• Bulletin of the Korean Mathematical Society
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• v.53 no.2
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• pp.519-530
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• 2016

In this paper, we study surfaces of revolution in the three dimensional pseudo-Galilean space. We classify surfaces of revolution generated by a non-isotropic curve in terms of the Gauss map and the Laplacian of the surface. Furthermore, we give the classification of surfaces of revolution generated by an isotropic curve satisfying pointwise 1-type Gauss map equation.

• Journal of the Korean Mathematical Society
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• v.42 no.3
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• pp.447-455
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• 2005

In this article, we introduce the notion of pointwise 1-type Gauss map of the first and second kinds and study surfaces of revolution with such Gauss map. Our main results state that surfaces of revolution with pointwise 1-type Gauss map of the first kind coincide with surfaces of revolution with constant mean curvature; and the right cones are the only rational surfaces of revolution with pointwise 1-type Gauss map of the second kind.

• Journal of the Korean Mathematical Society
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• v.53 no.4
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• pp.909-928
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• 2016

Chen and Ishikawa studied the surfaces of revolution of the polynomial and the rational kind of finite type in Euclidean 3-space $E^3$ [10]. Here, the pedal of revolution surfaces of the polynomial and the rational kind are discussed. Also, as a special case of general revolution surfaces, the sphere and catenoid are studied for the kind of finite type.

• Journal of the Chungcheong Mathematical Society
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• v.25 no.4
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• pp.677-686
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• 2012

In this paper, we investigate the surfaces of revolution with light-like axis satisfying some equation in terms of a position vector field and the Laplacian with respect to the non-degenerate third fundamental form in Minkowski 3-space. As a result, we give some special example of the surfaces of revolution with light-like axis.

• Bulletin of the Korean Mathematical Society
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• v.50 no.4
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• pp.1061-1067
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• 2013

In this paper, we study surfaces of revolution without parabolic points in 3-Euclidean space $\mathbb{R}^3$, satisfying the condition ${\Delta}^{II}G=f(G+C)$, where ${\Delta}^{II}$ is the Laplace operator with respect to the second fundamental form, $f$ is a smooth function on the surface and C is a constant vector. Our main results state that surfaces of revolution without parabolic points in $\mathbb{R}^3$ which satisfy the condition ${\Delta}^{II}G=fG$, coincide with surfaces of revolution with non-zero constant Gaussian curvature.

• Bulletin of the Korean Mathematical Society
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• v.54 no.6
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• pp.2081-2089
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• 2017

In this paper we generalize 3-dimensional Bour's Theorem to the case of 4-dimension. We proved that a helicoidal surface in $\mathbb{R}^4$ is isometric to a family of surfaces of revolution in $\mathbb{R}^4$ in such a way that helices on the helicoidal surface correspond to parallel circles on the surfaces of revolution. Moreover, if the surfaces are required further to have the same Gauss map, then they are hyperplanar and minimal. Parametrizations for such minimal surfaces are given explicitly.

• The Pure and Applied Mathematics
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• v.19 no.1
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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.

• Kyungpook Mathematical Journal
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• v.60 no.3
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• pp.585-597
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• 2020

In this study, we classify surfaces of revolution of Type 1 in the three dimensional Galilean space 𝔾₃ in terms of the position vector field, Gauss map, and Laplacian operator of the first and the second fundamental forms on the surface. Furthermore, we give a classification of surfaces of revolution of Type 1 generated by a non-isotropic curve satisfying the pointwise 1-type Gauss map equation.

• Honam Mathematical Journal
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• v.38 no.1
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• pp.151-167
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• 2016

In Minkowskian 3-spacetime $L^3$ we find timelike or spacelike surface of revolution for the given Gauss curvature K = -1, 0, 1 and mean curvature H = 0. In fact, we set up the surface of revolution with the time axis for z-axis to be able to draw those surfaces on standard pictures in Minkowskian 3-spacetime $L^3$.

• Bulletin of the Korean Mathematical Society
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• v.53 no.1
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• pp.227-245
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• 2016

A generalized constant ratio surface (GCR surface) is defined by the property that the tangential component of the position vector is a principal direction at each point on the surface, see [8] for details. In this paper, by solving some differential equations, a complete classification of Lorentz GCR surfaces in the three-dimensional Minkowski space is presented. Moreover, it turns out that a flat Lorentz GCR surface is an open part of a cylinder, apart from a plane and a CMC Lorentz GCR surface is a surface of revolution.