Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

REAL HYPERSURFACES OF TYPE B IN COMPLEX TWO-PLANE GRASSMANNIANS RELATED TO THE REEB VECTOR

  • Lee, Hyun-Jin (Graduate School of Electrical Engineering & Computer Science Kyungpook National University) ;
  • Suh, Young-Jin (Department of Mathematics Kyungpook National University)
  • Received : 2008.12.11
  • Published : 2010.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we give a new characterization of real hypersurfaces of type B, that is, a tube over a totally geodesic $\mathbb{Q}P^n$ in complex two-plane Grassmannians $G_2(\mathbb{C}^{m+2})$, where m = 2n, with the Reeb vector $\xi$ belonging to the distribution $\mathfrak{D}$, where $\mathfrak{D}$ denotes a subdistribution in the tangent space $T_xM$ such that $T_xM$ = $\mathfrak{D}{\bigoplus}\mathfrak{D}^{\bot}$ for any point $x{\in}M$ and $\mathfrak{D}^{\bot}=Span{\xi_1,\;\xi_2,\;\xi_3}$.

Keywords

References

  1. J. Berndt, Riemannian geometry of complex two-plane Grassmannians, Rend. Sem. Mat. Univ. Politec. Torino 55 (1997), no. 1, 19-83.
  2. J. Berndt, Real hypersurfaces with constant principal curvatures in complex space forms, Proceedings of the Tenth International Workshop on Differential Geometry, 1–2, Kyungpook Nat. Univ., Taegu, 2006.
  3. J. Berndt and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians, Monatsh. Math. 127 (1999), no. 1, 1–14.
  4. J. Berndt and Y. J. Suh, Real hypersurfaces with isometric Reeb flow in complex two-plane Grassmannians, Monatsh. Math. 137 (2002), no. 2, 87–98.
  5. T. E. Cecil and P. J. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans. Amer. Math. Soc. 269 (1982), no. 2, 481-499. https://doi.org/10.2307/1998460
  6. I. Jeong and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with Lie ${\xi}$-parallel normal Jacobi operator, J. Korean Math. Soc. 45 (2008), no. 4, 1113-1133. https://doi.org/10.4134/JKMS.2008.45.4.1113
  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/10.2307/2000565
  8. Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with commuting shape operator, Bull. Austral. Math. Soc. 68 (2003), no. 3, 379-393. https://doi.org/10.1017/S0004972700037795
  9. 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
  10. R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 10 (1973), 495-506.
  11. K. Yano and M. Kon, CR Submanifolds of Kaehlerian and Sasakian Manifolds, Progress in Mathematics, 30. Birkhauser, Boston, Mass., 1983.

