Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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KENMOTSU MANIFOLDS SATISFYING THE FISCHER-MARSDEN EQUATION

  • Chaubey, Sudhakar Kr (Section of Mathematics Department of Information Technology Shinas College of Technology) ;
  • De, Uday Chand (Department of Pure Mathematics University of Calcutta) ;
  • Suh, Young Jin (Department of Mathematics Kyungpook National University)
  • Received : 2019.09.03
  • Accepted : 2020.01.02
  • Published : 2021.05.01
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Abstract

The present paper deals with the study of Fischer-Marsden conjecture on a Kenmotsu manifold. It is proved that if a Kenmotsu metric satisfies 𝔏*g(λ) = 0 on a (2n + 1)-dimensional Kenmotsu manifold M2n+1, then either ξλ = -λ or M2n+1 is Einstein. If n = 1, M3 is locally isometric to the hyperbolic space H3 (-1).

Keywords

Acknowledgement

The authors express their sincere thanks to the referees and the Editor for providing the valuable suggestions in the improvement of the paper. First author acknowledges authority of Shinas College of Technology for their continuous support and encouragement to carry out this research work.

References

  1. K. Arslan, R. Ezentas, I. Mihai, and C. Murathan, Contact CR-warped product submanifolds in Kenmotsu space forms, J. Korean Math. Soc. 42 (2005), no. 5, 1101-1110. https://doi.org/10.4134/JKMS.2005.42.5.1101
  2. A. Basari and C. Murathan, On generalized φ-recurrent Kenmotsu manifolds, Fen Derg. 3 (2008), no. 1, 91-97.
  3. A. L. Besse, Einstein manifolds, reprint of the 1987 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2008.
  4. D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics, Vol. 509, Springer-Verlag, Berlin, 1976.
  5. W. M. Boothby and H. C. Wang, On contact manifolds, Ann. of Math. (2) 68 (1958), 721-734. https://doi.org/10.2307/1970165
  6. J.-P. Bourguignon, Une stratification de l'espace des structures riemanniennes, Compositio Math. 30 (1975), 1-41.
  7. C. Calin, Kenmotsu manifolds with η-parallel Ricci tensor, Bull. Soc. Math. Banja Luka 10 (2003), 10-15.
  8. P. Cernea and D. Guan, Killing fields generated by multiple solutions to the Fischer-Marsden equation, Internat. J. Math. 26 (2015), no. 4, 1540006, 18 pp. https://doi.org/10.1142/S0129167X15400066
  9. S. K. Chaubey and R. H. Ojha, On the m-projective curvature tensor of a Kenmotsu manifold, Differ. Geom. Dyn. Syst. 12 (2010), 52-60.
  10. S. K. Chaubey and C. S. Prasad, On generalized φ-recurrent Kenmotsu manifolds, TWMS J. Appl. Eng. Math. 5 (2015), no. 1, 1-9.
  11. J. T. Cho, Contact 3-manifolds with the Reeb flow symmetry, Tohoku Math. J. (2) 66 (2014), no. 4, 491-500. https://doi.org/10.2748/tmj/1432229193
  12. J. T. Cho, Reeb flow symmetry on almost cosymplectic three-manifolds, Bull. Korean Math. Soc. 53 (2016), no. 4, 1249-1257. https://doi.org/10.4134/BKMS.b150656
  13. 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
  14. J. Corvino, Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Comm. Math. Phys. 214 (2000), no. 1, 137-189. https://doi.org/10.1007/PL00005533
  15. U. C. De and K. Mandal, The Fischer-Marsden conjecture on almost Kenmotsu manifolds, Quaestiones Mathematicae (2018), 1-9. http://dx.doi.org/10.2989/16073606.2018.1533499
  16. U. C. De and G. Pathak, On 3-dimensional Kenmotsu manifolds, Indian J. Pure Appl. Math. 35 (2004), no. 2, 159-165.
