COMMUTING STRUCTURE JACOBI OPERATOR FOR HOPF HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS

Title & Authors
COMMUTING STRUCTURE JACOBI OPERATOR FOR HOPF HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS
Jeong, Im-Soon; Suh, Young-Jin; Yang, Hae-Young;

Abstract
In this paper we give a non-existence theorem for Hopf real hypersurfaces in complex two-plane Grassmannians $\small{G_2(\mathbb{C}^{m+2})}$ satisfying the condition that the structure Jacobi operator $\small{R_{\xi}}$ commutes with the 3-structure tensors $\small{{\phi}_i}$, i
Keywords
real hypersurfaces;complex two-plane Grassmannians;commuting structure Jacobi operator;geodesic Reeb flow;
Language
English
Cited by
References
1.
D. V. Alekseevskii, Compact quaternion spaces, Funkcional. Anal. i Prilozen 2 (1968), no. 2, 11–20

2.
J. Berndt, Real hypersurfaces in quaternionic space forms, J. Reine Angew. Math. 419 (1991), 9–26

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.
U.-H. Ki, J. D. Perez, F. G. Santos, and Y. J. Suh, Real hypersurfaces in complex space forms with $\xi$-parallel Ricci tensor and structure Jacobi operator, J. Korean Math. Soc. 44 (2007), no. 2, 307–326

6.
M. Kimura, Real hypersurfaces and complex submanifolds in complex projective space, Trans. Amer. Math. Soc. 296 (1986), no. 1, 137–149

7.
J. D. P´erez and Y. J. Suh, Real hypersurfaces of quaternionic projective space satisfying $\nabla_U_i$R=0, Differential Geom. Appl. 7 (1997), no. 3, 211–217

8.
J. D. Perez, F. G. Santos, and Y. J. Suh, Real hypersurfaces in complex projective space whose structure Jacobi operator is Lie $\xi$-parallel, Differential Geom. Appl. 22 (2005), no. 2, 181–188

9.
J. D. Perez, F. G. Santos, Real hypersurfaces in complex projective space whose structure Jacobi operator is D-parallel, Bull. Belg. Math. Soc. Simon Stevin 13 (2006), no. 3, 459–469

10.
J. D. Perez and Y. J. Suh, The Ricci tensor of real hypersurfaces in complex two-plane Grassmannians, J. Korean Math. Soc. 44 (2007), no. 1, 211–235

11.
Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with parallel shape operator, Bull. Austral. Math. Soc. 67 (2003), no. 3, 493–502

12.
Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with commuting shape operator, Bull. Austral. Math. Soc. 68 (2003), no. 3, 379–393

13.
Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with parallel shape operator. II, J. Korean Math. Soc. 41 (2004), no. 3, 535–565

14.
Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with vanishing Lie derivative, Canad. Math. Bull. 49 (2006), no. 1, 134–143

15.
Y. J. Suh, Real hypersurfaces of type B in complex two-plane Grassmannians, Monatsh. Math. 147 (2006), no. 4, 337–355