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Mathematics 2018, 6(8), 130; https://doi.org/10.3390/math6080130

Hypersurfaces with Generalized 1-Type Gauss Maps

1
Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 52828, Korea
2
Department of Mathematics, Chonnam National University, Gwangju 61186, Korea
3
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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Abstract

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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