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Uğur Dursun - One of the best experts on this subject based on the ideXlab platform.
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pseudo spherical submanifolds with 1 type pseudo spherical Gauss Map
Results in Mathematics, 2017Co-Authors: Burcu Bektaş, Elif Özkara Canfes, Uğur DursunAbstract:In this work, we study pseudo-Riemannian submanifolds of a pseudo-sphere with 1-type pseudo-spherical Gauss Map. First, we classify Lorentzian surfaces in a 4-dimensional pseudo-sphere \({\mathbb{S}^4_s(1)}\) with index s, \({s=1, 2}\), and having harmonic pseudo-spherical Gauss Map. Then we give a characterization theorem for pseudo-Riemannian submanifolds of a pseudo-sphere \({\mathbb{S}^{m-1}_s(1)\subset\mathbb{E}^m_s}\) with 1-type pseudo-spherical Gauss Map, and we classify spacelike surfaces and Lorentzian surfaces in the de Sitter space \({\mathbb{S}^4_1(1)\subset\mathbb{E}^5_1}\) with 1-type pseudo-spherical Gauss Map. Finally, according to the causal character of the mean curvature vector we obtain the classification of submanifolds of a pseudo-sphere having 1-type pseudo-spherical Gauss Map with nonzero constant component in its spectral decomposition.
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Pseudo-Spherical Submanifolds with 1-Type Pseudo-Spherical Gauss Map
Results in Mathematics, 2016Co-Authors: Burcu Bektaş, Elif Özkara Canfes, Uğur DursunAbstract:In this work, we study pseudo-Riemannian submanifolds of a pseudo-sphere with 1-type pseudo-spherical Gauss Map. First, we classify Lorentzian surfaces in a 4-dimensional pseudo-sphere (Formula presented.) with index s, (Formula presented.), and having harmonic pseudo-spherical Gauss Map. Then we give a characterization theorem for pseudo-Riemannian submanifolds of a pseudo-sphere (Formula presented.) with 1-type pseudo-spherical Gauss Map, and we classify spacelike surfaces and Lorentzian surfaces in the de Sitter space (Formula presented.) with 1-type pseudo-spherical Gauss Map. Finally, according to the causal character of the mean curvature vector we obtain the classification of submanifolds of a pseudo-sphere having 1-type pseudo-spherical Gauss Map with nonzero constant component in its spectral decomposition.Publisher's Versio
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pseudo spherical submanifolds with 1 type pseudo spherical Gauss Map
arXiv: Differential Geometry, 2015Co-Authors: Burcu Bektaş, Elif Özkara Canfes, Uğur DursunAbstract:In this work, we study the pseudo-Riemannian submanifolds of a pseudo-sphere with 1-type pseudo-spherical Gauss Map. First, we classify the Lorentzian surfaces in a 4-dimensional pseudo-sphere $\mathbb{S}^4_s(1)$ with index s, $s=1, 2$, and having harmonic pseudo-spherical Gauss Map. Then we give a characterization theorem for pseudo-Riemannian submanifolds of a pseudo-sphere $\mathbb{S}^{m-1}_s(1)\subset\mathbb{E}^m_s$ with 1-type pseudo-spherical Gauss Map, and we classify spacelike surfaces and Lorentzian surfaces in the de Sitter space $\mathbb{S}^4_1(1)\subset\mathbb{E}^5_1$ with 1-type pseudo-spherical Gauss Map. Finally, according to the causal character of the mean curvature vector we obtain the classification of submanifolds of a pseudo-sphere having 1-type pseudo-spherical Gauss Map with nonzero constant component in its spectral decomposition.
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Hyperbolic submanifolds with finite type hyperbolic Gauss Map
International Journal of Mathematics, 2015Co-Authors: Uğur Dursun, Rüya YeğinAbstract:We study submanifolds of hyperbolic spaces with finite type hyperbolic Gauss Map. First, we classify the hyperbolic submanifolds with 1-type hyperbolic Gauss Map. Then we prove that a non-totally umbilical hypersurface Mn with nonzero constant mean curvature in a hyperbolic space has 2-type hyperbolic Gauss Map if and only if M has constant scalar curvature. We also classify surfaces with constant mean curvature in the hyperbolic space having 2-type hyperbolic Gauss Map. Moreover we show that a horohypersphere in has biharmonic hyperbolic Gauss Map.
