RECURRENT JACOBI OPERATOR OF REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS

Title & Authors
RECURRENT JACOBI OPERATOR OF REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS
Jeong, Imsoon; Perez, Juan De Dios; Suh, Young Jin;

Abstract
In this paper we give a non-existence theorem for Hopf hypersurfaces in the complex two-plane Grassmannian $\small{G_2({\mathbb{C}}^{m+2})}$ with re-current normal Jacobi operator $\small{{\bar{R}}_N}$.
Keywords
real hypersurfaces;complex two-plane Grassmannians;recurrent normal Jacobi operator;distribution;Hopf hypersurfaces;
Language
English
Cited by
1.
Real Hypersurfaces in Complex Hyperbolic Two-Plane Grassmannians with Commuting Structure Jacobi Operators, Mediterranean Journal of Mathematics, 2016, 13, 5, 3389
2.
Real hypersurfaces in complex two-plane Grassmannians with recurrent Ricci tensor, International Journal of Geometric Methods in Modern Physics, 2015, 12, 09, 1550086
3.
Real hypersurfaces in the complex quadric with parallel normal Jacobi operator, Mathematische Nachrichten, 2017, 290, 2-3, 442
References
1.
J. Berndt, Real hypersurfaces in quaternionic space forms, J. Reine Angew. Math. 419 (1991), 9-26.

2.
J. Berndt, Riemannian geometry of complex two-plane Grassmannians, Rend. Sem. Mat. Univ. Politec. Torino 55 (1997), no. 1, 19-83.

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. Hamada, On real hypersurfaces of a complex projective space with recurrent second fundamental tensor, J. Ramanujan Math. Soc. 11 (1996), no. 2, 103-107.

6.
T. Hamada, On real hypersurfaces of a complex projective space with recurrent Ricci tensor, Glasg. Math. J. 41 (1999), no. 3, 297-302.

7.
T. Hamada, Note on real hypersurfaces of complex space forms with recurrent Ricci tensor, Differ. Geom. Dyn. Syst. 5 (2003), no. 1, 27-30.

8.
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.

9.
I. Jeong, H. J. Kim, and Y. J. Suh, Real hypersurfaces in complex two-plane Grassman-nians with parallel normal Jacobi operator, Publ. Math. Debrecen 76 (2010), no. 1-2, 203-218.

10.
I. Jeong, H. Lee, and Y. J. Suh, Hopf hypersurfaces in complex two-plane Grassmannians with Lie parallel normal Jacobi operator, Bull. Korean Math. Soc. 48 (2011), no. 2, 427-444.

11.
I. Jeong, J. D. Perez, and Y. J. Suh, Real hypersurfaces in complex two-plane Grass-mannians with commuting normal Jacobi operator, Acta Math. Hungar. 117 (2007), no. 3, 201-217.

12.
S. Kim, H. Lee, and H. Y. Yang, Real hypersurfaces in complex two-plane Grassman-nians with recurrent shape operator, Bull. Malays. Math. Sci. Soc. (2) 34 (2011), no. 2, 295-305.

13.
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol. I, Wiley Classics Library Edition Publ., 1996.

14.
H. Lee and Y. J. Suh, Real hypersurfaces of type B in complex two-plane Grassmannians related to the Reeb vector, Bull. Korean Math. Soc. 47 (2010), no. 3, 551-561.

15.
J. D. Perez and F. G. Santos, Real hypersurfaces in complex projective space with re-current structure Jacobi operator, Differential Geom. Appl. 26 (2008), no. 2, 218-223.