Hypersurfaces with Generalized 1-Type Gauss Maps

Dae Won Yoon, Dong-Soo Kim, Young Ho Kim and Jae Won Lee
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Dae Won Yoon: Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 52828, Korea
Dong-Soo Kim: Department of Mathematics, Chonnam National University, Gwangju 61186, Korea
Young Ho Kim: Department of Mathematics, Kyungpook National University, Daegu 41566, Korea
Jae Won Lee: Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 52828, Korea

Mathematics, 2018, vol. 6, issue 8, 1-14

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, E n , is said to be of generalized 1-type if, for the Laplace operator, Δ , on the submanifold, it satisfies Δ G = f G + g C , 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 E 3 . Second, we show that the Gauss map of any cylindrical surface in E 3 is of the generalized 1-type. Third, we prove that there are no tangent developable surfaces with generalized 1-type Gauss maps in E 3 , except planes. Finally, we show that cylindrical hypersurfaces in E n + 2 always have generalized 1-type Gauss maps.

Keywords: conical surface; developable surface; generalized 1-type Gauss map; cylindrical hypersurface (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2018
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