  17. A. E. Fischer and J. E. Marsden, Manifolds of Riemannian metrics with prescribed scalar curvature, Bull. Amer. Math. Soc. 80 (1974), 479-484. https://doi.org/10.1090/S0002-9904-1974-13457-9
  18. G. Ghosh and U. C. De, Kenmotsu manifolds with generalized Tanaka-Webster connection, Publ. Inst. Math. (Beograd) (N.S.) 102(116) (2017), 221-230. https://doi.org/10.2298/pim1716221g
  19. J.-B. Jun, U. C. De, and G. Pathak, On Kenmotsu manifolds, J. Korean Math. Soc. 42 (2005), no. 3, 435-445. https://doi.org/10.4134/JKMS.2005.42.3.435
  20. K. Kenmotsu, A class of almost contact Riemannian manifolds, Tohoku Math. J. (2) 24 (1972), 93-103. https://doi.org/10.2748/tmj/1178241594
  21. O. Kobayashi, A differential equation arising from scalar curvature function, J. Math. Soc. Japan 34 (1982), no. 4, 665-675. https://doi.org/10.2969/jmsj/03440665
  22. J. Lafontaine, Sur la geometrie d'une generalisation de l'equation differentielle d'Obata, J. Math. Pures Appl. (9) 62 (1983), no. 1, 63-72.
  23. K. Matsumoto, I. Mihai, and M. H. Shahid, Certain submanifolds of a Kenmotsu manifold, in The Third Pacific Rim Geometry Conference (Seoul, 1996), 183-193, Monogr. Geom. Topology, 25, Int. Press, Cambridge, MA, 1998.
  24. A. Mustafa, A. De, and S. Uddin, Characterization of warped product submanifolds in Kenmotsu manifolds, Balkan J. Geom. Appl. 20 (2015), no. 1, 86-97.
  25. M. F. Naghi, I. Mihai, S. Uddin, and F. R. Al-Solamy, Warped product skew CR-submanifolds of Kenmotsu manifolds and their applications, Filomat 32 (2018), no. 10, 3505-3528. https://doi.org/10.2298/fil1810505n
  26. C. Ozgur, On weakly symmetric Kenmotsu manifolds, Differ. Geom. Dyn. Syst. 8 (2006), 204-209.
  27. D. S. Patra and A. Ghosh, The Fischer-Marsden conjecture and contact geometry, Period. Math. Hungar. 76 (2018), no. 2, 207-216. https://doi.org/10.1007/s10998-017-0220-1
  28. G. Pitis, A remark on Kenmotsu manifolds, Bul. Univ. Bra,sov Ser. C 30 (1988), 31-32.
  29. D. G. Prakasha, P. Veeresha, and Venkatesha, The Fischer-Marsden conjecture on non-Kenmotsu (κ, µ) 0-almost Kenmotsu manifolds, J. Geom. 110 (2019), no. 1, Art. 1, 9 pp. https://doi.org/10.1007/s00022-018-0457-8
  30. S. Sasaki, On differentiable manifolds with certain structures which are closely related to almost contact structure. I, Tohoku Math. J. (2) 12 (1960), 459-476. https://doi.org/10.2748/tmj/1178244407
  31. S. Sular and C. Ozgur, On some submanifolds of Kenmotsu manifolds, Chaos Solitons Fractals 42 (2009), no. 4, 1990-1995. https://doi.org/10.1016/j.chaos.2009.03.185
  32. S. Sular, C. Ozgur, and C. Murathan, Pseudoparallel anti-invariant submanifolds of Kenmotsu manifolds, Hacet. J. Math. Stat. 39 (2010), no. 4, 535-543.
  33. T. Takahashi, Sasakian φ-symmetric spaces, Tohoku Math. J. (2) 29 (1977), no. 1, 91-113. https://doi.org/10.2748/tmj/1178240699
  34. Y. Wang, Yamabe solitons on three-dimensional Kenmotsu manifolds, Bull. Belg. Math. Soc. Simon Stevin 23 (2016), no. 3, 345-355. http://projecteuclid.org/euclid.bbms/1473186509 https://doi.org/10.36045/bbms/1473186509
  35. Y. Wang, Three-dimensional almost Kenmotsu manifolds with η-parallel Ricci tensor, J. Korean Math. Soc. 54 (2017), no. 3, 793-805. https://doi.org/10.4134/JKMS.j160252
  36. Y. Wang, Cotton tensors on almost coKahler 3-manifolds, Ann. Polon. Math. 120 (2017), no. 2, 135-148. https://doi.org/10.4064/ap170410-3-10
  37. Y. Zhao, W. Wang, and X. Liu, Trans-Sasakian 3-manifolds with Reeb flow invariant Ricci operator, Mathematics 6 (2018), 246-252. https://doi:10.3390/math6110246