• Journal of the Korean Mathematical Society
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• v.32 no.3
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• pp.471-482
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• 1995

A complex n-dimensional Kaehlerian manifold of constant holomorphic sectional curvature c is called a comples space form, which is denoted by $M_n(c)$. A complete and simply connected complex space form consists of a complex projective space $P_nC$, a complex Euclidean space $C^n$ or a complex hyperbolic space $H_nC$, according as c > 0, c = 0 or c < 0. The induced almost contact metric structure of a real hypersurface M of $M_n(c)$ is denoted by $(\phi, \zeta, \eta, g)$.

• Journal of the Korean Mathematical Society
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• v.54 no.3
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• pp.967-986
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• 2017

We give the necessary condition for an operator defined on a cartesian product of $c_0(\mathcal{X})$ spaces to be summing or dominated and we show that for the multiplication operators this condition is also sufficient. By using these results, we show that ${\Pi}_s(c_0,{\ldots},c_0;c_0)$ contains a copy of $l_s(l^m_2{\mid}m{\in}\mathbb{N})$ for s > 2 or a copy of $1_s(l^m_1{\mid}{\in}\mathbb{N})$, for any $l{\leq}S$ < ${\infty}$. Also ${\Delta}_{s_1,{\ldots},s_n}(c_0,{\ldots},c_0;c_0)$ contains a copy of $l_{{\upsilon}_n(s_1,{\ldots},s_n)}$ if ${\upsilon}_n(s_1,{\ldots},s_n){\leq}2$ or a copy of $l_{{\upsilon}_n(s_1,{\ldots},s_n)}(l^m_2{\mid}m{\in}\mathbb{N})$ if 2 < ${\upsilon}_n(s_1,{\ldots},s_n)$, where ${\frac{1}{{\upsilon}_n(s_1,{\ldots},s_n})}={\frac{1}{s_1}}+{\cdots}+{\frac{1}{s_n}}$. We find also the necessary and sufficient conditions for bilinear operators induced by some method of summability to be 1-summing or 2-dominated.

• Bulletin of the Korean Mathematical Society
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• v.26 no.2
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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.

• Korean Journal of Mathematics
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• v.18 no.2
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• pp.195-200
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• 2010

Let C be a closed convex set in ${\mathbb{S}}^m$ or ${\mathbb{H}}^m$. Assume that ${\Sigma}$ is an n-dimensional compact minimal submanifold outside C such that ${\Sigma}$ is orthogonal to ${\partial}C$ along ${\partial}{\Sigma}{\cap}{\partial}C$ and ${\partial}{\Sigma}$ lies on a geodesic sphere centered at a fixed point $p{\in}{\partial}{\Sigma}{\cap}{\partial}C$ and that r is the distance in ${\mathbb{S}}^m$ or ${\mathbb{H}}^m$ from p. We make use of a modified volume $M_p({\Sigma})$ of ${\Sigma}$ and obtain a sharp relative isoperimetric inequality $$\frac{1}{2}n^n{\omega}_nM_p({\Sigma})^{n-1}{\leq}Vol({\partial}{\Sigma}{\sim}{\partial}C)^n$$, where ${\omega}_n$ is the volume of a unit ball in ${\mathbb{R}}^n$ Equality holds if and only if ${\Sigma}$ is a totally geodesic half ball centered at p.

• Bulletin of the Korean Mathematical Society
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• v.26 no.1
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• pp.69-74
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• 1989

Recently one of the present authors [2] asserted that a real hypersurface of a complex space form M$^{n}$ (c), c.neq.0, is of cyclic parallel if and only if AJ=JA and he showed also a complete and connected cyclic-parallel real hypersurface of M$^{n}$ (c), is congruent to type $A_{1}$, $A_{2}$ or A according as c>0 or c<0. A real hypersurface of a complex space form M$^{n}$ (c) is said to be covariantly cyclic constant if the cyclic sum of covariant derivative of the second fundamental form is constant. The purpose of the present paper is to extend theorem 3 and 4 in [2] when the hypersurfaces are of coveriantly cyclic constant, that is a real hypersurface of a complex space form M$^{n}$ (c), c.neq.0, is of covariantly cyclic constant if an only if AJ=JA, and a complete and connected covariantly cyclic constant real hypersurface of M$^{n}$ (c) is congruent to type $A_{1}$, $A_{2}$ or a according as c>0 or c<0.

