• Korean Journal of Mathematics
• /
• 제17권4호
• /
• pp.481-485
• /
• 2009

Gabai's sutured manifold theory has produced many remarkable results in knot theory. Let M be the compact oriented 3-manifold and (M, ${\gamma}$) be sutured manifold. The aim of this note is to show that there exist a sutured manifold decomposition and a surface of M which defines a sutured manifold decomposition.

• 대한수학회보
• /
• 제43권4호
• /
• pp.679-692
• /
• 2006

On a compact oriented n-dimensional manifold $(M^n,\;g)$, it has been conjectured that a metric g satisfying the critical point equation (2) should be Einstein. In this paper, we prove that if a manifold $(M^4,\;g)$ is a 4-dimensional oriented compact warped product, then g can not be a solution of CPE with a non-zero solution function f.

• East Asian mathematical journal
• /
• 제36권5호
• /
• pp.593-604
• /
• 2020

Let the circle act on a compact oriented manifold with a discrete fixed point set. At each fixed point, there are positive integers called weights, which describe the local action of S¹ near the fixed point. In this paper, we provide the author's original proof that only uses the Atiyah-Singer index formula for the classification of the weights at the fixed points if the dimension of the manifold is 4 and there are at most 4 fixed points, which made the author possible to give a classification for any finite number of fixed points.

• Korean Journal of Mathematics
• /
• 제13권1호
• /
• pp.35-41
• /
• 2005

Let (M, $g$) be a compact oriented 4-dimensional Riemannian manifold whose sectional curvature $k$ satisfies $1{\geq}k{\geq}0.1714$. We show that M is topologically $S^4$ or ${\pm}\mathbb{C}\mathbb{P}^2$.

• 대한수학회논문집
• /
• 제28권1호
• /
• pp.183-189
• /
• 2013

Let W be a parallelizable compact oriented manifold of dimension $n$ with boundary ${\partial}W=M$. We define the so-called Gauss map $f:M{\rightarrow}S^{n-1}$ using a framing of TW and show that the degree of $f$ is equal to Euler-Poincar$\acute{e}$ number ${\chi}(W)$, regardless of the specific framing. As a special case, we get a Hopf theorem.

• 호남수학학술지
• /
• 제30권4호
• /
• pp.677-683
• /
• 2008

On a compact n-dimensional manifold M, it has been conjectured that a critical point metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of volume 1, should be Einstein. The purpose of the present paper is to prove that a 3-dimensional manifold (M,g) is isometric to a standard sphere if ker $s^*_g{{\neq}}0$ and there is a lower Ricci curvature bound. We also study the structure of a compact oriented stable minimal surface in M.

• Korean Journal of Mathematics
• /
• 제13권1호
• /
• pp.51-59
• /
• 2005

Let (M, $g$) be a compact oriented self-dual 4-dimensional Einstein manifold with positive sectional curvature. Then we show that, up to rescaling and isometry, (M, $g$) is $S^4$ or $\mathbb{C}\mathbb{P}_2$, with their cannonical metrics.

• 대한수학회논문집
• /
• 제13권4호
• /
• pp.839-845
• /
• 1998

For an oriented compact, connected Seifert fibred 3-manifold M with nonempty boundary, we construct a simplicial complex using the equivalence classes of marked annulus systems and show that it is contractible.

• 충청수학회지
• /
• 제11권1호
• /
• pp.185-192
• /
• 1998

We briefly review the $S^1$-equivariant cohomology theory of a finite dimensional compact oriented $S^1$-manifold and extend our discussion in infinite dimensional case.

• 대한수학회보
• /
• 제46권6호
• /
• pp.1213-1219
• /
• 2009

Let M$^n$ be a complete oriented non-compact minimally immersed submanifold in a complete Riemannian manifold N$^{n+p}$ of nonnegative curvature. We prove that if M is super-stable, then there are no non-trivial L$^2$ harmonic one forms on M. This is a generalization of the main result in [8].