• Journal of the Korean Mathematical Society
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• v.43 no.6
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• pp.1183-1197
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• 2006

Let X be a smooth, closed, connected spin 4-manifold with $b_1(X)=0$ and signature ${\sigma}-(X)$. In this paper we use Seiberg-Witten theory to prove that if X admits a spin alternating $A_4$ action, then $b^+_2(X)$ ${\geq}$ |${\sigma}{(X)}$|/8+3 under some non-degeneracy conditions.

• Kyungpook Mathematical Journal
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• v.55 no.4
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• pp.753-771
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• 2015

A complete invariant defined for (closed connected orientable) 3-manifolds is an invariant defined for the 3-manifolds such that any two 3-manifolds with the same invariant are homeomorphic. Further, if the 3-manifold itself can be reconstructed from the data of the complete invariant, then it is called a characteristic invariant defined for the 3-manifolds. In a previous work, a characteristic lattice point invariant defined for the 3-manifolds was constructed by using an embedding of the prime links into the set of lattice points. In this paper, a characteristic rational invariant defined for the 3-manifolds called the characteristic genus defined for the 3-manifolds is constructed by using an embedding of a set of lattice points called the PDelta set into the set of rational numbers. The characteristic genus defined for the 3-manifolds is also compared with the Heegaard genus, the bridge genus and the braid genus defined for the 3-manifolds. By using this characteristic rational invariant defined for the 3-manifolds, a smooth real function with the definition interval (-1, 1) called the characteristic genus function is constructed as a characteristic invariant defined for the 3-manifolds.

• Bulletin of the Korean Mathematical Society
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• v.31 no.1
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• pp.15-23
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• 1994

Throughout this paper, we are working on the complex number field C. The aim of this paper is to explain the applications of Theorem 2 in .cint. 1. In the surface theory, the adjoint linear system has played important roles and many tools have been developed to understand it. In the cases of higher dimensional varieties, we don't have any useful tools so far. Theorem 2 implies that it is enough to compute the dimension of the adjoint linear system to check the birationality. We can compute, somehow, the dimension of the adjoint linear system. For example, we can get an information about $h^{0}$ (X, $O_{x}$( $K_{x}$ + D)) from Euler characteristic of vertical bar $K_{X}$ + D vertical bar and some vanishing theorems. We are going to show the applications of Theorem 2 to smooth three-folds and smooth fourfold, specially, of general type with a nef canonical divisor, smooth Fano variety, and Calabi-Yau manifold. Our main results are Theorem A and Theorem B. Most of birationality problems in Theorem A and Theorem B have been studied. (see Ando [1] and Matsuki [4] for the detail matters.) But Theorem 2 gives short and easy proofs in the cases of dimension 3 and improves the previously known results in the cases of dimension 4.4. 4.4.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.4
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• pp.735-743
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• 2011

Let f be a diffeomorphism of a closed n-dimensional smooth manifold. In this paper, we introduce the notion of $C^1$-stably periodic shadowing property for a closed f-invariant set, and prove that for a transitive set ${\Lambda}$, if f has the $C^1$-stably periodic shadowing property on ${\Lambda}$, then ${\Lambda}$ admits a dominated splitting.

• Bulletin of the Korean Mathematical Society
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• v.35 no.4
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• pp.757-767
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• 1998

For a Riemannian manifold $(M^n, g)$ we consider the space $V(M^n, g)$ of all smooth functions on $M^n$ whose Hessian is proportional to the metric tensor $g$. It is well-known that if $M^n$ is a space form then $V(M^n)$ is of dimension n+2. In this paper, conversely, we prove that if $V(M^n)$ is of dimension $\ge{n+1}$, then $M^n$ is a Riemannian space form.