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Hyperbolic submanifolds with finite type hyperbolic Gauss Map
International Journal of Mathematics, 2015Co-Authors: Uğur Dursun, Rüya YeğinAbstract:We study submanifolds of hyperbolic spaces with finite type hyperbolic Gauss Map. First, we classify the hyperbolic submanifolds with 1-type hyperbolic Gauss Map. Then we prove that a non-totally umbilical hypersurface Mn with nonzero constant mean curvature in a hyperbolic space [Formula: see text] has 2-type hyperbolic Gauss Map if and only if M has constant scalar curvature. We also classify surfaces with constant mean curvature in the hyperbolic space [Formula: see text] having 2-type hyperbolic Gauss Map. Moreover we show that a horohypersphere in [Formula: see text] has biharmonic hyperbolic Gauss Map.
Young Ho Kim - One of the best experts on this subject based on the ideXlab platform.
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Some Classification of Canal Surfaces with the Gauss Map
Bulletin of the Malaysian Mathematical Sciences Society, 2018Co-Authors: Jinhua Qian, Young Ho KimAbstract:In this paper, we study canal surfaces in the Euclidean 3-space $$\mathbb {E}^{3}$$ in terms of their Gauss Map $$\mathbb {G}$$ . We obtain a complete classification of canal surfaces whose Gauss Maps are of the so-called pointwise 1-type, i.e., the Gauss Map $$\mathbb {G}$$ satisfies $$\Delta \mathbb {G}=f(\mathbb {G}+C)$$ for a nonzero smooth function f and a constant vector C, where $$\Delta $$ denotes the Laplace operator.
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Gauss Map and its applications on ruled submanifolds in minkowski space
Symmetry, 2018Co-Authors: Sun Mi Jung, Young Ho KimAbstract:We study ruled submanifolds in Minkowski space in regard to the Gauss Map satisfying some partial differential equation. As a generalization of usual cylinders, cones and null scrolls in a three-dimensional Minkowski space, a cylinder over a space curve, a product manifold of a right cone and a k-plane, a product manifold of a hyperbolic cone and a k-plane which look like kinds of cylinders over cones in 3-space, and the generalized B-scroll kind in Minkowski space are characterized with the partial differential equation regarding the Gauss Map, where k is a positive integer.
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classifications of canal surfaces with l1 pointwise 1 type Gauss Map
Milan Journal of Mathematics, 2015Co-Authors: Jinhua Qian, Young Ho KimAbstract:In this paper, we study canal surfaces in the Euclidean 3-space \({\mathbb{E}^{3}}\) in terms of their Gauss Map. We obtain a complete classification of such surfaces whose Gauss Map G satisfies \({{\Box G = f(G + C)}}\) for a non-zero smooth function f and a constant vector C, where \({\Box}\) denotes the Cheng-Yau operator.
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cheng yau operator and Gauss Map of surfaces of revolution
arXiv: Differential Geometry, 2014Co-Authors: Dong Soo Kim, Jong Ryul Kim, Young Ho KimAbstract:We study the Gauss Map $G$ of surfaces of revolution in the 3-dimensional Euclidean space ${\mathbb{E}^3}$ with respect to the so called Cheng-Yau operator $\square$ acting on the functions defined on the surfaces. As a result, we establish the classification theorem that the only surfaces of revolution with Gauss Map $G$ satisfying $\square G=AG$ for some $3\times3$ matrix $A$ are the planes, right circular cones, circular cylinders and spheres.
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SOME CLASSIFICATIONS OF RULED SUBMANIFOLDS IN MINKOWSKI SPACE AND THEIR Gauss Map
Taiwanese Journal of Mathematics, 2014Co-Authors: Dong Soo Kim, Young Ho Kim, Sun Mi JungAbstract:Ruled submanifolds of Minkowski space with finite-type Gauss Map are studied. Not having a parallel in Euclidean space, ruled submanifolds with degenerate rulings in Minkowski space drew our attention. We show that if non-cylindrical ruled submanifolds with non-degenerate rulings or ruled submanifolds with degenerate rulings have finite-type Gauss Map, the Gauss Map is one of the following: (1) harmonic; (2) of the so-called finite rank; (3) of null 2-type. For ruled submanifolds with degenerate rulings, we set up a relationship between finite-type immersions and immersions with finite-type Gauss Map and introduce new examples of ruled submanifolds with degenerate rulings. We also characterize minimal ruled submanifolds with degenerate rulings in terms of finite-type Gauss Map.
