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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, 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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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.
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Surfaces in the Euclidean Space E4 with Pointwise 1-Type Gauss Map
Hacettepe Journal of Mathematics and Statistics, 2011Co-Authors: Uğur Dursun, Güler Gürpınar ArsanAbstract:In this article we study surfaces in Euclidean space E 4 with pointwise 1-type Gauss map. We give a characterization of surfaces in E 4 with a pointwise 1-type Gauss map of the first kind. We conclude that an oriented non-minimal surface M in E 4 has a pointwise 1-type Gauss map of the first kind if and only if M is a surface in a 3-sphere of E 4 with constant mean curvature. We also obtain a characterization for non-planar minimal surfaces in E 4 with pointwise 1-type Gauss map of the second kind. Further we give a partial classification of surfaces in E 4 in terms of the pointwise 1-type Gauss map of the second kind.
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Hypersurfaces with Pointwise 1-type Gauss Map in Lorentz-Minkowski Space
Proceedings of the Estonian Academy of Sciences, 2009Co-Authors: Uğur DursunAbstract:Hypersurfaces of a Lorentz-Minkowski space L(superscript n+1) with pointwise 1-type Gauss map are characterized. We prove that an oriented hypersurface M(superscript q) in L(superscript n+1) has pointwise 1-type Gauss map of the first kind if and only if M(superscript q) has constant mean curvature and conclude that all oriented isoparametric hypersurfaces in L(superscript n+1) have 1-type Gauss map. Then we classify rational rotation hypersurfaces of L(superscript n+1) with pointwise 1-type Gauss map and give some examples.
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, 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
Elif Özkara Canfes - 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, 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
Rüya Yeğin - One of the best experts on this subject based on the ideXlab platform.
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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.
Dong Soo Kim - One of the best experts on this subject based on the ideXlab platform.
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Hypersurfaces with Generalized 1-Type Gauss Maps
Mathematics, 2018Co-Authors: Dae Won Yoon, Dong Soo Kim, Young Ho Kim, Jae Won LeeAbstract:In this paper, we study submanifolds in a Euclidean space with a generalized 1-type Gauss map. The Gauss map, G, of a submanifold in the n-dimensional Euclidean space, En, is said to be of generalized 1-type if, for the Laplace operator, Δ, on the submanifold, it satisfies ΔG=fG+gC, where C is a constant vector and f and g are some functions. The notion of a generalized 1-type Gauss map is a generalization of both a 1-type Gauss map and a pointwise 1-type Gauss map. With the new definition, first of all, we classify conical surfaces with a generalized 1-type Gauss map in E3. Second, we show that the Gauss map of any cylindrical surface in E3 is of the generalized 1-type. Third, we prove that there are no tangent developable surfaces with generalized 1-type Gauss maps in E3, except planes. Finally, we show that cylindrical hypersurfaces in En+2 always have generalized 1-type Gauss maps.
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Gauss Maps of Ruled Submanifolds and Applications II
Taiwanese Journal of Mathematics, 2016Co-Authors: Dong Soo Kim, Sun Mi Jung, Young Ho Kim, Dae Won YoonAbstract:The notion of pointwise $1$-type Gauss map was derived from the ordinary finite type Gauss map and it gives an interesting geometric properties on surfaces of $3$-dimensional Euclidean space. In particular, the helicoid and the right cone of $3$-dimensional Euclidean space are characterized by pointwise $1$-type Gauss map. Inspired by such a study, in this paper, we completely classify ruled submanifolds of Euclidean space with pointwise $1$-type Gauss map.
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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.
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helicoidal surfaces with pointwise 1 type Gauss map
Journal of Korean Medical Science, 2009Co-Authors: Miekyung Choi, Dong Soo Kim, Young Ho KimAbstract:The helicoidal surfaces with pointwise 1-type or harmonic Gauss map in Euclidean 3-space are studied. The notion of pointwise 1- type Gauss map is a generalization of usual sense of 1-type Gauss map. In particular, we prove that an ordinary helicoid is the only genuine he- licoidal surface of polynomial kind with pointwise 1-type Gauss map of the flrst kind and a right cone is the only rational helicoidal surface with pointwise 1-type Gauss map of the second kind. Also, we give a charac- terization of rational helicoidal surface with harmonic or pointwise 1-type Gauss map.
