Submanifolds of Codimension 3 in a Complex Space Form with Commuting Structure Jacobi Operator
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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) for c 6= 0. We denote byS and Rξ be the Ricci tensor of M and the structure Jacobi operator in the direction ofthe structure vector ξ, respectively. Suppose that the third fundamental form t satisfiesdt(X, Y ) = 2θg(φX, Y ) for a certain scalar θ 6= 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 Sξ = g(Sξ, ξ)ξ,then M is a real hypersurface in Mn(c) (⊂ Mn+1(c)) provided that ¯r − 2(n − 1)c ≤ 0,where ¯r denotes the scalar curvature of M.
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) for c 6= 0. We denote byS and Rξ be the Ricci tensor of M and the structure Jacobi operator in the direction ofthe structure vector ξ, respectively. Suppose that the third fundamental form t satisfiesdt(X, Y ) = 2θg(φX, Y ) for a certain scalar θ 6= 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 Sξ = g(Sξ, ξ)ξ,then M is a real hypersurface in Mn(c) (⊂ Mn+1(c)) provided that ¯r − 2(n − 1)c ≤ 0,where ¯r denotes the scalar curvature of M.
참고문헌 (25)
- A. Bejancu, "CR-submanifolds of a Kahler manifold I", Proc. Amer. Math. Soc, 69, pp.135~142 (1978)
- J. Berndt, "Real hypersurfaces with constant principal curvatures in a complex hyper-bolic space", J. Reine Angew. Math, 395, pp.132~141 (1989)
- J. Berndt, "Cohomogeneity one actions on non compact symmetric space of rank one", Trans. Amer. Math. Soc, 359, pp.3425~3438 (2007)
- D. E. Blair, "Semi-invariant immersion", Kodai Math. Sem. Rep, 27, pp.313~319 (1976)
- T. E. Cecil, "Focal sets and real hypersurfaces in complex projective space", Trans. Amer. Math. Soc, 269, pp.481~499 (1982)
- T. E. Cecil, "Geometry of Hypersurfaces" (2015), Springer
- J. T. Cho, "Real hypersurfaces of a complex projective space in terms of the Jacobi operators", Acta Math. Hungar, 80, pp.155~167 (1998)
- J. T. Cho, "Real hypersurfaces in complex space forms with Reeb-flows symmetric Jacobi operator", Canadian Math. Bull, 51, pp.359~371 (2008)
- J. Erbacher, "Reduction of the codimension of an isometric immersion", J. Differential Geom, 5, pp.333~340 (1971)
- 허종임, "Semi-invariant submanifolds of codimension 3 of a complex projective space in terms of the Jacobi operator", 대한수학회보, 42(1), pp.93~119 (2005), 대한수학회
- S. Kawamoto, "Codimension reduction for real submanifolds of a complex hyperbolic space", Rev. Mat. Univ. Complut. Madrid, 7, pp.119~128 (1994)
- U-H. Ki, "Jacobi operators on a semi-invariant submanifolds of codimen-sion 3 in a complex projective space", Nihonkai Math J, 14, pp.1~16 (2003)
- U-H. Ki, "Submanifolds of codimension 3 admitting almost contact metric structure in a complex projective space", Nihonkai Math J, 11, pp.57~86 (2000)
- M. Kimura, "Real hypersurfaces and complex submanifolds in complex projective space", Trans. Amer. Math. Soc, 296, pp.137~149 (1986)
- S. Montiel, "On some real hypersurfaces of a complex hyperbolic space", Geometriae Dedicata, 20, pp.245~261 (1986)
- R. Niebergall, "Tight and Taut submanifolds", pp.233~305 (1998), Cambridge University Press
- M. Okumura, "On some real hypersurfaces of a complex projective space", Trans. Amer. Math. Soc, 212, pp.355~364 (1973)
- M. Okumura, "Normal curvature and real submanifold of the complex projective space", Geom. Dedicata, 7, pp.509~517 (1978)
- M. Okumura, "Codimension reduction problem for real submanifolds of complex pro-jective space", Colloz. Math. Soc. Jámos Bolyai, 56, pp.574~585 (1989)
- H. Song, "Some differential-geometric properties of R-spaces", Tsukuba J. Math, 25, pp.279~298 (2001)
- R. Takagi, "On homogeneous real hypersurfaces in a complex projective space", Osaka J. Math, 19, pp.495~506 (1973)
- R. Takagi, "Real hypersurfaces in a complex projective space with constant principal curvatures I, II", J. Math. Soc. Japan, 27, pp.43~53 (1975)
- Y. Tashiro, "Relations between the theory of almost complex spaces and that of almost contact spaces", Sugaku, 16, pp.34~61 (1964)
- K. Yano, "On (f, g, u, v, w, λ, µ, ν)-structure satisfying λ2 + µ2 + ν2 = 1", Kodai Math. Sem. Rep, 56, pp.574~585 (1978)
- K. Yano, "CR submanifolds of Kaehlerian and Sasakian manifolds" (1983), Birkhauser