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
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,
, is said to be of generalized 1-type if, for the Laplace operator,
, on the submanifold, it satisfies
, where C
is a constant vector and f
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
. Second, we show that the Gauss map of any cylindrical surface in
is of the generalized 1-type. Third, we prove that there are no tangent developable surfaces with generalized 1-type Gauss maps in
, except planes. Finally, we show that cylindrical hypersurfaces in
always have generalized 1-type Gauss maps.
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MDPI and ACS Style
Yoon, D.W.; Kim, D.-S.; Kim, Y.H.; Lee, J.W. Hypersurfaces with Generalized 1-Type Gauss Maps. Mathematics 2018, 6, 130.
Yoon DW, Kim D-S, Kim YH, Lee JW. Hypersurfaces with Generalized 1-Type Gauss Maps. Mathematics. 2018; 6(8):130.
Yoon, Dae W.; Kim, Dong-Soo; Kim, Young H.; Lee, Jae W. 2018. "Hypersurfaces with Generalized 1-Type Gauss Maps." Mathematics 6, no. 8: 130.
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