Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster 𝔇 ⊥ -parallel structure Jacobi operator

Eunmi Pak; Young Suh

Open Mathematics (2014)

  • Volume: 12, Issue: 12, page 1840-1851
  • ISSN: 2391-5455

Abstract

top
Regarding the generalized Tanaka-Webster connection, we considered a new notion of 𝔇 ⊥ -parallel structure Jacobi operator for a real hypersurface in a complex two-plane Grassmannian G 2(ℂm+2) and proved that a real hypersurface in G 2(ℂm+2) with generalized Tanaka-Webster 𝔇 ⊥ -parallel structure Jacobi operator is locally congruent to an open part of a tube around a totally geodesic quaternionic projective space ℍP n in G 2(ℂm+2), where m = 2n.

How to cite

top

Eunmi Pak, and Young Suh. "Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster \[\mathfrak {D}^ \bot \] -parallel structure Jacobi operator." Open Mathematics 12.12 (2014): 1840-1851. <http://eudml.org/doc/269674>.

@article{EunmiPak2014,
abstract = {Regarding the generalized Tanaka-Webster connection, we considered a new notion of \[\mathfrak \{D\}^ \bot \] -parallel structure Jacobi operator for a real hypersurface in a complex two-plane Grassmannian G 2(ℂm+2) and proved that a real hypersurface in G 2(ℂm+2) with generalized Tanaka-Webster \[\mathfrak \{D\}^ \bot \] -parallel structure Jacobi operator is locally congruent to an open part of a tube around a totally geodesic quaternionic projective space ℍP n in G 2(ℂm+2), where m = 2n.},
author = {Eunmi Pak, Young Suh},
journal = {Open Mathematics},
keywords = {Complex two-plane Grassmannian; Hopf hypersurface; Generalized Tanaka-Webster connection; Structure Jacobi operator; complex two-plane Grassmannian; generalized Tanaka-Webster connection; structure Jacobi operator},
language = {eng},
number = {12},
pages = {1840-1851},
title = {Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster \[\mathfrak \{D\}^ \bot \] -parallel structure Jacobi operator},
url = {http://eudml.org/doc/269674},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Eunmi Pak
AU - Young Suh
TI - Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster \[\mathfrak {D}^ \bot \] -parallel structure Jacobi operator
JO - Open Mathematics
PY - 2014
VL - 12
IS - 12
SP - 1840
EP - 1851
AB - Regarding the generalized Tanaka-Webster connection, we considered a new notion of \[\mathfrak {D}^ \bot \] -parallel structure Jacobi operator for a real hypersurface in a complex two-plane Grassmannian G 2(ℂm+2) and proved that a real hypersurface in G 2(ℂm+2) with generalized Tanaka-Webster \[\mathfrak {D}^ \bot \] -parallel structure Jacobi operator is locally congruent to an open part of a tube around a totally geodesic quaternionic projective space ℍP n in G 2(ℂm+2), where m = 2n.
LA - eng
KW - Complex two-plane Grassmannian; Hopf hypersurface; Generalized Tanaka-Webster connection; Structure Jacobi operator; complex two-plane Grassmannian; generalized Tanaka-Webster connection; structure Jacobi operator
UR - http://eudml.org/doc/269674
ER -

