Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

GENERALIZED m-QUASI-EINSTEIN STRUCTURE IN ALMOST KENMOTSU MANIFOLDS

  • Mohan Khatri (Department of Mathematics and Computer Science Mizoram University) ;
  • Jay Prakash Singh (Department of Mathematics and Computer Science Mizoram University)
  • Received : 2022.05.20
  • Accepted : 2022.10.11
  • Published : 2023.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

The goal of this paper is to analyze the generalized m-quasi-Einstein structure in the context of almost Kenmotsu manifolds. Firstly we showed that a complete Kenmotsu manifold admitting a generalized m-quasi-Einstein structure (g, f, m, λ) is locally isometric to a hyperbolic space ℍ2n+1(-1) or a warped product ${\tilde{M}}{\times}{_{\gamma}{\mathbb{R}}$ under certain conditions. Next, we proved that a (κ, µ)'-almost Kenmotsu manifold with h' ≠ 0 admitting a closed generalized m-quasi-Einstein metric is locally isometric to some warped product spaces. Finally, a generalized m-quasi-Einstein metric (g, f, m, λ) in almost Kenmotsu 3-H-manifold is considered and proved that either it is locally isometric to the hyperbolic space ℍ3(-1) or the Riemannian product ℍ2(-4) × ℝ.

Keywords

Acknowledgement

The first author is thankful to the Department of Science and Technology, New Delhi, India for financial support in the form of INSPIRE Fellowship (DST/INSPIRE Fellowship/2018/IF180830).

