Regarding the generalized Tanaka-Webster connection, we considered a new notion of -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 -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.
@article{bwmeta1.element.doi-10_2478_s11533-014-0447-5,
author = {Eunmi Pak and Young Suh},
title = {Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster \[\mathfrak {D}^ \bot \]
-parallel structure Jacobi operator},
journal = {Open Mathematics},
volume = {12},
year = {2014},
pages = {1840-1851},
zbl = {1296.53120},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-014-0447-5}
}
Eunmi Pak; Young Suh. Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster \[\mathfrak {D}^ \bot \]
-parallel structure Jacobi operator. Open Mathematics, Tome 12 (2014) pp. 1840-1851. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-014-0447-5/
[1] Alekseevskii D. V., Compact quaternion spaces, Funct. Anal. Appl., 1968, 2, 11–20
[2] Berndt J., Riemannian geometry of complex two-plane Grassmannian, Rend. Sem. Mat. Univ. Politec. Torino, 1997, 55, 19–83 | Zbl 0909.53038
[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 | Zbl 0920.53016
[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 | Zbl 1015.53034
[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 | Zbl 1265.53057
[6] Jeong I., Machado C. J. G., Pérez J. D. and Suh Y. J., Real hypersurfaces in complex two-plane Grassmannians with -parallel structure Jacobi operator, Internat. J. Math., 2011, 22, 655–673 http://dx.doi.org/10.1142/S0129167X11006957 | Zbl 1219.53054
[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 | Zbl 1144.53069
[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] 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 | Zbl 1206.53064
[10] Pak E. and Suh Y. J., Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster Reeb parallel structure Jacobi operator, (Submitted) | Zbl 1296.53120
[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 | Zbl 1130.53039
[12] Pérez J. D. and Suh Y. J., Real hypersurfaces of quaternionic projective space satisfying , Differential Geom. Appl., 1997, 7, 211–217 http://dx.doi.org/10.1016/S0926-2245(97)00003-X
[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 | Zbl 1156.53034
[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 | Zbl 1197.53071
[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 | Zbl 1209.53046
[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 | Zbl 1293.53071
[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 | Zbl 1259.53052
[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 | Zbl 1279.53052
[19] Tanaka N., On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Jpn. J. Math., 1976, 2, 131–190 | Zbl 0346.32010
[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 | Zbl 0677.53043
[21] Webster S.M., Pseudo-Hermitian structures on a real hypersurface, J. Differential Geom., 1978, 13, 25–41 | Zbl 0379.53016