• Communications of the Korean Mathematical Society
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• v.18 no.1
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• pp.87-94
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• 2003

We use geometric properties of Gromov-Hausdorff-convergence to present a way to construct rough but natural invariants of metric geometry.

• Journal of KIISE:Computer Systems and Theory
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• v.31 no.5_6
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• pp.288-296
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• 2004

This paper proposes a new mesh simplification algorithm using differential error metric. Many simplification algorithms make use of a distance error metric, but it is hard to measure an accurate geometric error for the high-curvature region even though it has a small distance error measured in distance error metric. This paper proposes a new differential error metric that results in unifying a distance metric and its first and second order differentials, which become tangent vector and curvature metric. Since discrete surfaces may be considered as piecewise linear approximation of unknown smooth surfaces, theses differentials can be estimated and we can construct new concept of differential error metric for discrete surfaces with them. For our simplification algorithm based on iterative edge collapses, this differential error metric can assign the new vertex position maintaining the geometry of an original appearance. In this paper, we clearly show that our simplified results have better quality and smaller geometry error than others.

• Journal of applied mathematics & informatics
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• v.41 no.1
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• pp.193-203
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• 2023

In view of solution to the Hilbert fourth problem, the present study engages to investigate the projectively flat special (𝛼, 𝛽)-metric and the generalised first approximate Matsumoto (𝛼, 𝛽)-metric, where 𝛼 is a Riemannian metric and 𝛽 is a differential one-form. Further, we concluded that 𝛼 is locally Projectively flat and have 𝛽 is parallel with respect to 𝛼 for both the metrics. Also, we obtained necessary and sufficient conditions for the aforementioned metrics to be locally projectively flat.

• Journal of the Korean Mathematical Society
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• v.57 no.6
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• pp.1335-1345
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• 2020

We will show a rigidity of a Kähler potential of the Poincaré metric with a constant length differential.

• Nonlinear Functional Analysis and Applications
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• v.27 no.3
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• pp.663-677
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• 2022

This present paper extends some fixed point theorems in rectangular b-metric spaces using subadditive altering distance and establishing the existence and uniqueness of fixed point for Kannan type mappings. Non-trivial examples are further provided to support the hypotheses of our results.

• Bulletin of the Korean Mathematical Society
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• v.28 no.2
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• pp.231-241
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• 1991

Recently, many mathematicians([NT], [Ka], [TV], [CW], etc.) studied foliated structures on a smooth manifold with the viewpoint of transversal differential geometry. In this paper, we shall discuss certain hermitian foliations F on a riemannian manifold with a bundle-like metric, that is, their transversal bundles to F have hermitian structures.

• Bulletin of the Korean Mathematical Society
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• v.61 no.3
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• pp.849-866
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• 2024

In this paper, we study the rectifying curves in multiplicative Euclidean space of dimension 3, i.e., those curves for which the position vector always lies in its rectifying plane. Since the definition of rectifying curve is affine and not metric, we are directly able to perform multiplicative differential-geometric concepts to investigate such curves. By several characterizations, we completely classify the multiplicative rectifying curves by means of the multiplicative spherical curves.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.1095-1103
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• 2016

B. Y. Chen introduced a series of curvature invariants, known as Chen invariants, and proved sharp estimates for these intrinsic invariants in terms of the main extrinsic invariant, the squared mean curvature, for submanifolds in Riemannian space forms. Special classes of submanifolds in Sasakian manifolds play an important role in contact geometry. F. Defever, I. Mihai and L. Verstraelen [8] established Chen first inequality for C-totally real submanifolds in Sasakian space forms. Also, the differential geometry of slant submanifolds has shown an increasing development since B. Y. Chen defined slant submanifolds in complex manifolds as a generalization of both holomorphic and totally real submanifolds. The slant submanifolds of an almost contact metric manifolds were defined and studied by A. Lotta, J. L. Cabrerizo et al. A Chen first inequality for slant submanifolds in Sasakian space forms was established by A. Carriazo [4]. In this article, we improve this Chen first inequality for special contact slant submanifolds in Sasakian space forms.

• Communications of the Korean Mathematical Society
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• v.35 no.2
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• pp.613-623
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• 2020

The notion of quasi-Einstein metric in theoretical physics and in relation with string theory is equivalent to the notion of Ricci soliton in differential geometry. Quasi-Einstein metrics or Ricci solitons serve also as solution to Ricci flow equation, which is an evolution equation for Riemannian metrics on a Riemannian manifold. Quasi-Einstein metrics are subject of great interest in both mathematics and theoretical physics. In this paper the notion of Ricci 𝜌-soliton as a generalization of Ricci soliton is defined. We are motivated by the Ricci-Bourguignon flow to define this concept. We show that if a 3-dimensional almost Kenmotsu Einstein manifold M is a 𝜌-soliton, then M is a Kenmotsu manifold of constant sectional curvature -1 and the 𝜌-soliton is expanding with λ = 2.