• 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.

• 대한수학회지
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• 제33권1호
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• pp.101-119
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• 1996

A complex $n(\geq 2)$-dimensional Kaehlerian manifold of constant holomorphic sectional curvature c is called a complex space form, which is denoted by $M_n(c)$.

• 호남수학학술지
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• 제28권4호
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• pp.573-589
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• 2006

Let M be a real hypersurface with almost contact metric structure $({\phi},{\xi},{\eta},g)$ in a non flat complex space form $M_n(c)$. In this paper, we prove that if the structure Jacobi operator $R_{\xi}$ is ${\xi}$-parallel and the Ricci tensor S commutes with the structure operator $\phi$, then a real hypersurface in $M_n(c)$ is a Hopf hypersurface. Further, we characterize such Hopf hypersurface in $M_n(c)$.

• 대한수학회보
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• 제26권2호
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• pp.159-164
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• 1989

Let M be a hypersurface of dimension n(.geq.2) in an (n+1)-dimensional real space form over bar M(c) with constant curvature c and H the second fundamental tensor of M. M is said to be birecurrent if here exists a covariant tensor field .alpha. of order 2 such that .del.$^{2}$H=H .alpha., where .del. is the connection of M. Also, M is said to be recurrent if there exists a 1-form .betha. such that .del.H=H .betha.. Matsuyama [2] recently proved that a recurrent hypersurface M in a real space form is locally symmetric and a complete irreducible birecurrent hypersurface M in a real space form is recurrent. The main purpose of this paper is to characterize the birecurrent or recurrent hypersurface M of a Riemannian manifold with constant curvature c and to prove that M is classified as a cylinder, $M^{n}$ (c) or ( $c_{1}$)* $M^{n-r}$ ( $c_{2}$) where 1/ $c_{1}$+1/ $c_{2}$=1/c.

• Kyungpook Mathematical Journal
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• 제56권4호
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• pp.1207-1235
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• 2016

Let M be a real hypersurface with constant mean curvature in a complex space form $M_n(c),c{\neq}0$. In this paper, we prove that if the structure Jacobi operator $R_{\xi}= R({\cdot},{\xi}){\xi}$ with respect to the structure vector field ${\xi}$ is ${\phi}{\nabla}_{\xi}{\xi}$-parallel and $R_{\xi}$ commute with the structure tensor field ${\phi}$, then M is a homogeneous real hypersurface of Type A.

• Kyungpook Mathematical Journal
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• 제56권2호
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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.

• 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.

• 대한수학회보
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• 제41권4호
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• pp.609-618
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• 2004

Pseudo-parallel real hypersurfaces in complex space forms can be defined as an extrinsic analogues of pseudo-symmetric real hypersurfaces, that generalize the notion of semi-symmetric real hypersurface. In this paper a classification of the pseudo-parallel real hypersurfaces in a non-flat complex space forms is obtained.

• 대한수학회논문집
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• 제9권2호
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• pp.369-383
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• 1994

A complex n-dimensional Kahler manifold of constant holomorphic sectional curvature c is called a complex space form, which is denoted by $M_{n}$ (c). A complete and simply connected complex space form consists of a complex projective space $P_{n}$ C, a complex Euclidean space $C^{n}$ or a complex hyperbolic space $H_{n}$ C, according as c > 0, c = 0 or c < 0.(omitted)

• 대한수학회보
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• 제44권1호
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• pp.157-172
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• 2007

In this paper, we prove that if the structure Jacobi operator $R_{\xi}-parallel\;and\;R_{\xi}$ commutes with the Ricci tensor S, then a real hypersurface with non-negative scalar curvature of a nonflat complex space form $M_{n}(C)$ is a Hopf hypersurface. Further, we characterize such Hopf hypersurface in $M_{n}(C)$.