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Real Hypersurfaces in Complex Two-Plane Grassmannians with Commuting Jacobi Operators

  • Eunmi Pak
  • Young Jin Suh
  • Changhwa Woo
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 106)

Abstract

In this paper, we have considered new commuting conditions, that is, (R ξ ϕ)S = S(R ξ ϕ) (resp. \((\bar{R}_{N}\phi )S = S(\bar{R}_{N}\phi )\)) between the Jacobi operators R ξ (resp. \(\bar{R}_{N}\)), the structure tensor field ϕ and the Ricci tensor S for real hypersurfaces M in \(G_{2}(\mathbb{C}^{m+2})\). With such a condition we give a complete classification of Hopf hypersurfaces M in \(G_{2}(\mathbb{C}^{m+2})\).

Keywords

Ricci Tensor Real Hypersurface Shape Operator Jacobi Operator Geometric Quantity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This work was supported by Grant Proj. No. NRF-2011-220-C00002 from National Research Foundation of Korea. The first author by Grant Proj. No. NRF-2012- R1A2A2A01043023 and the third author supported by NRF Grant funded by the Korean Government (NRF-2013-Fostering Core Leaders of Future Basic Science Program).

References

  1. 1.
    Alekseevskii, D.V.: Compact quaternion spaces. Funct. Anal. Appl. 2, 106–114 (1968)CrossRefGoogle Scholar
  2. 2.
    Berndt, J., Suh, Y.J.: Real hypersurfaces in complex two-plane Grassmannians. Monatsh. Math. 127, 1–14 (1999)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Berndt, J., Suh, Y.J.: Real hypersurfaces with isometric Reeb flow in complex two-plane Grassmannians. Monatsh. Math. 137, 87–98 (2002)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Jeong, I., Machado, C.J.G., Pérez, J.D., Suh, Y.J.: Real hypersurfaces in complex two-plane Grassmannians with \(\mathfrak{D}^{\perp }\)-parallel structure Jacobi operator. Int. J. Math. 22, 655–673 (2011)CrossRefMATHGoogle Scholar
  5. 5.
    Lee, H., Suh, Y.J.: Real hypersurfaces of type B in complex two-plane Grassmannians related to the Reeb vector. Bull. Korean Math. Soc. 47, 551–561 (2010)Google Scholar
  6. 6.
    Pérez, J.D., Suh, Y.J.: The Ricci tensor of real hypersurfaces in complex two-plane Grassmannians. J. Korean Math. Soc. 44, 211–235 (2007)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Pérez, J.D., Jeong, I., Suh, Y.J.: Real hypersurfaces in complex two-plane Grassmannians with commuting normal Jacobi operator. Acta Math. Hungar. 117, 201–217 (2007)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Suh, Y.J.: Real hypersurfaces in complex two-plane Grassmannians with commuting Ricci tensor. J. Geom. Phys. 60, 1792–1805 (2010)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Suh, Y.J., Yang, H.Y.: Real hypersurfaces in complex two-plane Grassmannians with commuting structure Jacobi operator. Bull. Korean Math. Soc. 45, 495–507 (2008)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Kyungpook National UniversityDaeguSouth Korea

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