References

  1. A. Barros and E. Ribeiro, Jr., Integral formulae on quasi-Einstein manifolds and applications, Glasg. Math. J. 54 (2012), no. 1, 213-223. https://doi.org/10.1017/S0017089511000565 
  2. A. Barros and E. Ribeiro, Jr., Characterizations and integral formulae for generalized m-quasi-Einstein metrics, Bull. Braz. Math. Soc. (N.S.) 45 (2014), no. 2, 325-341. https://doi.org/10.1007/s00574-014-0051-0 
  3. 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 
  4. H.-D. Cao, Recent progress on Ricci solitons, in Recent advances in geometric analysis, 1-38, Adv. Lect. Math. (ALM), 11, Int. Press, Somerville, MA, 2010. 
  5. J. Case, The nonexistence of quasi-Einstein metrics, Pacific J. Math. 248 (2010), no. 2, 277-284. https://doi.org/10.2140/pjm.2010.248.277 
  6. J. Case, Y.-J. Shu, and G. Wei, Rigidity of quasi-Einstein metrics, Differential Geom. Appl. 29 (2011), no. 1, 93-100. https://doi.org/10.1016/j.difgeo.2010.11.003 
  7. G. Catino, Generalized quasi-Einstein manifolds with harmonic Weyl tensor, Math. Z. 271 (2012), no. 3-4, 751-756. https://doi.org/10.1007/s00209-011-0888-5 
  8. X. Chen, Quasi-Einstein structures and almost cosymplectic manifolds, Rev. R. Acad. Cienc. Exactas F'is. Nat. Ser. A Mat. RACSAM 114 (2020), no. 2, Paper No. 72, 14 pp. https://doi.org/10.1007/s13398-020-00801-x 
  9. J. T. Cho and M. Kimura, Reeb flow symmetry on almost contact three-manifolds, Differential Geom. Appl. 35 (2014), suppl., 266-273. https://doi.org/10.1016/j.difgeo.2014.05.002 
  10. U. C. De, S. K. Chaubey, and Y. J. Suh, Gradient Yamabe and gradient m-quasi Einstein metrics on three-dimensional cosymplectic manifolds, Mediterr. J. Math. 18 (2021), no. 3, Paper No. 80, 14 pp. https://doi.org/10.1007/s00009-021-01720-w 
  11. G. Dileo and A. M. Pastore, Almost Kenmotsu manifolds and local symmetry, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 2, 343-354. http://projecteuclid.org/euclid.bbms/1179839227  https://doi.org/10.36045/bbms/1179839227
  12. G. Dileo and A. M. Pastore, Almost Kenmotsu manifolds and nullity distributions, J. Geom. 93 (2009), no. 1-2, 46-61. https://doi.org/10.1007/s00022-009-1974-2 
  13. A. Ghosh, Quasi-Einstein contact metric manifolds, Glasg. Math. J. 57 (2015), no. 3, 569-577. https://doi.org/10.1017/S0017089514000494 
  14. A. Ghosh, Generalized m-quasi-Einstein metric within the framework of Sasakian and K-contact manifolds, Ann. Polon. Math. 115 (2015), no. 1, 33-41. https://doi.org/10.4064/ap115-1-3 
  15. A. Ghosh, Ricci soliton and Ricci almost soliton within the framework of Kenmotsu manifold, Carpathian Math. Publ. 11 (2019), no. 1, 56-69. https://doi.org/10.15330/cmp.11.1.59-69 
  16. A. Ghosh, m-quasi-Einstein metric and contact geometry, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 113 (2019), no. 3, 2587-2600. https://doi.org/10.1007/s13398-019-00642-3 
  17. A. Ghosh, m-quasi-Einstein metrics satisfying certain conditions on the potential vector field, Mediterr. J. Math. 17 (2020), no. 4, Paper No. 115, 17 pp. https://doi.org/10.1007/s00009-020-01558-8 
  18. Z. Hu, D. Li, and J. Xu, On generalized m-quasi-Einstein manifolds with constant scalar curvature, J. Math. Anal. Appl. 432 (2015), no. 2, 733-743. https://doi.org/10.1016/j.jmaa.2015.07.021 
  19. Z. Hu, D. Li, and S. Zhai, On generalized m-quasi-Einstein manifolds with constant Ricci curvatures, J. Math. Anal. Appl. 446 (2017), no. 1, 843-851. https://doi.org/10.1016/j.jmaa.2016.09.019 
  20. M. Kanai, On a differential equation characterizing a Riemannian structure of a manifold, Tokyo J. Math. 6 (1983), no. 1, 143-151. https://doi.org/10.3836/tjm/1270214332 
  21. K. Kenmotsu, A class of almost contact Riemannian manifolds, Tohoku Math. J. (2) 24 (1972), 93-103. https://doi.org/10.2748/tmj/1178241594 
  22. M. Limoncu, Modifications of the Ricci tensor and applications, Arch. Math. (Basel) 95 (2010), no. 2, 191-199. https://doi.org/10.1007/s00013-010-0150-0 
  23. J. Milnor, Curvatures of left invariant metrics on Lie groups, Advances in Math. 21 (1976), no. 3, 293-329. https://doi.org/10.1016/S0001-8708(76)80002-3 
  24. A. M. Pastore and V. Saltarelli, Generalized nullity distributions on almost Kenmotsu manifolds, Int. Electron. J. Geom. 4 (2011), no. 2, 168-183. https://doi.org/10.4310/sii.2011.v4.n2.a13 
  25. D. S. Patra, A. Ghosh, and A. Bhattacharyya, The critical point equation on Kenmotsu and almost Kenmotsu manifolds, Publ. Math. Debrecen 97 (2020), no. 1-2, 85-99.  https://doi.org/10.5486/PMD.2020.8702
  26. D. Perrone, Almost contact metric manifolds whose Reeb vector field is a harmonic section, Acta Math. Hungar. 138 (2013), no. 1-2, 102-126. https://doi.org/10.1007/s10474-012-0228-1 
  27. 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.  https://doi.org/10.2422/2036-2145.2011.4.01
  28. L. Vanhecke and D. Janssens, Almost contact structures and curvature tensors, Kodai Math. J. 4 (1981), no. 1, 1-27. http://projecteuclid.org/euclid.kmj/1138036310  https://doi.org/10.2996/kmj/1138036310
  29. Y. Wang, Gradient Ricci almost solitons on two classes of almost Kenmotsu manifolds, J. Korean Math. Soc. 53 (2016), no. 5, 1101-1114. https://doi.org/10.4134/JKMS.j150416 
  30. Y. Wang, Conformally flat almost Kenmotsu 3-manifolds, Mediterr. J. Math. 14 (2017), no. 5, Paper No. 186, 16 pp. https://doi.org/10.1007/s00009-017-0984-9 
  31. Y. Wang and X. Liu, On almost Kenmotsu manifolds satisfying some nullity distributions, Proc. Nat. Acad. Sci. India Sect. A 86 (2016), no. 3, 347-353. https://doi.org/10.1007/s40010-016-0274-0