• Bulletin of the Korean Mathematical Society
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• v.35 no.1
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• pp.25-32
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• 1998

Let N be the 3-dimensional Heisenberg group equipped with a left-invariant metric on N. We characterize the Jacobi fields and the conjegate points along a geodesic on N, which points out that Theorem 4 of [1] is not correct.

• Journal of the Chungcheong Mathematical Society
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• v.11 no.1
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• pp.143-150
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• 1998

In this paper, we investigate some properties of Jacobi fields and conjugate points in a complete Riemannian manifold M. Also we get a necessary and sufficient condition about a geodesic without conjugate points in the manifold with non-negative curvature.

• Communications of the Korean Mathematical Society
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• v.9 no.2
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• pp.385-391
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• 1994

In this report, given a Riemannian foliation F on a Riemannian manifold, we introduce the concept of F-Jacobi fields along normal geodesics to investigate geometric properties of the leaves of F.(omitted)

• Honam Mathematical Journal
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• v.38 no.2
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• pp.385-391
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• 2016

We consider Jacobi filds as the first derivatives for ${\varepsilon}$, the energy of harmonic extensions, in a given manifold. In this paper we see that the Jacobi fild is bounded by the given boundary map. Here we give no restriction concerned with the curvature for the given manifold.

• Kyungpook Mathematical Journal
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• v.63 no.2
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• pp.287-311
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• 2023

In this paper, we introduce the notion of cyclic structure Jacobi semi-symmetric real hypersurfaces in the complex hyperbolic quadric Q^m* = SO⁰_2,m/SO₂SO_m. We give a classifiction of when real hypersurfaces in the complex hyperbolic quadric Q^m* having 𝔄-principal or 𝔄-isotropic unit normal vector fields have cyclic structure Jacobi semi-symmetric tensor.

• Bulletin of the Korean Mathematical Society
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• v.60 no.4
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• pp.849-861
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• 2023

In this paper, we introduce the notion of semi-symmetric structure Jacobi operator for Hopf real hypersufaces in the complex quadric Q^m = SO_m+2/SO_mSO₂. Next we prove that there does not exist any Hopf real hypersurface in the complex quadric Q^m = SO_m+2/SO_mSO₂ with semi-symmetric structure Jacobi operator. As a corollary, we also get a non-existence property of Hopf real hypersurfaces in the complex quadric Q^m with either symmetric (parallel), or recurrent structure Jacobi operator.

• 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 ${\bar{r}}-2(n-1)c{\leq}0$.

• Kyungpook Mathematical Journal
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• v.56 no.2
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• pp.541-575
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• 2016

Let M be a real hypersurface of a complex space form with almost contact metric structure (${\phi}$, ${\xi}$, ${\eta}$, g). In this paper, we prove that if the structure Jacobi operator $R_{\xi}= R({\cdot},{\xi}){\xi}$ is ${\phi}{\nabla}_{\xi}{\xi}$-parallel and $R_{\xi}$ commute with the structure tensor ${\phi}$, then M is a homogeneous real hypersurface of Type A provided that $TrR_{\xi}$ is constant.

• Communications of the Korean Mathematical Society
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• v.37 no.1
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• pp.229-257
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• 2022

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 A and R_𝜉 the shape operator in the direction of distinguished normal vector field and the structure Jacobi operator with respect to the structure vector 𝜉, 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_𝜉A = AR_𝜉 and at the same time ∇_𝜉R_𝜉 = 0 on M, then M is a Hopf hypersurface of type (A) provided that the scalar curvature s of M holds s - 2(n - 1)c ≤ 0.

• East Asian mathematical journal
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• v.38 no.1
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• pp.43-63
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• 2022

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 R_ξ the structure Jacobi operator with respect to the structure vector field ξ and by ${\bar{r}}$ the scalar curvature of M. Suppose that R_ξ is φ∇_ξξ-parallel and at the same time 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, then M is a Hopf hypersurface of type (A) in M_n+1(c) provided that ${\bar{r}-2(n-1)c}$ ≤ 0.