Cited by

  1. Real hypersurfaces in complex two-plane Grassmannians with harmonic curvature vol.100, pp.1, 2013, https://doi.org/10.1016/j.matpur.2012.10.010
  2. Real Hypersurfaces in Complex Two-Plane Grassmannians Whose Jacobi Operators Corresponding to -Directions are of Codazzi Type vol.01, pp.03, 2011, https://doi.org/10.4236/apm.2011.13015
  3. Hopf Hypersurfaces in Complex Two-Plane Grassmannians with Reeb Parallel Shape Operator vol.38, pp.2, 2015, https://doi.org/10.1007/s40840-014-0039-3
  4. REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS SOME OF WHOSE JACOBI OPERATORS ARE ξ-INVARIANT vol.23, pp.03, 2012, https://doi.org/10.1142/S0129167X1100746X
  5. Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition vol.62, pp.3, 2012, https://doi.org/10.1007/s10587-012-0049-y
  6. Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster $$\mathfrak{D}^ \bot$$-parallel structure Jacobi operator vol.12, pp.12, 2014, https://doi.org/10.2478/s11533-014-0447-5
  7. Pseudo anti-commuting and Ricci soliton real hypersurfaces in complex two-plane Grassmannians vol.86, 2014, https://doi.org/10.1016/j.geomphys.2014.08.011
  8. REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS WITH GENERALIZED TANAKA–WEBSTER 𝔇⊥-PARALLEL SHAPE OPERATOR vol.09, pp.04, 2012, https://doi.org/10.1142/S0219887812500326
  9. Semi-parallel real hypersurfaces in complex two-plane Grassmannians vol.34, 2014, https://doi.org/10.1016/j.difgeo.2014.03.011
  10. Levi-Civita and generalized Tanaka–Webster covariant derivatives for real hypersurfaces in complex two-plane Grassmannians vol.194, pp.3, 2015, https://doi.org/10.1007/s10231-014-0405-7
  11. REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS WHOSE SHAPE OPERATOR IS OF CODAZZI TYPE IN GENERALIZED TANAKA-WEBSTER CONNECTION vol.52, pp.1, 2015, https://doi.org/10.4134/BKMS.2015.52.1.057
  12. Real Hypersurfaces in Complex Two-Plane Grassmannians with GTW Reeb Lie Derivative Structure Jacobi Operator vol.13, pp.3, 2016, https://doi.org/10.1007/s00009-015-0535-1
  13. REAL HYPERSURFACES OF TYPE A IN COMPLEX TWO-PLANE GRASSMANNIANS RELATED TO THE NORMAL JACOBI OPERATOR vol.49, pp.2, 2012, https://doi.org/10.4134/BKMS.2012.49.2.423
  14. Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster parallel normal Jacobi operator vol.65, pp.1, 2015, https://doi.org/10.1007/s10587-015-0169-2
  15. Real hypersurfaces of complex two-plane Grassmannians with certain parallel conditions 2017, https://doi.org/10.1007/s00022-017-0402-2
  16. Real hypersurfaces in complex two-plane Grassmannians with Reeb parallel Ricci tensor vol.64, 2013, https://doi.org/10.1016/j.geomphys.2012.10.005
  17. Ricci Semi-symmetric Hypersurfaces in Complex Two-Plane Grassmannians vol.40, pp.3, 2017, https://doi.org/10.1007/s40840-016-0372-9
  18. Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka–Webster D-parallel shape operator vol.33, pp.1, 2017, https://doi.org/10.1007/s10114-016-4738-6
  19. Real Hypersurfaces in Complex Two-plane Grassmannians with $${\mathfrak{D}^{\bot}}$$ -Parallel Shape Operator vol.64, pp.3-4, 2013, https://doi.org/10.1007/s00025-013-0317-7
  20. Hopf Hypersurfaces in Complex Grassmannians of Rank Two vol.71, pp.3-4, 2017, https://doi.org/10.1007/s00025-016-0601-4
  21. Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition II vol.64, pp.1, 2014, https://doi.org/10.1007/s10587-014-0089-6
  22. Generalized Tanaka-Webster and Levi-Civita connections for normal Jacobi operator in complex two-plane Grassmannians vol.65, pp.2, 2015, https://doi.org/10.1007/s10587-015-0196-z
  23. Semi-parallelism of normal Jacobi operator for Hopf hypersurfaces in complex two-plane Grassmannians vol.172, pp.2, 2013, https://doi.org/10.1007/s00605-013-0553-7
  24. Real hypersurfaces in complex two-plane Grassmannians with recurrent Ricci tensor vol.12, pp.09, 2015, https://doi.org/10.1142/S0219887815500863
  25. RECURRENT JACOBI OPERATOR OF REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS vol.50, pp.2, 2013, https://doi.org/10.4134/BKMS.2013.50.2.525
  26. Real hypersurfaces in complex two-plane Grassmannians with generalized Tanaka–Webster Reeb parallel shape operator vol.171, pp.3-4, 2013, https://doi.org/10.1007/s00605-013-0475-4
  27. Semi-parallel symmetric operators for Hopf hypersurfaces in complex two-plane Grassmannians vol.177, pp.4, 2015, https://doi.org/10.1007/s00605-015-0778-8
  28. On the Riemannian Curvature Tensor of a Real Hypersurface in Complex Two-Plane Grassmannians 2017, https://doi.org/10.1007/s40840-017-0500-1
  29. Real Hypersurfaces in Complex Two-Plane Grassmannians with Reeb Parallel Structure Jacobi Operator vol.57, pp.04, 2014, https://doi.org/10.4153/CMB-2013-018-3
  30. Real Hypersurfaces in Complex Two-Plane Grassmannians with GTW Harmonic Curvature vol.58, pp.04, 2015, https://doi.org/10.4153/CMB-2015-039-7
  31. Real Hypersurfaces in Complex Two-plane Grassmannians with Reeb Parallel Ricci Tensor in the GTW Connection vol.59, pp.04, 2016, https://doi.org/10.4153/CMB-2016-035-x