• Title/Summary/Keyword: oriented manifold

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REMARKS ON THE SUTURED MANIFOLDS

  • Park, Ki Sung
    • Korean Journal of Mathematics
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    • v.17 no.4
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    • pp.481-485
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    • 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.

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FlexDesigner:Object-Oriented Non-manifold Modeling Kernel with Hierarchically Modularized Structure (FlexDesigner:계층적으로 모듈화된 주초의 객체 지향 방식 비다양체 모델링 커널)

  • 이강수;이건우
    • Korean Journal of Computational Design and Engineering
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    • v.2 no.4
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    • pp.222-236
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    • 1997
  • Conventional solid or surface modeling systems cannot represent both the complete solid model and the abstract model in a unified framework. Recently, non-manifold modeling systems are proposed to solve this problem. This paper describes FlexDesigner, an open kernel system for modeling non-manifold models. It summarizes the data structure for non-manifold models, system design methodology, system modularization, and the typical characteristics of each module in the system. A data structure based on partial-topological elements is adopted to represent the relationship among topological elements. It is efficient in the usage of memory and has topological completeness compared with other published data structures. It can handle many non-manifold situations such as isolate vertices, dangling edges, dangling faces, a mixed dimensional model, and a cellular model. FlexDesigner is modularized hierarchically and designed by the object-oriented methodology for reusability. FlexDesigner is developed using the C++ and OpenGL on both SGI workstation and IBM PC.

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CIRCLE ACTIONS ON ORIENTED MANIFOLDS WITH FEW FIXED POINTS

  • Jang, Donghoon
    • East Asian mathematical journal
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    • v.36 no.5
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    • pp.593-604
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    • 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.

THE CRITICAL POINT EQUATION ON A FOUR DIMENSIONAL WARPED PRODUCT MANIFOLD

  • Hwang, Seung-Su;Chang, Jeong-Wook
    • Bulletin of the Korean Mathematical Society
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    • v.43 no.4
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    • pp.679-692
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    • 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.

A PINCHING THEOREM FOR RIEMANNIAN 4-MANIFOLD

  • Ko, Kwanseok
    • Korean Journal of Mathematics
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    • v.13 no.1
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    • pp.35-41
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    • 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 ${\pm}\mathbb{C}\mathbb{P}^2$.

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SELF-DUAL EINSTEIN MANIFOLDS OF POSITIVE SECTIONAL CURVATURE

  • Ko, Kwanseok
    • Korean Journal of Mathematics
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    • v.13 no.1
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    • pp.51-59
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    • 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.

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TRIANGULATIONS OF SEIFERT FIBERED 3-MANIFOLDS

  • Hong, Sung-Bok;Jeong, Myung-Hwa;SaKong, Jung-Sook
    • Communications of the Korean Mathematical Society
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    • v.13 no.4
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    • pp.839-845
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    • 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.

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A FREE ℤp-ACTION AND THE SEIBERG-WITTEN INVARIANTS

  • Nakamura, Nobuhiro
    • Journal of the Korean Mathematical Society
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    • v.39 no.1
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    • pp.103-117
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    • 2002
  • We consider the situation that ${\mathbb{Z}_p}\;=\;{\mathbb{Z}/p\mathbb{Z}}$ acts freely on a closed oriented 4-manifold X with ${b_2}^{+}\;{\geq}\;2$. In this situation, we study the relation between the Seiberg-Witten invariants of X and those of the quotient manifold $X/{\mathbb{Z}}_p$. We prove that the invariants of X are equal to those of $X/{\mathbb{Z}}_p$ modulo p.

ON THE STRUCTURE OF THE FUNDAMENTAL GROUP OF MANIFOLDS WITH POSITIVE SCALAR CURVATURE

  • Kim, Jin-Hong;Park, Han-Chul
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.1
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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.

ON THE STRUCTURE OF MINIMAL SUBMANIFOLDS IN A RIEMANNIAN MANIFOLD OF NON-NEGATIVE CURVATURE

  • Yun, Gab-Jin;Kim, Dong-Ho
    • Bulletin of the Korean Mathematical Society
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    • v.46 no.6
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    • pp.1213-1219
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    • 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].