• Bulletin of the Korean Mathematical Society
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• v.48 no.2
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• pp.427-444
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• 2011

In this paper we give some non-existence theorems for Hopf hypersurfaces in the complex two-plane Grassmannian $G_2(\mathbb{C}^{m+2})$ with Lie parallel normal Jacobi operator $\bar{R}_N$ and totally geodesic D and $D^{\bot}$ components of the Reeb flow.

• Journal of the Korean Mathematical Society
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• v.44 no.6
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• pp.1363-1372
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• 2007

We investigate the containment measure of one domain to contain in another domain in a plane $X^{\kappa}$ of constant curvature. We obtain some Bonnesen-type inequalities involving the area, length, radius of the inscribed and the circumscribed disc of a domain D in $X^{\kappa}$.

• Korean Journal of Mathematics
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• v.25 no.4
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• pp.513-535
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• 2017

As a continuation to the study in [8, 12, 15, 17], we construct bi-conservative central normal (BCN) and bi-conservative asymptomatic normal (BAN) ruled surfaces in Euclidean 3-space ${\mathbb{E}}^3$. For such surfaces, local study is given and some examples are constructed using computer aided geometric design (CAGD).

• Korean Journal of Mathematics
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• v.28 no.3
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• pp.555-571
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• 2020

The aim of the present paper is to study pseudoparallel invariant submanifold of a (𝜅, 𝜇)-paracontact metric manifold. We consider pseudoparallel, Ricci-generalized pseudoparallel and 2-Ricci generalized pseudo parallel invariant submanifolds of a (𝜅, 𝜇)-paracontact metric manifold and we obtain new results contribute to geometry.

• Communications of the Korean Mathematical Society
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• v.10 no.4
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• pp.931-941
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• 1995

The definitions and techniques, which deals with homotopic harmonic maps from a compact Riemannian manifold into a compact metric space, developed by N. J. Korevaar and R. M. Schoen [7] can be applied to more general situations. In this paper, we prove that for a complicated domain, possibly noncompact Riemannian manifold with infinitely generated fundamental group, the existence of homotopic harmonic maps can be proved if the initial map is simple in some sense.

• Bulletin of the Korean Mathematical Society
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• v.37 no.3
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• pp.587-599
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• 2000

In this paper we study some nonpositively curved Riemannian manifolds acted on by a Lie group of isometries with principal orbits of codimension one. Among other results it is proved that if the universal covering manifold satisfies some conditions then every nonexceptional singular orbit is a totally geodesic submanifold. When M is flat and is not toruslike, it is proved that either each orbit is isometric to $R^k\timesT^m$or there is a singular orbit. If the singular orbit is unique and nonexceptional, then it is isometric to $R^k\timesT^m$.

• Journal of the Korean Mathematical Society
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• v.32 no.4
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• pp.835-848
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• 1995

Let $M_n$(c) be an n-dimensional complete and simply connected Kahlerian manifold of constant holomorphic sectional curvature c, which is called a complex space form. Then according to c > 0, c = 0 or c < 0 it is a complex projective space $P_nC$, a complex Euclidean space $C^n$ or a complex hyperbolic space $H_nC$.

• Communications of the Korean Mathematical Society
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• v.27 no.3
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• pp.591-602
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• 2012

In present paper we establish conditions for the integrability of various distributions of GCR-lightlike submanifolds and obtain conditions for the distributions to define totally geodesic foliations in GCR-lightlike submanifolds.

• Bulletin of the Korean Mathematical Society
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• v.46 no.3
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• pp.447-461
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• 2009

In this paper we give a non-existence theorem for Hopf real hypersurfaces in complex two-plane Grassmannians $G_2(\mathbb{C}^{m+2})$ satisfying the condition that the structure Jacobi operator $R_{\xi}$ commutes with the 3-structure tensors ${\phi}_i$, i = 1, 2, 3.

• Communications of the Korean Mathematical Society
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• v.29 no.1
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• pp.123-130
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• 2014

In this note, we investigate stable minimal hypersurfaces with weighted Poincar$\acute{e}$ inequality. We show that we still get the vanishing property without assuming that the hypersurfaces is non-totally geodesic. This generalizes a result in [2].