• Communications of the Korean Mathematical Society
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• v.23 no.3
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• pp.421-426
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• 2008

Let $M^n$ be a complete immersed super stable minimal submanifold in $\mathbb{R}^{n+p}$ with fiat normal bundle. We prove that if M has finite total $L^2$ norm of its second fundamental form, then M is an affine n-plane. We also prove that any complete immersed super stable minimal submanifold with flat normal bundle has only one end.

• Journal of the Korean Mathematical Society
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• v.55 no.5
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• pp.1103-1129
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• 2018

In this paper, we show several vanishing type theorems for p-harmonic ${\ell}$-forms on Riemannian manifolds ($p{\geq}2$). First of all, we consider complete non-compact immersed submanifolds $M^n$ of $N^{n+m}$ with flat normal bundle, we prove that any p-harmonic ${\ell}$-form on M is trivial if N has pure curvature tensor and M satisfies some geometric conditions. Then, we obtain a vanishing theorem on Riemannian manifolds with a weighted $Poincar{\acute{e}}$ inequality. Final, we investigate complete simply connected, locally conformally flat Riemannian manifolds M and point out that there is no nontrivial p-harmonic ${\ell}$-form on M provided that the Ricci curvature has suitable bound.

• Bulletin of the Korean Mathematical Society
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• v.52 no.1
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• pp.301-312
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• 2015

In this paper, we study a class of spacelike rotational surfaces in the Minkowski 4-space $\mathbb{E}^4_1$ with meridian curves lying in 2-dimensional spacelike planes and having pointwise 1-type Gauss map. We obtain all such surfaces with pointwise 1-type Gauss map of the first kind. Then we prove that the spacelike rotational surface with flat normal bundle and pointwise 1-type Gauss map of the second kind is an open part of a spacelike 2-plane in $\mathbb{E}^4_1$.

• Communications of the Korean Mathematical Society
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• v.21 no.4
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• pp.739-747
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• 2006

By using O(m+n, 1)/$O(m){\times}O(n,1)-system$, we give an analytic version of Ribaucour transformation for flat m-dimensional submanifolds in $\mathbb{H}^{m+n}$ with flat, non-degenerate normal bundle and free of weak umbilics, where $n{\geq}m-1$.

• Bulletin of the Korean Mathematical Society
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• v.31 no.1
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• pp.25-33
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• 1994

In [2] we proved that if the minimum of the sectional curvature of a compact real minimal hypersurface of CP$^{m}$ is 1/(2m-1), then M is the geodesic hypersphere. This result was generalized in [8] to the case of M is a generic submanifold with flat normal connection. The purpose of the present paper is to prove a following generalization of theorems in [2] and [8].

• Honam Mathematical Journal
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• v.33 no.4
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• pp.557-573
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• 2011

Let G be a compact and connected semisimple Lie group, H a closed subgroup, g (resp. h) the Lie algebra of G (resp. H), B the Killing form of g, g the normal metric on the homogeneous space G/H which is induced by -B. Let D be an invarint connection with Weyl structure (D, g, ${\omega}$) in the tangent bundle over the normal homogeneous Riemannian manifold (G/H, g) which is projectively flat. Then, the affine connection D on (G/H, g) is a Yang-Mills connection if and only if D is the Levi-Civita connection on (G/H, g).

• KOREAN JOURNAL OF CROP SCIENCE
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• v.31 no.2
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• pp.167-172
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• 1986

This experiment was carried out to obtain some information about the growth habit and morphological and anatomical characteristics of fasciated soybean, var. Buchai-kong. The results obtained are summarized as follows: 1. Main stern of fasciated soybean, var. Buchai-kong was gradually broadened and became flat, on which leaves emerged irregularly compared with normal soybean, var. Danyeop-kong. The number of leaves on main stern was more in Buchai-kong, but the total number of leaves per plant was not considerably different each other. 2. The degree of fasciation (max./min. in diameter at the broadened part) in main stern of Buchai-kong was about 3.1, and was related to the amount of growth of main stern and branches of Danyeop-kong. Stern fasciation types found in Buchai-kong were linear (the most common) and bifurcated with regard to branching of main stern tip. 3. The number of branches per plant was much less in Buchai-kong than in Danyeop-kong. There were fas-ciated branches and fasciated petiole in Buchai-kong. 4. About 75 percent of pods of Buchai-kong were concentrated on the top of stern, which showed strong pod clustering. 5. Seed yield per plant of Buchai-kong was about 60 percent of that of Danyeop-kong, and the decrease in seed yield was mostly affected by the decrease in seed weight and seeds per pod. 6. The number of vascular bundles of transection of mature stern was more in Buchai-kong, but Danyeop-kong developed the better vascular bundle.