• 호남수학학술지
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• 제41권4호
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• pp.669-682
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• 2019

The objective of this paper is to introduce and study Einstein-like para-Kenmotsu manifolds. For a para-Kenmotsu manifold to be Einstein-like, a necessary and sufficient condition in terms of its curvature tensor is obtained. We also obtain the scalar curvature of an Einstein-like para-Kenmotsu manifold. A necessary and sufficient condition for an almost para-contact metric hypersurface of a locally product Riemannian manifold to be para-Kenmotsu is derived and it is shown that the para-Kenmotsu hypersurface of a locally product Riemannian manifold of almost constant curvature is always Einstein.

• International Journal of Precision Engineering and Manufacturing
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• 제1권2호
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• pp.105-114
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• 2000

An output feedback control (OFC) is presented for a linear stochastic system with known disturbance and applied to a flexible spacecraft for the reduction of residual vibration while allowing the natural deflection during operation. By converting the tracking problem into regulator problem, the OFC minimizes the expected value of a guadratic objective function composing of error stats which always remain on the intersection of sliding hypersurfaces. For the numerical evaluation with a flexible spacecraft, a large slewing maneuver strategy is devised with a tracking model for nominal trajectory and start-cost-stop strategy for economical maneuver in conjunction with the input shaping technique. The performance and efficacy of the proposed control scheme are illustrated with the comparison of different maneuver strategies.

• 대한수학회보
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• 제42권2호
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• pp.337-358
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• 2005

Let M be a real hypersurface with almost contact metric structure $(\phi,\;\xi,\;\eta,\;g)$ in a nonflat complex space form $M_n(c)$. In this paper, we prove that if the structure Jacobi operator $R_\xi$ commutes with both the structure tensor $\phi$ and the Ricc tensor S, then M is a Hopf hypersurface in $M_n(c)$ provided that the mean curvature of M is constant or $g(S\xi,\;\xi)$ is constant.

• 대한수학회논문집
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• 제35권4호
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• pp.1203-1219
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• 2020

Jin [7] defined a new connection on semi-Riemannian manifolds, which is a non-symmetric and non-metric connection. He said that this connection is an (ℓ, m)-type connection. Jin also studied lightlike hypersurfaces of an indefinite trans-Sasakian manifold with an (ℓ, m)-type connection in [7]. We study further the geometry of this subject. In this paper, we study generic lightlike submanifolds of an indefinite trans-Sasakian manifold endowed with an (ℓ, m)-type connection.

• 대한수학회지
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• 제40권3호
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• pp.461-486
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• 2003

It is known that the CR geometries of Levi non-degen-erate hypersurfaces in $\C^n$ and of the elliptic or hyperbolic CR submanifolds of codimension two in $\C^4$ share many common features. In this paper, a special class of normalized coordinates is introduced for any CR manifold M which is one of the above three kinds and it is shown that the explicit expression in these coordinates of an isotropy automorphism $f{\in}Aut(M)_o {\subset}Aut(M),\;o{\in}M$, is equal to the expression of a corresponding element of the automorphism group of the homogeneous model. As an application of this property, an extension theorem for CR maps is obtained.

• 대한수학회보
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• 제48권1호
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• pp.129-140
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• 2011

The aim of this paper is to study the structure of the fundamental group of a closed oriented Riemannian manifold with positive scalar curvature. To be more precise, let M be a closed oriented Riemannian manifold of dimension n (4 $\leq$ n $\leq$ 7) with positive scalar curvature and non-trivial first Betti number, and let be $\alpha$ non-trivial codimension one homology class in $H_{n-1}$(M;$\mathbb{R}$). Then it is known as in [8] that there exists a closed embedded hypersurface $N_{\alpha}$ of M representing $\alpha$ of minimum volume, compared with all other closed hypersurfaces in the homology class. Our main result is to show that the fundamental group ${\pi}_1(N_{\alpha})$ is always virtually free. In particular, this gives rise to a new obstruction to the existence of a metric of positive scalar curvature.

• 대한수학회지
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• 제50권3호
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• pp.479-491
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• 2013

Let M be a smooth real hypersurface in complex space of dimension $n$, $n{\geq}3$, and assume that the Levi-form at $z_0$ on M has at least $(q+1)$-positive eigenvalues, $1{\leq}q{\leq}n-2$. We estimate solutions of the local $\bar{\partial}$-closed extension problem near $z_0$ for $(p,q)$-forms in Sobolev spaces. Using this result, we estimate the local solution of tangential Cauchy-Riemann equation near $z_0$ in Sobolev spaces.

• 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.

• 대한수학회지
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• 제44권2호
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• pp.407-442
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• 2007

The study of Euclidean submanifolds with finite type "classical" Gauss map was initiated by B.-Y. Chen and P. Piccinni in [11]. On the other hand, it was believed that for spherical sub manifolds the concept of spherical Gauss map is more relevant than the classical one (see [20]). Thus the purpose of this article is to initiate the study of spherical submanifolds with finite type spherical Gauss map. We obtain several fundamental results in this respect. In particular, spherical submanifolds with 1-type spherical Gauss map are classified. From which we conclude that all isoparametric hypersurfaces of $S^{n+1}$ have 1-type spherical Gauss map. Among others, we also prove that Veronese surface and equilateral minimal torus are the only minimal spherical surfaces with 2-type spherical Gauss map.

• 대한수학회보
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• 제58권1호
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• pp.21-30
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• 2021

In this paper, we prove some rigidity results about embedded minimal hypersurface M ⊂ ℝn+1 with compact ∂M that has one end which is regular at infinity. We first show that if M ⊂ ℝn+1 meets a hyperplane in a constant angle ≥ ��/2, then M is part of an n-dimensional catenoid. We show that if M meets a sphere in a constant angle and ∂M lies in a hemisphere determined by the hyperplane through the center of the sphere and perpendicular to the limit normal vector nM of the end, then M is part of either a hyperplane or an n-dimensional catenoid. We also show that if M is tangent to a C2 convex hypersurface S, which is symmetric about a hyperplane P and nM is parallel to P, then M is also symmetric about P. In special, if S is rotationally symmetric about the xn+1-axis and nM = en+1, then M is also rotationally symmetric about the xn+1-axis.