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Mathematics 2018, 6(8), 130;

Hypersurfaces with Generalized 1-Type Gauss Maps

Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 52828, Korea
Department of Mathematics, Chonnam National University, Gwangju 61186, Korea
Department of Mathematics, Kyungpook National University, Daegu 41566, Korea
Author to whom correspondence should be addressed.
Received: 18 May 2018 / Revised: 18 July 2018 / Accepted: 23 July 2018 / Published: 26 July 2018
(This article belongs to the Special Issue Differential Geometry)
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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. View Full-Text
Keywords: conical surface; developable surface; generalized 1-type Gauss map; cylindrical hypersurface conical surface; developable surface; generalized 1-type Gauss map; cylindrical hypersurface

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Yoon, D.W.; Kim, D.-S.; Kim, Y.H.; Lee, J.W. Hypersurfaces with Generalized 1-Type Gauss Maps. Mathematics 2018, 6, 130.

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