• 호남수학학술지
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• 제43권2호
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• pp.289-304
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• 2021

In this paper, we establish several geometric characterizations focusing on the relationship between the squared norm of the second fundamental form and the warping function of SCR-lightlike warped product submanifolds in an indefinite Kaehler manifold. In particular, we find an estimate for the squared norm of the second fundamental form h in terms of the Hessian of the warping function λ for SCR-lightlike warped product submanifolds of an indefinite complex space form. Consequently, we derive an optimal inequality, namely $${\parallel}h{\parallel}^2{\geq}2q\{{\Delta}(ln{\lambda})+{\parallel}{\nabla}(ln{\lambda}){\parallel}^2+\frac{c}{2}p\}$$, for SCR-lightlike warped product submanifolds in an indefinite complex space form. We also provide one non-trivial example for this class of warped products in an indefinite Kaehler manifold.

• East Asian mathematical journal
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• 제37권1호
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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.

• 대한수학회보
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• 제26권1호
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• pp.75-82
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• 1989

It is recently proved by Aiyama and the authors [2] that a complete space-like complex submanifold of a complex space form $M^{n+p}$$_{p}$ (c') (c'.geq.0) is to totally geodesic. This is a complex version of the Bernstein-type theorem in the Minkowski space due to Calabi [4] and Cheng and Yau [5], which is generalized by Nishikawa[7] in the Lorentz manifold satisfying the strong energy condition. The purpose of this paper is to consider his result in the complex Lorentz manifold and is to prove the following.e following.

• 대한수학회보
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• 제47권4호
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• pp.701-714
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• 2010

In this paper, we study the geometry of real half lightlike submanifolds of an indefinite Kaehler manifold. The main result is a characterization theorem for screen conformal real half lightlike submanifolds of an indefinite complex space form.

• 대한수학회논문집
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• 제13권2호
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• pp.317-336
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• 1998

We study real hypersurfaces of a complex space form such that the Jacobi operator with respect to the structure vector field and the structure tensor $\phi$ on the real hypersurface commute.

• 대한수학회논문집
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• 제25권3호
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• pp.443-450
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• 2010

In this paper, we study the geometry of lightlike real hyper-surfaces of an indefinite Kaehler manifold. The main result is a characterization theorem for lightlike real hypersurfaces M of an indefinite complex space form $\bar{M}(c)$ such that the screen distribution is totally umbilic.

• Kyungpook Mathematical Journal
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• 제62권1호
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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.

• 대한수학회지
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• 제61권3호
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• pp.547-564
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• 2024

We study totally real and complex subsets of a right quarternionic vector space of dimension n + 1 with a Hermitian form of signature (n, 1) and extend these notions to right quaternionic projective space. Then we give a necessary and sufficient condition for a subset of a right quaternionic projective space to be totally real or complex in terms of the quaternionic Hermitian triple product. As an application, we show that the limit set of a non-elementary quaternionic Kleinian group 𝚪 is totally real (resp. commutative) with respect to the quaternionic Hermitian triple product if and only if 𝚪 leaves a real (resp. complex) hyperbolic subspace invariant.

• 대한수학회논문집
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• 제16권2호
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• pp.253-263
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• 2001

Niebergall and Ryan posed some open questions on 3-dimensional real hypersurfaces in complex space forms. In this paper we give affirmative answers to such kind of questions.

• KIEAE Journal
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• 제15권5호
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• pp.59-66
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• 2015

Purpose: Recently, Housing area plan has tried to reflect a public space of various forms. However, most of cases are indiscriminately developments, which don't reflect the diverse needs of residents. In addition, a subject of the public space is not clear and consistently stable of the management, so the residents don't have an interest enough to take advantage of it. Method: In order to make a plan of the public space of a residential complex, an architect designer needs to take a few things into consideration. One is a physical facility in needs. And other one is a management of keeping a social place stably and consistently, which allows residents to be able to interact with each other. It should be regarded for residents to form a sense of belonging, while minimizing the interface friction. When all these problems are fulfilled, an interaction will be made to improve the quality of the living environment. Result: Therefore, in this study, it is necessary to define the meaning of territoriality of the external public space of housing complex. This study makes it possible to improve the relationship of neighborhood and the quality of the life for the residents, depending on the time. This is the first step as 'the research for the territoriality of the external public space of housing complex', in order to define about the concept, function, characteristic of territory(the base of territoriality) and to establish the territoriality that can form the physical or psychological boundary in public space.