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http://dx.doi.org/10.7858/eamj.2020.027

STRUCTURE JACOBI OPERATOR OF SEMI-INVARINAT SUBMANIFOLDS IN COMPLEX SPACE FORMS  

KI, U-HANG (The National Academy of Sciences)
KIM, SOO JIN (Department of Mathematics Chosun University)
Publication Information
East Asian mathematical journal / v.36, no.3, 2020 , pp. 389-415 More about this Journal
Abstract
Let M be a semi-invariant submanifold of codimension 3 with almost contact metric structure (𝜙, ξ, η, g) in a complex space form Mn+1(c), c ≠ 0. We denote by Rξ and R'X be the structure Jacobi operator with respect to the structure vector ξ and be R'X = (∇XR)(·, X)X for any unit vector field X on M, respectively. Suppose that the third fundamental form t satisfies dt(X, Y) = 2𝜃g(𝜙X, Y) for a scalar 𝜃(≠ 2c) and any vector fields X and Y on M. In this paper, we prove that if it satisfies Rξ𝜙 = 𝜙Rξ and at the same time R'ξ = 0, then M is a Hopf real hypersurfaces of type (A), provided that the scalar curvature ${\bar{r}}$ of M holds ${\bar{r}}-2(n-1)c{\leq}0$.
Keywords
semi-invariant submanifold; distinguished normal; complex space form; Hopf real hypersurfaces; structure Jacobi operator; scalar curvature;
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1 T. E. Cecil and P. J. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans. Amer. Math. Soc. 269(1982), 481-499.   DOI
2 T. E. Cecil and P. J. Ryan, Geometry of Hypersurfaces, Springer (2015).
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.   DOI
4 U-H. Ki, Cyclic-parallel real hypersurfaces of a complex space form, Tsukuba J. Math. 12(1988), 259-268.   DOI
5 J. T. Cho and U-H. Ki, Jacobi operators on real hypersurfaces of a complex projective space, Tsukuha J. Math. 22(1997), 145-156.
6 J. Erbacher, Reduction of the codimension of an isometric immersion, J. Diff. Geom. 3(1971), 333-340.
7 J. I. Her, U-H. Ki and S.-B. Lee, Semi-invariant submanifolds of codimension 3 of a complex projective space in terms of the Jacobi operator, Bull. Korean Math. Soc. 42(2005), 93-119.   DOI
8 U-H. Ki, H. Kurihara and H. Song, Semi-invariant submanifolds of codimension 3 in complex space forms in terms of the structure Jacobi operator, to appear.
9 U-H. Ki and H. Song, Jacobi operators on a semi-invariant submanifolds of codimension 3 in a complex projective space, Nihonkai Math J. 14(2003), 1-16.
10 U-H. Ki and H. Song, Submanifolds of codimension 3 in a complex space form with commuting structure Jacobi operator, to appear.
11 U-H. Ki, H. Song and R. Takagi, Submanifolds of codimension 3 admitting almost contact metric structure in a complex projective space, Nihonkai Math J. 11(2000), 57-86.
12 U-H. Ki, Soo Jin Kim and S.-B. Lee, The structure Jacobi operator on real hypersurfaces in a nonflat complex space form, Bull. Korean Math. Soc. 42(2005), 337-358.   DOI
13 M. Kimura, Real hypersurfaces and complex submanifolds in complex projective space, Trans. Amer. Math. Soc. 296(1986), 137-149.   DOI
14 S. Montiel and A.Romero, On some real hypersurfaces of a complex hyperbolic space, Geom. Dedicata 20(1986), 245-261.   DOI
15 H. Song, Some differential-geometric properties of R-spaces, Tsukuba J. Math. 25(2001), 279-298.   DOI
16 R. Niebergall and P. J. Ryan, Real hypersurfaces in complex space form, in Tight and Taut submanifolds, Cambridge University Press : (1998(T. E. Cecil and S.-S. Chern eds.)), 233-305.
17 M. Okumura, On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc. 212(1973), 355-364.   DOI
18 M. Okumura, Normal curvature and real submanifold of the complex projective space, Geom. Dedicata 7(1978), 509-517.   DOI
19 R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 19(1973), 495-506.
20 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.   DOI
21 Y. Tashiro, Relations between the theory of almost complex spaces and that of almost contact spaces (in Japanese), Sugaku 16(1964), 34-61.
22 K. Yano, and U-H. Ki, On (f, g, u, v, w, ${\lambda}$, ${\mu}$, ${\nu}$)-structure satisfying ${\lambda}^2+{\mu}^2+{\nu}^2$ = 1, Kodai Math. Sem. Rep. 29(1978), 285-307.   DOI
23 K. Yano and M. Kon, CR submanifolds of Kaehlerian and Sasakian manifolds, Birkhauser (1983).
24 D. E. Blair, G. D. Ludden and K. Yano, Semi-invariant immersion, Kodai Math. Sem. Rep. 27(1976), 313-319.   DOI
25 A. Bejancu, CR-submanifolds of a Kahler manifold I, Proc. Amer. Math. Soc. 69(1978), 135-142.   DOI
26 J. Berndt, Real hypersurfaces with constant principal curvatures in a complex hyperbolic space, J. Reine Angew. Math. 395(1989), 132-141.
27 J. Berndt and L. Vanhecke, Two natural generalizations of locally symmetric spaces, Diff. Geom. Appl. 2(1992), 57-82.   DOI