THE STRUCTURE JACOBI OPERATOR ON REAL HYPERSURFACES IN A NONFLAT COMPLEX SPACE FORM

Title & Authors
THE STRUCTURE JACOBI OPERATOR ON REAL HYPERSURFACES IN A NONFLAT COMPLEX SPACE FORM
KI, U-HANG; KIM, SOO-JIN; LEE, SEONG-BAEK;

Abstract
Let M be a real hypersurface with almost contact metric structure $\small{(\phi,\;\xi,\;\eta,\;g)}$ in a nonflat complex space form $\small{M_n(c)}$. In this paper, we prove that if the structure Jacobi operator $\small{R_\xi}$ commutes with both the structure tensor $\small{\phi}$ and the Ricc tensor S, then M is a Hopf hypersurface in $\small{M_n(c)}$ provided that the mean curvature of M is constant or $\small{g(S\xi,\;\xi)}$ is constant.
Keywords
structure Jacobi operator;Ricci tensor;Hopf hypersurface;nonflat complex space form;
Language
English
Cited by
1.
ON THE STRUCTURE JACOBI OPERATOR AND RICCI TENSOR OF REAL HYPERSURFACES IN NONFLAT COMPLEX SPACE FORMS,;

호남수학학술지, 2010. vol.32. 4, pp.747-761
1.
The Ricci tensor and structure Jacobi operator of real hypersurfaces in a complex projective space, Journal of Geometry, 2009, 94, 1-2, 123
2.
ON THE STRUCTURE JACOBI OPERATOR AND RICCI TENSOR OF REAL HYPERSURFACES IN NONFLAT COMPLEX SPACE FORMS, Honam Mathematical Journal, 2010, 32, 4, 747
3.
Real Hypersurfaces of Nonflat Complex Projective Planes Whose Jacobi Structure Operator Satisfies a Generalized Commutative Condition, International Journal of Mathematics and Mathematical Sciences, 2016, 2016, 1
References
1.
J. Berndt, Real hypersurfaces with constant principal curvatures in a complex hyperbolic space, J. Reine Agnew. 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 of a complex projective space in terms of the Jacobi operators, Acta Math. Hungar. 80 (1998), 155-167

4.
J. T. Cho, Jacobi operators on real hypersurfaces of a complex projective space, Tsukuba J. Math. 22 (1998), 145-156

5.
U.-H. Ki, Cyclic-parallel real hypersurfaces of a complex space form, Tsukuba J. Math. 12 (1988), 259-268

6.
U.-H. Ki, H.-J. Kim, and A.-A. Lee, The Jacobi operator of real hypersurfaces in a complex space form, Comm. Korean Math. Soc. 13 (1998), 545-560

7.
U.-H. Ki, S.-J. Kim, and S.-B. Lee, Some characterizations of a real hypersurfaces of type A, Kyungpook Math. J. 31 (1991), 73-82

8.
U.-H. Ki, A. A. Lee, and S.-B. Lee, On real hypersurfaces of a complex space form in terms of Jacobi operators, Comm. Korean Math. Soc. 13 (1998), 317-336

9.
U.-H. Ki and H. Song, Jacobi operators on a semi-invariant submanifold of codimension 3 in a complex projective space, Nihonkai Math. J. 14 (2003), 1-16

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.
M. Kimura and S. Maeda, Lie derivatives on real hypersurfaces in a complex projective space, Czechoslovak Math. J. 45 (1995), 135-148

13.
S. Montiel, Real hypersurfaces of a complex hyperbolic space, J. Math. Soc. Japan 37 (1985), 515-535

14.
S. Montiel and A. Romero, On some real hypersurfaces of a complex hyperbolic space, Geometriae Dedicata 20 (1986), 245-261

15.
R. Niebergall and P. J. Ryan, Real hypersurfaces in complex space forms, in Tight and Taut submanifolds, Cambridge Univ. Press (1998(T.E. Cecil and S.S. Chern, eds.)), 233-305

16.
M. Okumura, On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc. 212 (1975), 355-364

17.
R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 10 (1973), 495-506

18.
R. Takagi, Real hypersurfaces in a complex projective space with constant principal curvatures I, II, J. Math. Soc. Japan 27 (1975), 43-53, 507-516

19.
K. Yano and M. Kon, CR submanifolds of Kaehlerian and Sasakian manifolds, Birkhauser, 1983