edition, Progress in Mathematics, 203, Birkhauser Boston, Ltd., Boston, MA, 2010. https://doi.org/10.1007/978-0-8176-4959-3
  6. D. E. Blair, T. Koufogiorgos, and B. J. Papantoniou, Contact metric manifolds satisfying a nullity condition, Israel J. Math. 91 (1995), no. 1-3, 189-214. https://doi.org/10.1007/BF02761646
  7. X. Dai, Y. Zhao, and U. C. De, *-Ricci soliton on (κ, µ)'-almost Kenmotsu manifolds, Open Math. 17 (2019), no. 1, 874-882. https://doi.org/10.1515/math-2019-0056
  8. U. C. De, A. M. Blaga, A. Sarkar, and T. Mandal, *-Ricci-Yamabe solitons on almost coKahler manifolds, Communicated.
  9. U. C. De, S. K. Chaubey, and Y. J. Suh, A note on almost co-Kahler manifolds, Int. J. Geom. Methods Mod. Phys. 17 (2020), no. 10, 2050153, 14 pp. https://doi.org/10.1142/S0219887820501534
  10. D. Dey, Sasakian 3-metric as a *-conformal Ricci soliton represents a Berger sphere, Bull. Korean Math. Soc. 59 (2022), no. 1, 101-110. https://doi.org/10.4134/BKMS.b210125
  11. T. Dutta, N. Basu, and A. Bhattacharyya, Almost conformal Ricci solituons on 3-dimensional trans-Sasakian manifold, Hacet. J. Math. Stat. 45 (2016), no. 5, 1379-1391.
  12. T. Dutta, N. Basu, and A. Bhattacharyya, Conformal Ricci soliton in Lorentzian α-Sasakian manifolds, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 55 (2016), no. 2, 57-70.
  13. T. Dutta and A. Bhattacharyya, Ricci soliton and conformal Ricci soliton in Lorentzian β-Kenmotsu manifold, Int. J. Math. Combin. 2 (2018), 1-12.
  14. A. E. Fischer, An introduction to conformal Ricci flow, Classical Quantum Gravity 21 (2004), no. 3, S171-S218. https://doi.org/10.1088/0264-9381/21/3/011
  15. T. Hamada, Real hypersurfaces of complex space forms in terms of Ricci *-tensor, Tokyo J. Math. 25 (2002), no. 2, 473-483. https://doi.org/10.3836/tjm/1244208866
  16. R. S. Hamilton, The Ricci flow on surfaces, in Mathematics and general relativity (Santa Cruz, CA, 1986), 237-262, Contemp. Math., 71, Amer. Math. Soc., Providence, RI, 1988. https://doi.org/10.1090/conm/071/954419
  17. A. Haseeb and R. Prasad, *-conformal η-Ricci solitons in ⲉ-Kenmotsu manifolds, Publ. Inst. Math. (Beograd) (N.S.) 108(122) (2020), 91-102. https://doi.org/10.2298/PIM2022091H
  18. G. Kaimakamis and K. Panagiotidou, *-Ricci solitons of real hypersurfaces in non-flat complex space forms, J. Geom. Phys. 86 (2014), 408-413. https://doi.org/10.1016/j.geomphys.2014.09.004
  19. C. Ozgur, On Ricci solitons with a semi-symmetric metric connection, Filomat 35 (2021), no. 11, 3635-3641. https://doi.org/10.2298/FIL2111635O
  20. S. Pigola, M. Rigoli, M. Rimoldi, and A. G. Setti, Ricci almost solitons, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 10 (2011), no. 4, 757-799.
  21. S. Roy, S. Dey, A. Bhattacharyya, and S. K. Hui, *-conformal η-Ricci soliton on Sasakian manifold, Asian-Eur. J. Math. 15 (2022), no. 2, Paper No. 2250035, 17 pp. https://doi.org/10.1142/S1793557122500358
  22. A. Sardar, M. N. I. Khan, and U. C. De, η - *-Ricci solitons and almost coKahler manifolds, Mathematics 9 (2021), 3200.
  23. A. Sarkar and G. Biswas, *-Ricci solitons on three dimensional trans-Sasakian manifolds, Math. Student 88 (2019), no. 3-4, 153-164.
  24. A. Sarkar and G. Biswas, Ricci solitons on three dimensional generalized Sasakian space forms with quasi Sasakian metric, Afr. Mat. 31 (2020), no. 3-4, 455-463. https://doi.org/10.1007/s13370-019-00735-7
  25. R. Sharma, Certain results on K-contact and (k, µ)-contact manifolds, J. Geom. 89 (2008), no. 1-2, 138-147. https://doi.org/10.1007/s00022-008-2004-5
  26. Y. J. Suh and U. C. De, Yamabe solitons and Ricci solitons on almost co-Kahler manifolds, Canad. Math. Bull. 62 (2019), no. 3, 653-661. https://doi.org/10.4153/s0008439518000693
  27. S. Tachibana, On almost-analytic vectors in almost-Kahlerian manifolds, Tohoku Math. J. (2) 11 (1959), 247-265. https://doi.org/10.2748/tmj/1178244584
  28. S. Tanno, Some transformations on manifolds with almost contact and contact metric structures. II, Tohoku Math. J. (2) 15 (1963), 322-331. https://doi.org/10.2748/tmj/1178243768
  29. W. Wang, Almost cosymplectic (k, µ)-metrics as η-Ricci solitons, J. Nonlinear Math. Phys. 29 (2022), no. 1, 58-72. https://doi.org/10.1007/s44198-021-00019-4
  30. Y. Wang, Ricci solitons on 3-dimensional cosymplectic manifolds, Math. Slovaca 67 (2017), no. 4, 979-984. https://doi.org/10.1515/ms-2017-0026
  31. Y. Wang, Ricci solitons on almost co-Kahler manifolds, Canad. Math. Bull. 62 (2019), no. 4, 912-922. https://doi.org/10.4153/s0008439518000632
  32. K. Yano, Integral formulas in Riemannian geometry, Pure and Applied Mathematics, No. 1, Marcel Dekker, Inc., New York, 1970.