• East Asian mathematical journal
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• v.17 no.1
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• pp.103-114
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• 2001

When the target manifold is certain Kenmotsu manifolds, we characterize invariant immersions, tangential anti-invariant immersions and normal anti-invariant immersions by the spectra of the Jacobi operator.

• East Asian mathematical journal
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• v.40 no.1
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• pp.1-23
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• 2024

Let M be a semi-invariant submanifold of codimension 3 with almost contact metric structure (𝜙, 𝜉, 𝜂, g) in a complex space form M_n+1(c). We denote by A, K and L the second fundamental forms with respect to the unit normal vector C, D and E respectively, where C is the distinguished normal vector, and by R_𝜉 = R(𝜉, ·)𝜉 the structure Jacobi operator. 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 , and at the same time R_𝜉K = KR_𝜉 and ∇_{𝜙∇𝜉𝜉}R_𝜉 = 0. In this paper, we prove that if it satisfies ∇_𝜉R_𝜉 = 0 on M, then M is a real hypersurface of type (A) in M_n(c) provided that the scalar curvature $\bar{r}$ of M holds $\bar{r}-2(n-1)c{\leq}0$.

• East Asian mathematical journal
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• v.36 no.3
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• pp.389-415
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• 2020

Let M be a semi-invariant submanifold of codimension 3 with almost contact metric structure (𝜙, ξ, η, g) in a complex space form M_n+1(c), c ≠ 0. We denote by R_ξ 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 R_ξ𝜙 = 𝜙R_ξ 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 ${\bar{r}}-2(n-1)c{\leq}0$.

• East Asian mathematical journal
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• v.38 no.5
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• pp.549-581
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• 2022

Let M be a semi-invariant submanifold with almost contact metric structure (𝜙, 𝜉, 𝜂, g) of codimension 3 in a complex space form M_n+1(c), c≠ 0. We denote by S and R_𝜉 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 R_𝜉S = SR_𝜉 and at the same time R_𝜉𝜙 = 𝜙R_𝜉, then M is a Hopf hypersurface of type (A) provided that the scalar curvature s of M holds s - 2(n - 1)c ≤ 0.

• Kyungpook Mathematical Journal
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• v.62 no.1
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• pp.133-166
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• 2022

Let M be a semi-invariant submanifold with almost contact metric structure (𝜙, 𝜉, 𝜂, g) of codimension 3 in a complex space form M_n+1(c) for c ≠ 0. We denote by S and R_𝜉 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 if it satisfies R_𝜉𝜙 = 𝜙R_𝜉 and at the same time S𝜉 = g(S𝜉, 𝜉)𝜉, then M is a real hypersurface in M_n(c) (⊂ M_n+1(c)) provided that $\bar{r}-2(n-1)c{\leq}0$, where $\bar{r}$ denotes the scalar curvature of M.

• Communications of the Korean Mathematical Society
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• v.12 no.2
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• pp.373-382
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• 1997

When the target manifold is a Sasakian or cosympletic space form, we characterize invariant immersions, tangential anti-invariant immersions and normal anti-invariant immersions by the spectra of the Jacobi operator.

• East Asian mathematical journal
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• v.37 no.1
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• pp.41-77
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• 2021

Let M be a semi-invariant submanifold of codimension 3 with almost contact metric structure (��, ξ, η, g) in a complex space form Mn+1(c), c ≠ 0. We denote by Rξ = R(·, ξ)ξ and A(i) be Jacobi operator with respect to the structure vector field ξ and be the second fundamental form in the direction of the unit normal C(i), respectively. Suppose that the third fundamental form t satisfies dt(X, Y ) = 2��g(��X, Y ) for certain scalar ��(≠ 2c)and any vector fields X and Y and at the same time Rξ is ��∇ξξ-parallel, then M is a Hopf hypersurface in Mn(c) provided that it satisfies RξA(1) = A(1)Rξ, RξA(2) = A(2)Rξ and ${\bar{r}}-2(n-1)c{\leq}0$, where ${\bar{r}}$ denotes the scalar curvature of M.

• Communications of the Korean Mathematical Society
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• v.9 no.4
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• pp.917-926
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• 1994

Let ($M, G_M, F$) be a (p+q)-dimensional Riemannian manifold with a foliation F of codimension q and a bundle-like metric $g_M$ with respect to F ([9]). Aside from the Laplacian $\bigtriangleup_g$ associated to the metric g, there is another differnetial operator, the Jacobi operator $J_D$, which is a second order elliptic operator acting on sections of the normal bundle. Its spectrum isdiscrete as a consequence of the compactness of M. The study of the spectrum of $\bigtriangleup_g$ acting on functions or forms has attracted a lot of attention. In this point of view, the present authors [7] have studied the spectrum of the Laplacian and the curvature of a compact orientable cosymplectic manifold. On the other hand, S. Nishikawa, Ph. Tondeur and L. Vanhecke [6] studied the spectral geometry for Riemannian foliations. The purpose of the present paper is to study the relation between two spectra and the transversal geometry of cosymplectic foliations. We shall be in $C^\infty$-category. Manifolds are assumed to be connected.