• 대한수학회보
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• 제34권4호
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• pp.535-547
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• 1997

The (2, 2, 2, 2)-orbifold is a 2-dimensional orbifold with four order 2 cone points having 2-sphere as an underlying space. The (2, 2, 2, 2)-orbifold admits different geometric structures. The purpose of this paper is to find some real profective structures on the (2, 2, 2, 2)-orbifold.

• East Asian mathematical journal
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• 제32권5호
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• pp.717-728
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• 2016

In this paper, we give a necessary and sufficient condition for an essential simple loop on a 2-bridge sphere in an even Heckoid orbifold for the trivial knot to be null-homotopic, peripheral or torsion in the orbifold. We also give a necessary and sufficient condition for two essential simple loops on a 2-bridge sphere in an even Heckoid orbifold for the trivial knot to be homotopic in the orbifold.

• 대한수학회보
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• 제34권4호
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• pp.549-560
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• 1997

For positive integers $n_i \geq 2, i = 1, 2, 3, 4$, such that $\Sigma \frac{n_i}{1} < 2$, there exists a quadrilateral $P = P_1 P_2 P_3 P_4$ in the hyperbolic plane $H^2$ with the interior angle $\frac{n_i}{\pi}$ at $P_i$. Let $\Gamma \subset Isom(H^2)$ be the (discrete) group generated by reflections in each side of $P$. Then the quotient space $H^2/\gamma$ is a differentiable orbifold of type $(^* n_1 n_2 n_3 n_4)$. It will be shown that the deformation space of $Rp^2$-structures on this orbifold can be mapped continuously and bijectively onto the cell of dimension 4 - \left$\mid$ {i$\mid$n_i = 2} \right$\mid$$.

• 대한수학회논문집
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• 제13권2호
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• pp.345-358
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• 1998

Let (p/q,n) denote the orbifold with its underlying space $S^3$ and a rational knot or link p/q as its singular set with a cyclic isotropy group of order n. In this paper we shall show the geometrical realization for the case (7/3,n) for all $n \geq 3$.

• 대한수학회논문집
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• 제20권4호
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• pp.803-812
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• 2005

We construct an infinite family of closed 3-manifolds M(2m+ 1, n, k) which are identification spaces of certain polyhedra P(2m+ 1, n, k), for integers $m\;\ge\;1,\;n\;\ge\;3,\;and\;k\;\ge\;2$. We prove that they are (n / d)- fold cyclic coverings of the 3-sphere branched over certain links $L_{(m,d)}$, where d = gcd(n, k), by handle decomposition of orbifolds. This generalizes the results in [3] and [2] as a particular case m = 2.

• Kyungpook Mathematical Journal
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• 제56권2호
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• pp.577-582
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• 2016

In this paper we consider all orientation-preserving ${\mathbb{Z}}_4$-actions on 3-dimensional handlebodies $V_g$ of genus g > 0. We study the graph of groups (${\Gamma}(v)$, G(v)), which determines a handlebody orbifold $V({\Gamma}(v),G(v)){\simeq}V_g/{\mathbb{Z}}_4$. This algebraic characterization is used to enumerate the total number of ${\mathbb{Z}}_4$ group actions on such handlebodies, up to equivalence.

• 대한수학회지
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• 제38권6호
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• pp.1167-1177
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• 2001

The groups generated by reflections in faces of Coxeter polyhedra in three-dimensional Thurstons spaces are considered. We develop a method for finding of finite index subgroups of Coxeter groups which uniformize three-dimensional manifolds obtained as two-fold branched coverings of manifolds of Heegaard genus one, that are lens spaces L(p, q) and the space S$^2$$\times$S$^1$.

• 대한수학회지
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• 제50권2호
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• pp.425-444
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• 2013

We construct some hyperbolic hyperelliptic space forms whose fundamental groups are generated by only two or three isometries. Each occurring group is obtained from a supergroup, which is an extended Coxeter group generated by plane re ections and half-turns. Then we describe covering properties and determine the isometry groups of the constructed manifolds. Furthermore, we give an explicit construction of space form of the second smallest volume nonorientable hyperbolic 3-manifold with one cusp.

• 대한수학회논문집
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• 제35권2호
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• pp.591-602
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• 2020

There is a well-known class of compact, complex, non-Kählerian manifolds constructed by Bosio, called the LVMB manifolds, which properly includes the Hopf manifold, the Calabi-Eckmann manifold, and the LVM manifolds. As in the case of LVM manifolds, these LVMB manifolds can admit a regular holomorphic foliation 𝓕. Moreover, later Meersseman showed that if an LVMB manifold is actually an LVM manifold, then the regular holomorphic foliation 𝓕 is actually transverse Kähler. The aim of this paper is to deal with a converse question and to give a simple and new proof of a well-known result of Cupit-Foutou and Zaffran. That is, we show that, when the holomorphic foliation 𝓕 on an LVMB manifold N is transverse Kähler with respect to a basic and transverse Kähler form and the leaf space N/𝓕 is an orbifold, N/𝓕 is projective, and thus N is actually an LVM manifold.