Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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REAL HYPERSURFACES WITH ∗-RICCI TENSORS IN COMPLEX TWO-PLANE GRASSMANNIANS

  • Chen, Xiaomin (College of Science China University of Petroleum-Beijing)
  • Received : 2016.05.12
  • Published : 2017.05.31
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Abstract

In this article, we consider a real hypersurface of complex two-plane Grassmannians $G_2({\mathbb{C}}^{m+2})$, $m{\geq}3$, admitting commuting ${\ast}$-Ricci and pseudo anti-commuting ${\ast}$-Ricci tensor, respectively. As the applications, we prove that there do not exist ${\ast}$-Einstein metrics on Hopf hypersurfaces as well as ${\ast}$-Ricci solitons whose potential vector field is the Reeb vector field on any real hypersurfaces.

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References

  1. J. Berndt and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians, Monatsh. Math. 127 (1999), no. 1, 1-14. https://doi.org/10.1007/s006050050018
  2. J. Berndt and Y. J. Suh, Real hypersurfaces with isometric Reeb flows in complex two-plane Grassmannians, Monatsh. Math. 137 (2002), no. 2, 87-98. https://doi.org/10.1007/s00605-001-0494-4
  3. 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
  4. R. Hamilton, The Ricci flow on surfaces, mathematics and general relativity, (Santa Cruz, CA, 1986), Contemp. Math. 71, pp. 237-262, Amer. Math. Soc., Providence, RI., 1988.
  5. I. Jeong and Y. J. Suh, Pseudo anti-commuting and Ricci soliton real hypersurfaces in complex two-plane Grassmannians, J. Geom. Phys. 86 (2014), 258-272. https://doi.org/10.1016/j.geomphys.2014.08.011
  6. G. Kaimakamis, K. Panagiotidou, -Ricci solitons of real hypersurfaces in non-flat com- plex space forms, J. Geom. Phys. 86 (2014), 408-413. https://doi.org/10.1016/j.geomphys.2014.09.004
  7. M. Kimura, Real hypersurfaces and complex submanifolds in complex projective space, Trans. Amer. Math. Soc. 296 (1986), no. 1, 137-149. https://doi.org/10.1090/S0002-9947-1986-0837803-2
  8. M. Kimura, Some real hypersurfaces of a complex projective space, Saitama Math. J. 5 (1987), 1-5.
  9. H. Lee and Y. J. Suh, Real hypersurfaces of type B in complex two-plane Grassmannians related to the Reeb vector, Bull. Korean Math. Soc. 47 (2010), no. 3, 551-561. https://doi.org/10.4134/BKMS.2010.47.3.551
  10. J. D. Perez and Y. J. Suh, Certain conditions on the Ricci tensor of real hypersurfaces in quaternionic projective space, Acta Math. Hungar. 91 (2001), no. 4, 343-356. https://doi.org/10.1023/A:1010676103922
  11. Y. J. Suh, Real hypersurfaces of type B in complex two-plane Grassmannians, Monatsh. Math. 147 (2006), no. 4, 337-355. https://doi.org/10.1007/s00605-005-0329-9
  12. Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with commuting Ricci tensor, J. Geom. Phys. 60 (2010), no. 11, 1792-1805. https://doi.org/10.1016/j.geomphys.2010.06.007
  13. Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with parallel Ricci tensor, Proc. Roy. Soc. Edinburgh 142A (2012), 1309-1324.

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