Luiz C B Da Silva - One of the best experts on this subject based on the ideXlab platform.
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invariant surfaces with coordinate finite type Gauss Map in simply isotropic space
Journal of Mathematical Analysis and Applications, 2020Co-Authors: Alev Kelleci, Luiz C B Da SilvaAbstract:Abstract We consider the extrinsic geometry of surfaces in simply isotropic space, a three-dimensional space equipped with a rank 2 metric of index zero. Since the metric is degenerate, a surface normal cannot be unequivocally defined based on metric properties only. To understand the contrast between distinct choices of an isotropic Gauss Map, here we study surfaces with a Gauss Map whose coordinates are eigenfunctions of the surface Laplace-Beltrami operator. We take into account two choices, the so-called minimal and parabolic normals, and show that when applied to simply isotropic invariant surfaces the condition that the coordinates of the corresponding Gauss Map are eigenfunctions leads to planes, certain cylinders, or surfaces with constant isotropic mean curvature. Finally, we also investigate (non-necessarily invariant) surfaces with harmonic Gauss Map and show this characterizes constant mean curvature surfaces.
Burcu Bektaş - One of the best experts on this subject based on the ideXlab platform.
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pseudo spherical submanifolds with 1 type pseudo spherical Gauss Map
Results in Mathematics, 2017Co-Authors: Burcu Bektaş, Elif Özkara Canfes, Uğur DursunAbstract:In this work, we study pseudo-Riemannian submanifolds of a pseudo-sphere with 1-type pseudo-spherical Gauss Map. First, we classify Lorentzian surfaces in a 4-dimensional pseudo-sphere \({\mathbb{S}^4_s(1)}\) with index s, \({s=1, 2}\), and having harmonic pseudo-spherical Gauss Map. Then we give a characterization theorem for pseudo-Riemannian submanifolds of a pseudo-sphere \({\mathbb{S}^{m-1}_s(1)\subset\mathbb{E}^m_s}\) with 1-type pseudo-spherical Gauss Map, and we classify spacelike surfaces and Lorentzian surfaces in the de Sitter space \({\mathbb{S}^4_1(1)\subset\mathbb{E}^5_1}\) with 1-type pseudo-spherical Gauss Map. Finally, according to the causal character of the mean curvature vector we obtain the classification of submanifolds of a pseudo-sphere having 1-type pseudo-spherical Gauss Map with nonzero constant component in its spectral decomposition.
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Pseudo-Spherical Submanifolds with 1-Type Pseudo-Spherical Gauss Map
Results in Mathematics, 2016Co-Authors: Burcu Bektaş, Elif Özkara Canfes, Uğur DursunAbstract:In this work, we study pseudo-Riemannian submanifolds of a pseudo-sphere with 1-type pseudo-spherical Gauss Map. First, we classify Lorentzian surfaces in a 4-dimensional pseudo-sphere (Formula presented.) with index s, (Formula presented.), and having harmonic pseudo-spherical Gauss Map. Then we give a characterization theorem for pseudo-Riemannian submanifolds of a pseudo-sphere (Formula presented.) with 1-type pseudo-spherical Gauss Map, and we classify spacelike surfaces and Lorentzian surfaces in the de Sitter space (Formula presented.) with 1-type pseudo-spherical Gauss Map. Finally, according to the causal character of the mean curvature vector we obtain the classification of submanifolds of a pseudo-sphere having 1-type pseudo-spherical Gauss Map with nonzero constant component in its spectral decomposition.Publisher's Versio
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pseudo spherical submanifolds with 1 type pseudo spherical Gauss Map
arXiv: Differential Geometry, 2015Co-Authors: Burcu Bektaş, Elif Özkara Canfes, Uğur DursunAbstract:In this work, we study the pseudo-Riemannian submanifolds of a pseudo-sphere with 1-type pseudo-spherical Gauss Map. First, we classify the Lorentzian surfaces in a 4-dimensional pseudo-sphere $\mathbb{S}^4_s(1)$ with index s, $s=1, 2$, and having harmonic pseudo-spherical Gauss Map. Then we give a characterization theorem for pseudo-Riemannian submanifolds of a pseudo-sphere $\mathbb{S}^{m-1}_s(1)\subset\mathbb{E}^m_s$ with 1-type pseudo-spherical Gauss Map, and we classify spacelike surfaces and Lorentzian surfaces in the de Sitter space $\mathbb{S}^4_1(1)\subset\mathbb{E}^5_1$ with 1-type pseudo-spherical Gauss Map. Finally, according to the causal character of the mean curvature vector we obtain the classification of submanifolds of a pseudo-sphere having 1-type pseudo-spherical Gauss Map with nonzero constant component in its spectral decomposition.
