• Journal of the Korean Mathematical Society
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• v.58 no.6
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• pp.1407-1419
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• 2021

The Iwasawa decomposition NAK of the Lie group G = SO₀(2, 1) with a left invariant metric produces Riemannian submersions G → N\G, G → A\G, G → K\G, and G → NA\G. For each of these, we calculate the curvature of the base space and the lifting of a simple closed curve to the total space G. Especially in the first case, the base space has a constant curvature 0; the holonomy displacement along a (null-homotopic) simple closed curve in the base space is determined only by the Euclidean area of the region surrounded by the curve.

• Journal of the Korean Mathematical Society
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• v.52 no.6
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• pp.1209-1251
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• 2015

The purpose of this paper is to classify all compact manifolds modeled on the 4-dimensional solvable Lie group $Sol_1^4$, and more generally, the crystallographic groups of $Sol_1^4$. The maximal compact subgroup of Isom($Sol_1^4$) is $D_4={\mathbb{Z}}_4{\rtimes}{\mathbb{Z}}_2$. We shall exhibit an infra-solvmanifold of $Sol_1^4$ whose holonomy is $D_4$. This implies that all possible holonomy groups do occur; the trivial group, ${\mathbb{Z}}_2$ (5 families), ${\mathbb{Z}}_4$, ${\mathbb{Z}}_2{\times}{\mathbb{Z}}_2$ (5 families), and ${\mathbb{Z}}_4{\rtimes}{\mathbb{Z}}_2$ (2 families).

• Journal of the Korean Mathematical Society
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• v.40 no.1
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• pp.109-128
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• 2003

In this paper, we show that the Euler characteristic of an even dimensional closed projectively flat manifold is equal to the total measure which is induced from a probability Borel measure on RP$^{n}$ invariant under the holonomy action, and then discuss its consequences and applications. As an application, we show that the Chen's conjecture is true for a closed affinely flat manifold whose holonomy group action permits an invariant probability Borel measure on RP$^{n}$ ; that is, such a closed affinly flat manifold has a vanishing Euler characteristic.

• Journal of the Korean Mathematical Society
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• v.54 no.2
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• pp.507-516
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• 2017

Real Bott manifolds is a class of flat manifolds with holonomy group $\mathbb{Z}^k_2$ of diagonal type. In this paper we formulate necessary and sufficient conditions of the existence of a Spin-structure on real Bott manifolds. It extends results of [9].

• Journal of the Korean Mathematical Society
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• v.42 no.3
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• pp.555-571
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• 2005

A pair of pants $\Sigma(0,3)$ is a building block of oriented surfaces. The purpose of this paper is to formulate the matrix presentations of elements of the Teichmuller space of a pair of pants. In the level of the matrix group $SL(2,\mathbb{R})$, we shall show that an odd number of traces of matrix presentations of the generators of the fundamental group of $\Sigma(0,3)$ should be negative.

• The Pure and Applied Mathematics
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• v.11 no.1
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• pp.73-88
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• 2004

A punctured torus $\Sigma(1,1)$ is a building block of oriented surfaces. The goal of this paper is to formulate the matrix presentations of elements of the Teichmuller space of a punctured torus. Let $\cal{C}$ be a matrix presentation of the boundary component of $\Sigma(1,1)$.In the level of the matrix group $\mathbb{SL}$($\mathbb2,R$) we shall show that the trace of $\cal{C}$ is always negative.

• Journal of the Chungcheong Mathematical Society
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• v.15 no.2
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• pp.57-66
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• 2003

The various deformation spaces associated with maximal geometric structures on closed oriented 3-manifolds was studied in [2], leaving out the geometry of $\mathbb{R}^3$. In this paper, we study the Weil spaces and Teichm$\ddot{u}$ller spaces of non-orientable 3-dimensional flat Riemannian manifolds. In particular, we find the Teichm$\ddot{u}$ller spaces are homeomorphic to the Euclidean spaces $\mathbb{R}^4$ or $\mathbb{R}^3$ depending on the holonomy group $\mathbb{Z}_2$ or $\mathbb{Z}_2{\times}\mathbb{Z}_2$ respectively.

• Communications of the Korean Mathematical Society
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• v.18 no.1
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• pp.95-104
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• 2003

The deformation spaces of the six orientable 3-dimensional flat Riemannian manifolds are studies. It is proved that the Teichmuller spaces are homeomorphic to the Euclidean spaces. To state more precisely, let $\Phi$ denote the holonomy group of the manifold. Then the Teichmuller space is homeomorphic to (1) ${\mathbb{R}}^6\;if\;\Phi$ is trivial, (2) ${\mathbb{R}}^4\;if\;\Phi$ is cyclic with order two, (3) ${\mathbb{R}}^2\;if\;\Phi$ is cyclic of order 3, 4 or 6, and (4) ${\mathbb{R}}^3\;if\;\Phi\;\cong\;{\mathbb{Z}_2}\;\times\;{\mathbb{Z}_2}$.