Browse > Article
http://dx.doi.org/10.7858/eamj.2022.033

COMMUTING STRUCTURE JACOBI OPERATOR FOR SEMI-INVARIANT SUBMANIFOLDS OF CODIMENSION 3 IN COMPLEX SPACE FORMS  

KI, U-Hang (The National Academy of Sciences)
SONG, Hyunjung (Department of Mathematics Hankuk University of Foreign Studies)
Publication Information
East Asian mathematical journal / v.38, no.5, 2022 , pp. 549-581 More about this Journal
Abstract
Let M be a semi-invariant submanifold with almost contact metric structure (𝜙, 𝜉, 𝜂, g) of codimension 3 in a complex space form Mn+1(c), c≠ 0. We denote by S and R𝜉 be the Ricci tensor of M and the structure Jacobi operator in the direction of the structure vector 𝜉, respectively. Suppose that the third fundamental form t satisfies dt(X, Y) = 2𝜃g(𝜙X, Y) for a certain scalar 𝜃(≠ 2c) and any vector fields X and Y on M. In this paper, we prove that M satisfies R𝜉S = SR𝜉 and at the same time R𝜉𝜙 = 𝜙R𝜉, then M is a Hopf hypersurface of type (A) provided that the scalar curvature s of M holds s - 2(n - 1)c ≤ 0.
Keywords
semi-invariant submanifold; distinguished normal; complex space form; structure Jacobi operator; Ricci tensor; scalar curvature; Hopf real hypersurfaces;
Citations & Related Records
연도 인용수 순위
  • Reference
1 U-H. Ki and S.J. Kim, Structure Jacobi operator of semi-invariant submanifolds in complex space forms, East Asian Math. J. 36 (2020), 380-415.
2 J. Berndt and H. Tamaru, Cohomogeneity one actions on non compact symmetric space of rank one, Trans. Amer. Math. Soc. 359(2007), 3425-3438.   DOI
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 H. Song, Some differential-geometric properties of R-spaces, Tsukuba J. Math. 25(2001), 279-298.
5 K. Yano and M. Kon, CR submanifolds of Kaehlerian and Sasakian manifolds, Birkhauser (1983).
6 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.
7 D. E. Blair, G. D. Ludden and K. Yano, Semi-invariant immersion, Kodai Math. Sem. Rep. 27(1976), 313-319.   DOI
8 U-H. Ki and H.J. Kim, Semi-invariant submanifolds with lift flat normal connection in a complex projective space, Kyungpook Math. J. 40(2000), 185-194.
9 U-H. Ki, S. Nagai and R. Takagi, The structure vector filed and structure Jacobi operator of real hypersurfaces in nonflat complex space forms, Geom. Dedicata 149(2010), 161-176.   DOI
10 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.
11 Y. Tashiro, Relations between the theory of almost complex spaces and that of almost contact spaces (in Japanese), Sugaku 16(1964), 34-61.
12 K. Yano and U-H. Ki, On (f, g, u, v, w, , λ, µ, ν)-structure satisfying λ2 + µ2 + ν2 = 1, Kodai Math. Sem. Rep. 29(1978), 285-307.   DOI
13 J. Berndt, Real hypersurfaces with constant principal curvatures in a complex hyperbolic space, J. Reine Angew. Math. 395(1989), 132-141.
14 M. Okumura, On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc. 212(1973), 355-364.   DOI
15 M. Okumura, Codimension reduction problem for real submanifolds of complex projective space. Colloq. Math Janos Bolyai. 56(1989), 574-585.
16 R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 19(1973), 495-506.
17 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.
18 A. Bejancu, CR-submanifolds of a Kahler manifold I, Proc. Amer. Math. Soc. 69(1978), 135-142.   DOI
19 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
20 T. E. Cecil and P. J. Ryan, Geometry of Hypersurfaces, Springer (2015).
21 J. T. Cho and U-H. Ki, Real hypersurfaces in complex space forms with Reeb-flows symmetric Jacobi operator, Canadian Math. Bull. 51(2008), 359-371.   DOI
22 U-H. Ki and H. Song, Semi-invariant submanifolds of condimension 3 in a complex space form with commuting structure Jacobi operator, Kyungpook Math. J. 62(2022), 133-166.
23 S. Kawamoto, Codimension reduction for real submanifolds of a complex hyperbolic space,Rev. Mat. Univ. Compul. Madrid 7(1994), 119-128.
24 U-H. Ki and H. Kurihara, Semi-invariant submanifolds of codimension 3 in a complex space form in terms of the structure Jacobi operators, Comm. Korean Math. Soc. 37(2022), 229-257   DOI
25 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.
26 M. Kimura, Real hypersurfaces and complex submanifolds in complex projective space, Trans. Amer. Math. Soc. 296(1986), 137-149.   DOI
27 S. Montiel and A.Romero, On some real hypersurfaces of a complex hyperbolic space, Geom. Dedicata 20(1986), 245-261.   DOI