• Communications of the Korean Mathematical Society
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• v.34 no.3
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• pp.771-782
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• 2019

We show that mappings preserving unit distance are close to two-isometries. We also prove that a mapping f is a linear isometry up to translation when f is a two-expansive surjective mapping preserving unit distance. Then we apply these results to consider two-isometries between normed spaces, strictly convex normed spaces and unital $C^*$-algebras. Finally, we propose some remarks and problems about generalized two-isometries on Banach spaces.

• Journal of the Korean Mathematical Society
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• v.54 no.1
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• pp.303-317
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• 2017

An isometry of hyperbolic space can be written as a composition of the reflection in the isometric sphere and two Euclidean isometries on the boundary at infinity. The isometric sphere is also used to construct the Ford fundamental domains for the action of discrete groups of isometries. In this paper, we study the isometric spheres of isometries acting on hyperbolic 4-space. This is a new phenomenon which occurs in hyperbolic 4-space that the two isometric spheres of a parabolic isometry can intersect transversally. We provide one geometric way to classify isometries of hyperbolic 4-space using the isometric spheres.

• Honam Mathematical Journal
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• v.30 no.1
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• pp.115-118
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• 2008

A bounded linear operator T on a complex separable Hilbert space H is called a two-isometry, if $T^{*2}T^2-2T^*T+1=0$. In this paper it is shown that every two-isometry is not supercyclic. This generalizes a result due to Ansari and Bourdon.

• Bulletin of the Korean Mathematical Society
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• v.53 no.2
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• pp.495-506
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• 2016

In this note we consider the monoid $\mathcal{PODI}_n$ of all monotone partial permutations on $\{1,{\ldots},n\}$ and its submonoids $\mathcal{DP}_n$, $\mathcal{POI}_n$ and $\mathcal{ODP}_n$ of all partial isometries, of all order-preserving partial permutations and of all order-preserving partial isometries, respectively. We prove that both the monoids $\mathcal{POI}_n$ and $\mathcal{ODP}_n$ are quotients of bilateral semidirect products of two of their remarkable submonoids, namely of extensive and of co-extensive transformations. Moreover, we show that $\mathcal{PODI}_n$ is a quotient of a semidirect product of $\mathcal{POI}_n$ and the group $\mathcal{C}_2$ of order two and, analogously, $\mathcal{DP}_n$ is a quotient of a semidirect product of $\mathcal{ODP}_n$ and $\mathcal{C}_2$.

• Bulletin of the Korean Mathematical Society
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• v.26 no.1
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• pp.47-52
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• 1989

Methods of Riemannian geometry has played an important role in the study of compact transformation groups. Every effective action of a compact Lie group on a differential manifold leaves a Riemannian metric invariant and the study of such actions reduces to the one involving the group of isometries of a Riemannian metric on the manifold which is, a priori, a Lie group under the compact open topology. Once an action of a compact Lie group is given an invariant metric is easily constructed by the averaging method and the Lie group is naturally imbedded in the group of isometries as a Lie subgroup. But usually this invariant metric has more symmetries than those given by the original action. Therefore the first question one may ask is when one can find a Riemannian metric so that the given action coincides with the action of the full group of isometries. This seems to be a difficult question to answer which depends very much on the orbit structure and the group itself. In this paper we give a sufficient condition that a subgroup action of a compact Lie group has an invariant metric which is not invariant under the full action of the group and figure out some aspects of the action and the orbit structure regarding the invariant Riemannian metric. In fact, according to our results, this is possible if there is a larger transformation group, containing the oringnal action and either having larger orbit somewhere or having exactly the same orbit structure but with an orbit on which a Riemannian metric is ivariant under the orginal action of the group and not under that of the larger one. Recently R. Saerens and W. Zame showed that every compact Lie group can be realized as the full group of isometries of Riemannian metric. [SZ] This answers a question closely related to ours but the situation turns out to be quite different in the two problems.

• Korean Journal of Computational Design and Engineering
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• v.19 no.2
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• pp.119-128
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• 2014

Computing the natural-looking interpolation of different shapes is a fundamental problem of computer graphics. It is proved by some researchers that such an interpolation can be achieved by pursuing the isometry. In this paper, a novel coordinate system that is invariant under isometries is defined. The coordinate system can easily be converted from the global vertex coordinates. Furthermore, the global coordinates can be efficiently recovered from the new coordinates by simply solving two sparse least-squares problems. Since the proposed coordinate system is invariant under isometries, then transformations such as global rigid trans-formations, articulated posture deformations, or any other isometric deformations, do not change the coordinate values. Therefore, shape interpolation can be done in this framework without being affected by the distortions caused by the isometry.

• Journal of the Korean Mathematical Society
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• v.45 no.1
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• pp.205-219
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• 2008

It is shown that if $A{\geq}0$ and T are two bounded linear operators on a complex Hilbert space H satisfying the inequality $T^*\;AT{\leq}A$ and the condition $AT=A^{1/2}TA^{1/2}$, then there exists the maximum reducing subspace for A and $A^{1/2}T$ on which the equality $T^*\;AT=A$ is satisfied. We concretely express this subspace in two ways, and as applications, we derive certain decompositions for quasinormal contractions. Also, some facts concerning the quasi-isometries are obtained.

• Bulletin of the Korean Mathematical Society
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• v.38 no.2
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• pp.413-425
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• 2001

We use the branching rules on SU(3) to show that if two Aloff-Wallach spaces $M_{k,l}\;and\;M_{k',l'}$ are isospectral for the Laplacian acting on smooth functions, they are isometric. We also show that 1 is the non-zero smallest eigenvalue among all Aloff-Wallach spaces and compute the multiplicities.

• Journal of the Korean Mathematical Society
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• v.56 no.4
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• pp.1131-1158
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• 2019

An ideal right-angled pentagon in hyperbolic 4-space ${\mathbb{H}}^4$ is a sequence of oriented geodesics ($L_1,{\ldots},L_5$) such that $L_i$ intersects $L_{i+1},i=1,{\ldots},4$, perpendicularly in ${\mathbb{H}}^4$ and the initial point of $L_1$ coincides with the endpoint of $L_5$ in the boundary at infinity ${\partial}{\mathbb{H}}^4$. We study the geometry of such pentagons and the various possible augmentations and prove identities for the associated quaternion half side lengths as well as other geometrically defined invariants of the configurations. As applications we look at two-generator groups ${\langle}A,B{\rangle}$ of isometries acting on hyperbolic 4-space such that A is parabolic, while B and AB are loxodromic.

• Journal of the Korean Mathematical Society
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• v.56 no.3
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• pp.595-622
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• 2019

An ideal right-angled pentagon in hyperbolic 4-space ${\mathbb{H}}^4$ is a sequence of oriented geodesics ($L_1,{\ldots},L_5$) such that Li intersects $L_{i+1},\;i=1,\;{\ldots},\;4$, perpendicularly in ${\mathbb{H}}^4$ and the initial point of $L_1$ coincides with the endpoint of $L_5$ in the boundary at infinity ${\partial}{\mathbb{H}}^4$. We study the geometry of such pentagons and the various possible augmentations and prove identities for the associated quaternion half side lengths as well as other geometrically defined invariants of the configurations. As applications we look at two-generator groups ${\langle}A,B{\rangle}$ of isometries acting on hyperbolic 4-space such that A is parabolic, while B and AB are loxodromic.