• Journal of the Korean Vacuum Society
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• v.17 no.3
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• pp.220-225
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• 2008

In this research, the P(VDF-TrFE) copolymer thin films were prepared by the physical vapor deposition and studied to their piezoelectric properties. In the case of a specimen produced by varying the deposition temperature from $260^{\circ}C$ to $300^{\circ}C$, its piezoelectric coefficient($d_{33}$) increased from 32.3pC/N to 36.28pC/N, and piezoelectric voltage coefficient($g_{33}$) from $793{\times}10^{-3}V{\cdot}m/N$ to $910.5{\times}10^{-3}V{\cdot}m/N$. On the basis of these experimental results, we concluded that the P(VDF-TrFE) copolymer thin film prepared at $300^{\circ}C$ showed the optimum piezoelectric properties. At the deposition temperature of $320^{\circ}C$, its piezoelectric coefficient(d33) decreased 25.3 pC/N and piezoelectric voltage coefficient($g_{33}$) $680{\times}10^{-3}V{\cdot}m/N$.

• Journal of the Korean Mathematical Society
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• v.34 no.2
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• pp.321-336
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• 1997

Let S be the Ricci curvature of an n-dimensional compact minimal totally real submanifold M of a quaternion projective space $HP^n (c)$ of quaternion sectional curvature c. We proved that if $S \leq \frac{16}{3(n -2)}c$, then either $S \equiv \frac{4}{n - 1}c$ (i.e. M is totally geodesic or $S \equiv \frac{16}{3(n - 2)}c$. All compact minimal totally real submanifolds of $HP^n (c)$ satisfy in $S \equiv \frac{16}{3(n - 2)}c$ are determined.

• Journal of the Korean Mathematical Society
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• v.44 no.1
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• pp.35-54
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• 2007

Let $M_c$ = M(2, 0, c) be the moduli space of O(l)-semistable rank 2 torsion-free sheaves with Chern classes $c_1=0\;and\;c_2=c$ on a K3 surface X, where O(1) is a generic ample line bundle on X. When $c=2n\geq4$ is even, $M_c$ is a singular projective variety equipped with a holomorphic symplectic structure on the smooth locus. In particular, $M_c$ has trivial canonical divisor. In [22], O'Grady asks if there is any symplectic desingularization of $M_{2n}$ for $n\geq3$. In this paper, we show that there is no crepant resolution of $M_{2n}$ for $n\geq3$. This obviously implies that there is no symplectic desingularization.

• Kyungpook Mathematical Journal
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• v.58 no.4
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• pp.637-650
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• 2018

In this paper first we show properties of isosymmetric operators given by M. Stankus [13]. Next we introduce an [m, C]-symmetric operator T on a complex Hilbert space H. We investigate properties of the spectrum of an [m, C]-symmetric operator and prove that if T is an [m, C]-symmetric operator and Q is an n-nilpotent operator, respectively, then T + Q is an [m + 2n - 2, C]-symmetric operator. Finally, we show that if T is [m, C]-symmetric and S is [n, D]-symmetric, then $T{\otimes}S$ is [m + n - 1, $C{\otimes}D$]-symmetric.

• Journal of the Korean Mathematical Society
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• v.40 no.3
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• pp.563-575
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• 2003

We show that if a connected Stein manifold M of dimension n has the holomorphic automorphism group Aut(M) isomorphic to $Aut(C^k {\times}(C^*)^{n - k})$ as topological groups, then M itself is biholomorphically equivalent to C^k{\times}(C^*)^{n - k}$. Besides, a new approach to the study of U(n)-actions on complex manifolds of dimension n is given.