Geodesic
Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster parallel normal Jacobi operator
Czechoslovak Mathematical Journal, Tome 65 (2015) no. 1, pp. 207-218 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

We study the classifying problem of immersed submanifolds in Hermitian symmetric spaces. Typically in this paper, we deal with real hypersurfaces in a complex two-plane Grassmannian $G_2({\mathbb C}^{m+2})$ which has a remarkable geometric structure as a Hermitian symmetric space of rank 2. In relation to the generalized Tanaka-Webster connection, we consider a new concept of the parallel normal Jacobi operator for real hypersurfaces in $G_2({\mathbb C}^{m+2})$ and prove non-existence of real hypersurfaces in $G_2({\mathbb C}^{m+2})$ with generalized Tanaka-Webster parallel normal Jacobi operator.
We study the classifying problem of immersed submanifolds in Hermitian symmetric spaces. Typically in this paper, we deal with real hypersurfaces in a complex two-plane Grassmannian $G_2({\mathbb C}^{m+2})$ which has a remarkable geometric structure as a Hermitian symmetric space of rank 2. In relation to the generalized Tanaka-Webster connection, we consider a new concept of the parallel normal Jacobi operator for real hypersurfaces in $G_2({\mathbb C}^{m+2})$ and prove non-existence of real hypersurfaces in $G_2({\mathbb C}^{m+2})$ with generalized Tanaka-Webster parallel normal Jacobi operator.
DOI : 10.1007/s10587-015-0169-2
Classification : 53C15, 53C40
Keywords: real hypersurface; complex two-plane Grassmannian; Hopf hypersurface; generalized Tanaka-Webster connection; normal Jacobi operator; generalized Tanaka-Webster parallel normal Jacobi operator 
@article{10_1007_s10587_015_0169_2,
     author = {Pak, Eunmi and de Dios P\'erez, Juan and Machado, Carlos J. G. and Woo, Changhwa},
     title = {Hopf hypersurfaces in complex two-plane {Grassmannians} with generalized {Tanaka-Webster} parallel normal {Jacobi} operator},
     journal = {Czechoslovak Mathematical Journal},
     pages = {207--218},
     year = {2015},
     volume = {65},
     number = {1},
     doi = {10.1007/s10587-015-0169-2},
     mrnumber = {3336034},
     zbl = {06433730},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-015-0169-2/}
}
TY  - JOUR
AU  - Pak, Eunmi
AU  - de Dios Pérez, Juan
AU  - Machado, Carlos J. G.
AU  - Woo, Changhwa
TI  - Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster parallel normal Jacobi operator
JO  - Czechoslovak Mathematical Journal
PY  - 2015
SP  - 207
EP  - 218
VL  - 65
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-015-0169-2/
DO  - 10.1007/s10587-015-0169-2
LA  - en
ID  - 10_1007_s10587_015_0169_2
ER  - 
%0 Journal Article
%A Pak, Eunmi
%A de Dios Pérez, Juan
%A Machado, Carlos J. G.
%A Woo, Changhwa
%T Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster parallel normal Jacobi operator
%J Czechoslovak Mathematical Journal
%D 2015
%P 207-218
%V 65
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-015-0169-2/
%R 10.1007/s10587-015-0169-2
%G en
%F 10_1007_s10587_015_0169_2
Pak, Eunmi; de Dios Pérez, Juan; Machado, Carlos J. G.; Woo, Changhwa. Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster parallel normal Jacobi operator. Czechoslovak Mathematical Journal, Tome 65 (2015) no. 1, pp. 207-218. doi: 10.1007/s10587-015-0169-2

[1] Alekseevskij, D. V.: Compact quaternion spaces. Funkts. Anal. Prilozh. 2 (1968), 11-20 Russian. | MR | Zbl

[2] Berndt, J., Suh, Y. J.: Real hypersurfaces with isometric Reeb flow in complex two-plane Grassmannians. Monatsh. Math. 137 (2002), 87-98. | DOI | MR | Zbl

[3] Berndt, J., Suh, Y. J.: Real hypersurfaces in complex two-plane Grassmannians. Monatsh. Math. 127 (1999), 1-14. | DOI | MR | Zbl

[4] Cho, J. T.: Levi-parallel hypersurfaces in a complex space form. Tsukuba J. Math. 30 (2006), 329-343. | DOI | MR | Zbl

[5] Cho, J. T.: CR structures on real hypersurfaces of a complex space form. Publ. Math. 54 (1999), 473-487. | MR | Zbl

[6] Pérez, J. de Dios, Jeong, I., Suh, Y. J.: Real hypersurfaces in complex two-plane Grassmannians with commuting normal Jacobi operator. Acta Math. Hung. 117 (2007), 201-217. | DOI | MR

[7] Pérez, J. de Dios, Suh, Y. J.: Real hypersurfaces of quaternionic projective space satisfying $\nabla_{U_i} R = 0$. Differ. Geom. Appl. 7 (1997), 211-217. | DOI | MR

[8] Jeong, I., Kim, H. J., Suh, Y. J.: Real hypersurfaces in complex two-plane Grassmannians with parallel normal Jacobi operator. Publ. Math. 76 (2010), 203-218. | MR | Zbl

[9] Jeong, I., Kimura, M., Lee, H., Suh, Y. J.: Real hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster Reeb parallel shape operator. Monatsh. Math. 171 (2013), 357-376. | DOI | MR | Zbl

[10] Jeong, I., Suh, Y. J.: Real hypersurfaces in complex two-plane Grassmannians with $\frak F$-parallel normal Jacobi operator. Kyungpook Math. J. 51 (2011), 395-410. | DOI | MR | Zbl

[11] 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 (2010), 551-561. | DOI | MR | Zbl

[12] Tanaka, N.: On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections. Jap. J. Math., new Ser. 2 (1976), 131-190. | DOI | MR | Zbl

[13] Tanno, S.: Variational problems on contact Riemannian manifolds. Trans. Am. Math. Soc. 314 (1989), 349-379. | DOI | MR | Zbl

[14] Webster, S. M.: Pseudo-Hermitian structures on a real hypersurface. J. Differ. Geom. 13 (1978), 25-41. | DOI | MR | Zbl

Cité par Sources :