• Bulletin of the Korean Mathematical Society
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• v.36 no.4
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• pp.799-808
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• 1999

We investigate when the .product of two smooth manifolds admits a weakly Lagrangian embedding. Assume M, N are oriented smooth manifolds of dimension m and n,. respectively, which admit weakly Lagrangian immersions into $C^m$ and $C^n$. If m and n are odd, then $M\;{\times}\;N$ admits a weakly Lagrangian embedding into $C^{m+n}$ In the case when m is odd and n is even, we assume further that $\chi$(N) is an even integer. Then $M\;{\times}\;N$ admits a weakly Lagrangian embedding into $C^{m+n}$. As a corollary, we obtain the result that $S^n_1\;{\times}\;S^n_2\;{\times}\;...{\times}\;S^n_k$, $\kappa$>1, admits a weakly Lagrang.ian embedding into $C^n_1+^n_2+...+^n_k$ if and only if some ni is odd.

• Bulletin of the Korean Mathematical Society
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• v.35 no.4
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• pp.809-817
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• 1998

We investigate when the product of two smooth manifolds admits a weakly Lagrangian embedding. Prove that, if $M^m$ and $N^n$ are smooth manifolds such that M admits a weakly Lagrangian embedding into ${\mathbb}C^m$ whose normal bundle has a nowhere vanishing section and N admits a weakly Lagrangian immersion into ${\mathbb}C^n$, then $M \times N$ admits a weakly Lagrangian embedding into ${\mathbb}C^{m+n}$. As a corollary, we obtain that $S^m {\times} S^n$ admits a weakly Lagrangian embedding into ${\mathbb}C^{m+n}$ if n=1,3. We investigate the problem of whether $S^m{\times}S^n$ in general admits a weakly Lagrangian embedding into ${\mathbb} C^{m+n}$.

• Honam Mathematical Journal
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• v.34 no.4
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• pp.513-517
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• 2012

Let L(M) be the bundle of all linear frames over a smooth manifold M, $u$ an arbitrarily given point of L(M), and ${\nabla}:\mathfrak{X}(M){\times}\mathfrak{X}(M){\rightarrow}\mathfrak{X}(M)$ a linear connection on M. Then the following result is well known: the horizontal subspace at the point $u$ may be written in terms of local coordinates of $u{\in}L(M)$ and Christoel's symbols defined by ${\nabla}$. This result is very fundamental on the study of the theory of connections. In this paper we show that the local expression of the horizontal subspace at the point u does not depend on the choice of a local coordinate system around the point $u{\in}L(M)$, which is rarely seen.

• Communications of the Korean Mathematical Society
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• v.23 no.4
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• pp.587-595
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• 2008

It is well known that critical points of the total scalar curvature functional S on the space of all smooth Riemannian structures of volume 1 on a compact manifold M are exactly the Einstein metrics. When the domain of S is restricted to the space of constant scalar curvature metrics, there has been a conjecture that a critical point is isometric to a standard sphere. In this paper we investigate the relationship between the first Betti number and stable minimal surfaces, and study the analytic properties of stable minimal surfaces in M for n = 3.

• Bulletin of the Korean Mathematical Society
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• v.38 no.2
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• pp.347-355
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• 2001

Let ($X, \omega$) be a closed symplectic 4-manifold. Let a finite cyclic group G act semifreely, holomorphically on X as isometries with fixed point set $\Sigma$(may be empty) which is a 2-dimension submanifold. Then there is a smooth structure on the quotient X'=X/G such that the projection $\pi$:X$\rightarrow$X' is a Lipschitz map. Let L$\rightarrow$X be the Spin$^c$ -structure on X pulled back from a Spin$^c$-structure L'$\rightarrow$X' and b_2^$+(X')>1. If the Seiberg-Witten invariant SW(L')$\neq$0 of L' is non-zero and $L=E\bigotimesK^-1\bigotimesE$ then there is a G-invariant pseudo-holomorphic curve u:$C\rightarrowX$,/TEX> such that the image u(C) represents the fundamental class of the Poincare dual $c_1$(E). This is an equivariant version of the Taubes' Theorem.