• Honam Mathematical Journal
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• 제22권1호
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• pp.83-90
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• 2000

We establish a sufficient condition for a simple closed curve in the Euclidean plain $\mathbb{R}^2$, the unit sphere $S^2$ or the hyperbolic plane $H^2$ to be the boundary of a metric ball.

• Honam Mathematical Journal
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• 제43권1호
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• pp.88-99
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• 2021

The purpose of this paper is to assign a movable frame to an arbitrary point of a rational Bézier curve on the 2-sphere S2 in Euclidean 3-space R3 to provide a better understanding of the geometry of the curve. Especially, we obtain the formula of geodesic curvature for a quadratic rational Bézier curve that allows a curve to be characterized on the surface. Moreover, we give some important results and relations for the Darboux frame and geodesic curvature of a such curve. Then, in specific case, given characterizations for the quadratic rational Bézier curve are illustrated on a unit 2-sphere.

• Journal for History of Mathematics
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• 제16권3호
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• pp.109-114
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• 2003

In this paper, we deal with partially conformal geodesic transformations in Kahler geometry by using Fermi coordinates when tile submanifold is a geodesic sphere. We derive the necessary and sufficient condition for tile existence of such transformation in terms of the Jacobi operator and its derivative.

• East Asian mathematical journal
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• 제21권1호
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• pp.113-122
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• 2005

Let $\mathbb{H}^3$ be the 3-dimensional Heisenberg group equipped with a left-invariant metric. In this paper, We characterize the Gaussian curvatures of the geodesic spheres on $\mathbb{H}^3$.

• Communications of the Korean Mathematical Society
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• 제25권1호
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• pp.83-96
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• 2010

Let ${\mathbb{H}}^{2n+1}$ be the (2n+1)-dimensional Heisenberg group equipped with a left-invariant metric. In this paper we study the Gaussian curvatures of the geodesic spheres and the volumes of geodesic balls in ${\mathbb{H}}^{2n+1}$.

• Honam Mathematical Journal
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• 제45권3호
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• pp.513-541
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• 2023

In this paper, we calculate Frenet frames, Frenet derivative formulas, curvatures, arc lengths, geodesic curvatures according to the Lorentzian 3-space ℝ³₁, Lorentzian sphere 𝕊²₁ and hyperbolic sphere ℍ²₀ of the spherical indicatrix curves of the spacelike Salkowski curve with the timelike principal normal in ℝ³₁ and draw the graphs of these indicatrix curves on the spheres.

• Advances in biomechanics and applications
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• 제1권2호
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• pp.85-93
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• 2014

In this paper, two procedures of enumerating the axial rotation are proposed using the unit sphere of the spherical rotation coordinate system specifying 3D rotation. If the trajectory of the movement is known, the integration of the axial component of the angular velocity plus the geometric effect equal to the enclosed area subtended by the geodesic path on the surface of the unit sphere. If the postures of the initial and final positions are known, the axial rotation is determined by the angular difference from the parallel transport along the geodesic path. The path dependency of the axial rotation of the three dimensional rigid body motion is due to the geometric effect corresponding to the closed loop discontinuity. Firstly, the closed loop discontinuity is examined for the infinitesimal region. The general closed loop discontinuity can be evaluated by the summation of those discontinuities of the infinitesimal regions forming the whole loop. This general loop discontinuity is equal to the surface area enclosed by the closed loop on the surface of the unit sphere. Using this quantification of the closed loop discontinuity of the axial rotation, the geometric effect is determined in enumerating the axial rotation. As an example, the axial rotation of the arm by the Codman's movement is evaluated, which other methods of enumerating the axial rotations failed.

• Bulletin of the Korean Mathematical Society
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• 제51권5호
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• pp.1375-1397
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• 2014

In this paper we establish codimension reduction theorem for submanifolds of a (4m+3)-dimensional unit sphere $S^{4m+3}$ with Sasakian 3-structure and apply it to submanifolds of a quaternionic projective space.

• Bulletin of the Korean Mathematical Society
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• 제36권2호
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• pp.395-402
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• 1999

We introduce the notion of strongly k-disk homogeneous apace and establish a characterization theorem. More specifically, we prove that any analytic Riemannian manifold (M,g) of dimension n which is strongly k-disk homogeneous with 2$\leq$k$\leq$n-1 is a space of constant curvature. Its K hler analog is obtained. The total mean curvature homogeneity of geodesic sphere in k-disk is also considered.

• The Pure and Applied Mathematics
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• 제19권1호
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• pp.1-5
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• 2012

We study surfaces of revolution in the three dimensional Euclidean space $\mathbb{R}^3$ with two distinct axes of revolution. As a result, we prove that if a connected surface in the three dimensional Euclidean space $\mathbb{R}^3$ admits two distinct axes of revolution, then it is either a sphere or a plane.