Jacobi Operators with Respect to the Reeb Vector Fields on Real Hypersurfaces in a Nonflat Complex Space Form

• Journal title : Kyungpook mathematical journal
• Volume 56, Issue 2,  2016, pp.541-575
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2016.56.2.541
Title & Authors
Jacobi Operators with Respect to the Reeb Vector Fields on Real Hypersurfaces in a Nonflat Complex Space Form
Ki, U-Hang; Kim, Soo Jin; Kurihara, Hiroyuki;

Abstract
Let M be a real hypersurface of a complex space form with almost contact metric structure ($\small{{\phi}}$, $\small{{\xi}}$, $\small{{\eta}}$, g). In this paper, we prove that if the structure Jacobi operator $R_{\xi} Keywords complex space form;real hypersurface;structure Jacobi operator;structure tensor;Ricci tensor; Language English Cited by 1. Structure Jacobi Operators of Real Hypersurfaces with Constant Mean Curvature in a Complex Space Form, Kyungpook mathematical journal, 2016, 56, 4, 1207 References 1. J. Berndt, Real hypersurfaces with constant principal curvatures in complex hyperblic spaces, J. Reine Angew. Math., 395(1989), 132-141. 2. T. E. Cecil and P. J. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans. Amer. Math. Soc., 269(1982), 481-499. 3. J. T. Cho and U-H. Ki, Real hypersurfaces in complex projective spaces in terms of Jacobi operators, Acta Math. Hungar., 80(1998), 155-167. 4. J. T. Cho and U-H. Ki, Real hypersurfaces in complex space form with Reeb flow symmetric Jacobi operator, Canadian Math. Bull., 51(2008), 359-371. 5. U-H. Ki, I. -B. Kim and D. H. Lim, Characterizations of real hypersurfaces of type A in a complex space form, Bull. Korean Math. Soc., 47(2010), 1-15. 6. U-H. Ki and H. Kurihara, Real hypersurfaces and${\xi}\$-parallel structure Jacobi operators in complex space forms, J. Korean Academy Sciences, Sciences Series, 48(2009), 53-78.

7.
U-H. Ki, H. Kurihara, S. Nagai and R. Takagi, Characterizations of real hypersurfaces of type A in a complex space form in terms of the structure Jacobi operator, Toyama Math. J., 32(2009), 5-23.

8.
U-H. Ki, H. Kurihara and R. Takagi, Jacobi operators along the structure flow on real hypersurfaces in a nonflat complex space form, Tsukuba J. Math., 33(2009), 39-56.

9.
U-H. Ki, S. Nagai and R. Takagi, The structure vector field and structure Jacobi operator of real hypersurfaces in non at complex space forms, Geom. Dedicata, 149(2010), 161-176.

10.
U-H. Ki and Y. J. Suh, On real hypersurfaces of a complex space form, Math J. Okayama Univ., 32(1990), 207-221.

11.
M. Kimura, Real hypersurfaces and complex submanifolds in complex projective space, Trans. Amer. Math. Soc., 296(1986), 137-149.

12.
S. Montiel and A. Romero, On some real hypersurfaces of a complex hyperblic space, Geom Dedicata, 20(1986), 245-261.

13.
M. Okumura, On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc., 212(1975), 355-364.

14.
M. Ortega, J. D. Perez and F. G. Santos, Non-existence of real hypersurfaces with parallel structure Jacobi operator in nonflat complex space forms, Rocky Mountain J. Math., 36(2006), 1603-1613.

15.
J. D. Perez, F. G. Santos and Y. J. Suh Real hypersurfaces in complex projective spaces whose structure Jacobi operator is D-parallel, Bull. Belg. Math. Soc., 13(2006), 459-469.

16.
J. D. Perez, F. G. Santos and Y. J. Suh Real hypersurfaces in nonflat complex space forms with commuting structure Jacobi operator, Houston J. Math., 33(2007), 1005-1009.

17.
R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math., 19(1973), 495-506.

18.
R. Takagi, Real hypersurfaces in a complex projective space with constant principal curvatures I, II, J. Math. Soc., 15(1975), 43-53, 507-516.