Kyungpook Mathematical Journal

  • 제63권3호
  • /
  • Pages.507-519
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  • 2023
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  • 1225-6951(pISSN)
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  • 0454-8124(eISSN)
경북대학교 자연과학대학 수학과 (Department of Mathematics, Kyungpook National University)
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On f-cosymplectic and (k, µ)-cosymplectic Manifolds Admitting Fischer -Marsden Conjecture

  • 투고 : 2022.08.10
  • 심사 : 2022.11.28
  • 발행 : 2023.09.30
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초록

The aim of this paper is to study the Fisher-Marsden conjucture in the frame work of f-cosymplectic and (k, µ)-cosymplectic manifolds. First we prove that a compact f-cosymplectic manifold satisfying the Fisher-Marsden equation R'*g = 0 is either Einstein manifold or locally product of Kahler manifold and an interval or unit circle S1. Further we obtain that in almost (k, µ)-cosymplectic manifold with k < 0, the Fisher-Marsden equation has a trivial solution.

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과제정보

This work was supported by University Grants Commission, New Delhi, India(UGC-Ref-No: 942/(CSIR-UGC NET DEC, 2018).

참고문헌

  1. N. Aktan, M. Yildirim and C. Murathan, Almost f-cosymplectic manifolds, Mediterr. J. Math., 11(2)(2014), 775-787. https://doi.org/10.1007/s00009-013-0329-2
  2. D. E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, Birkhauser Boston, Inc., Boston(2002).
  3. J. -P. Bourguignon, Une stratification de l'espace des structures Riemanniennes, Compositio Math., 30(1975), 1-41.
  4. B. Cappelletti-Montano, A. De Nicola and I. Yudin, A survey on cosymplectic geometry, Rev. Math. Phys., 25(10)(2013), 1343002, 55 pp.
  5. P. Cernea and D. Guan, Killing fields generated by multiple solutions to the Fischer-Marsden equation, Internat. J. Math., 26(4)(2015), 1540006, 18 pp.
  6. S. K. Chaubey, U. C. De and Y. J. Suh, Kenmotsu manifolds satisfying the Fischer-Marsden equation, J. Korean Math. Soc., 58(3)(2021), 597-607.
  7. X. Chen, Notes on Ricci solitons in f-cosymplectic manifolds, Zh. Mat. Fiz. Anal. Geom., 13(3)(2017), 242-253. https://doi.org/10.15407/mag13.03.242
  8. J. Corvino, Curvature deformation and a gluing construction for the Einstein constraint equations, Comm. Math. Phys., 214(1)(2000), 137-189. https://doi.org/10.1007/PL00005533
  9. P. Dacko, On almost cosymplectic manifolds with the structure vector field ξ belonging to the k-nullity distribution, Balkan J. Geom. Appl., 5(2)(2000), 47-60.
  10. 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
  11. O. Kobayashi, A differential equation arising from scalar curvature function, J. Math. Soc. Japan, 34(4)(1982), 665-675. https://doi.org/10.2969/jmsj/03440665
  12. J. Lafontaine, Sur la geometrie d'une generalisation de l'equation differentielle d'Obata, J. Math. Pures Appl., 62(1)(1983), 63-72.
  13. F. Li, Vacuum Static Spaces with Harmonic Curvature, Mathematical Sciences and Applications E-Notes. arXiv preprint arXiv:2102.01280.
  14. H. Oztrk, N. Aktan and C. Murathan, Almost α-cosymplectic (κ, µ, ν)-Spaces, arXiv preprint arXiv:1007.0527(2010).
  15. D. S. Patra and A. Ghosh, The Fischer-Marsden conjecture and contact geometry, Period. Math. Hungar., 76(2)(2018), 207-216. https://doi.org/10.1007/s10998-017-0220-1
  16. D. G. Prakasha, P. Veeresha and Venkatesha, The Fischer-Marsden conjecture on non-Kenmotsu (κ, µ)'-almost Kenmotsu manifolds, J. Geom., 110(1)(2019), 9 pp.
  17. Y. Shen, A note on Fischer-Marsden's conjecture, Proc. Amer. Math. Soc., 125(3)(1997), 901-905. https://doi.org/10.1090/S0002-9939-97-03635-6