Dae Won Yoon - One of the best experts on this subject based on the ideXlab platform.
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Classifications of Flat Surfaces with Generalized 1-Type Gauss Map in $${\mathbb L}^{3}$$
Mediterranean Journal of Mathematics, 2018Co-Authors: Dae Won YoonAbstract:In this paper, we study surfaces in the Minkowski 3-space $$\mathbb {L}^3$$ L 3 , such that $$\Delta G$$ Δ G is a linear combination of the Gauss Map G and a constant vector C , called generalized 1-type Gauss Map . First of all, we prove that all cylindrical surfaces in $$\mathbb {L}^3$$ L 3 have generalized 1-type Gauss Map. Second, we classify conical surfaces with generalized 1-type Gauss Map in $$\mathbb {L}^3$$ L 3 . After then, nonplanar tangent developable surfaces in the Minkowski 3-space $$\mathbb {L}^3$$ L 3 with generalized 1-type Gauss Map are open pieces of a Euclidean plane or a Minkowski plane. Finally, we show that a null scroll in the Minkowski 3-space $${\mathbb {L}}^3$$ L 3 has generalized 1-type Gauss Map if and only if it is either an open piece of a Minkowski plane or a B -scroll.
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classification of ruled surfaces with pointwise 1 type Gauss Map in minkowski 3 space
Taiwanese Journal of Mathematics, 2011Co-Authors: Miekyung Choi, Young Ho Kim, Dae Won YoonAbstract:We study the ruled surfaces in Minkowski 3-space with pointwise 1-type Gauss Map. As a result, we introduce some new examples of the ruled surfaces with pointwise 1-type Gauss Map in Minkowski 3-space.
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CLASSIFICATION OF RULED SURFACES WITH POINTWISE 1-TYPE Gauss Map
Taiwanese Journal of Mathematics, 2010Co-Authors: Miekyung Choi, Young Ho Kim, Dae Won YoonAbstract:Ruled surfaces with the Gauss Map satisfying a partial differential equation which is similar to an eigenvalue problem in a 3-dimensional Euclidean space are studied. Such a Gauss Map is said to be of pointwise 1-type, namely, the Gauss Map $G$ satisfies $\Delta G = f(G+C)$, where $\Delta$ is the Laplacian operator, $f$ is a non-zero function and $C$ is a constant vector. As a result, such ruled surfaces are completely determined by the function $f$ and the vector $C$ when their Gauss Map is of pointwise 1-type. New examples of ruled surfaces called cylinders of an infinite type and rotational ruled surfaces are introduced in this regard.
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On the Gauss Map of Ruled Surfaces in Minkowsi Space
Rocky Mountain Journal of Mathematics, 2005Co-Authors: Young Ho Kim, Dae Won YoonAbstract:In this paper, we study some characterization of ruled surfaces in Minkowski space in terms of the Gauss Map. We give new examples of cylindrical and noncylindrical ruled surfaces in a 4-dimensional Minkowski space with the pointwise 1-type Gauss Map.
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ruled surfaces with pointwise 1 type Gauss Map
Journal of Geometry and Physics, 2000Co-Authors: Young Ho Kim, Dae Won YoonAbstract:Abstract In this paper, we study ruled surfaces in a three-dimensional Minkowski space with pointwise 1-type Gauss Map and obtain the complete classification theorems for those. We also obtain a new characterization of minimal ruled surfaces in a three-dimensional Minkowski space.