References

top
  1. [1] Alekseevskii D. V., Compact quaternion spaces, Funct. Anal. Appl., 1968, 2, 11–20 
  2. [2] Berndt J., Riemannian geometry of complex two-plane Grassmannian, Rend. Sem. Mat. Univ. Politec. Torino, 1997, 55, 19–83 Zbl0909.53038
  3. [3] Berndt J. and Suh Y. J., Real hypersurfaces in complex two-plane Grassmannians, Monatsh. Math., 1999, 127, 1–14 http://dx.doi.org/10.1007/s006050050018 Zbl0920.53016
  4. [4] Berndt J. and Suh Y. J., Real hypersurfaces with isometric Reeb flow in complex two-plane Grassmannians, Monatsh. Math., 2002, 137, 87–98 http://dx.doi.org/10.1007/s00605-001-0494-4 Zbl1015.53034
  5. [5] Jeong I., Pérez J. D. and Suh Y. J., Real hypersurfaces in complex two-plane Grassmannians with parallel structure Jacobi operator, Acta Math. Hungar., 2009, 122, 173–186 http://dx.doi.org/10.1007/s10474-008-8004-y Zbl1265.53057
  6. [6] Jeong I., Machado C. J. G., Pérez J. D. and Suh Y. J., Real hypersurfaces in complex two-plane Grassmannians with 𝔇 ⊥ -parallel structure Jacobi operator, Internat. J. Math., 2011, 22, 655–673 http://dx.doi.org/10.1142/S0129167X11006957 Zbl1219.53054
  7. [7] Ki U-H., Pérez J. D., Santos F. G. and Suh Y. J., Real hypersurfaces in complex space forms with ξ-parallel Ricci tensor and structure Jacobi operator, J. Korean Math. Soc., 2007, 44, 307–326 http://dx.doi.org/10.4134/JKMS.2007.44.2.307 Zbl1144.53069
  8. [8] Kon M., Real hypersurfaces in complex space forms and the generalized-Tanaka-Webster connection, Proceeding of the 13th International Workshop on Differential Geometry anad Related Fields (5–7 Nov. 2009 Taegu Republic of Korea), National Institute of Mathematical Sciences, 2009, 145–159 
  9. [9] Lee H. and Suh Y. J., Real hypersurfaces of type B in complex two-plane Grassmannians related to the Reeb vector, Bull. Korean Math. Soc., 2010, 47, 551–561 http://dx.doi.org/10.4134/BKMS.2010.47.3.551 Zbl1206.53064
  10. [10] Pak E. and Suh Y. J., Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster Reeb parallel structure Jacobi operator, (Submitted) Zbl1296.53120
  11. [11] Pérez J. D., Santos F. G. and Suh Y. J., Real hypersurfaces in complex projective space whose structure Jacobi operator is D-parallel, Bull. Belg. Math. Soc. Simon Stevin, 2006, 13, 459–469 Zbl1130.53039
  12. [12] Pérez J. D. and Suh Y. J., Real hypersurfaces of quaternionic projective space satisfying ∇ U t R = 0 , Differential Geom. Appl., 1997, 7, 211–217 http://dx.doi.org/10.1016/S0926-2245(97)00003-X 
  13. [13] Pérez J. D. and Suh Y. J., The Ricci tensor of real hypersurfaces in complex two-plane Grassmannians, J. Korean Math. Soc., 2007, 44, 211–235 http://dx.doi.org/10.4134/JKMS.2007.44.1.211 Zbl1156.53034
  14. [14] Pérez J. D., Suh Y. J. and Watanabe Y., Generalized Einstein real hypersurfaces in complex two-plane Grassmannians, J. Geom. Phys., 2010, 60, 1806–1818 http://dx.doi.org/10.1016/j.geomphys.2010.06.017 Zbl1197.53071
  15. [15] Suh Y. J., Real hypersurfaces in complex two-plane Grassmannians with ξ-invariant Ricci tensor, J. Geom. Phys., 2011, 61, 808–814 http://dx.doi.org/10.1016/j.geomphys.2010.12.010 Zbl1209.53046
  16. [16] Suh Y. J., Real hypersurfaces in complex two-plane Grassmannians with parallel Ricci tensor, Proc. Roy. Soc. Edinburgh Sect. A., 2012, 142, 1309–1324 http://dx.doi.org/10.1017/S0308210510001472 Zbl1293.53071
  17. [17] Suh Y. J., Real hypersurfaces in complex two-plane Grassmannians with Reeb parallel Ricci tensor, J. Geom. Phys., 2013, 64, 1–11 http://dx.doi.org/10.1016/j.geomphys.2012.10.005 Zbl1259.53052
  18. [18] Suh Y. J., Real hypersurfaces in complex two-plane Grassmannians with harmonic curvature, J. Math. Pures Appl., 2013, 100, 16–33 http://dx.doi.org/10.1016/j.matpur.2012.10.010 Zbl1279.53052
  19. [19] Tanaka N., On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Jpn. J. Math., 1976, 2, 131–190 Zbl0346.32010
  20. [20] Tanno S., Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc., 1989, 314, 349–379 http://dx.doi.org/10.1090/S0002-9947-1989-1000553-9 Zbl0677.53043
  21. [21] Webster S.M., Pseudo-Hermitian structures on a real hypersurface, J. Differential Geom., 1978, 13, 25–41 Zbl0379.53016

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.