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Hypersurfaces with Generalized 1-Type Gauss Maps

Author

Listed:
  • 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)

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.

Suggested Citation

  • Dae Won Yoon & Dong-Soo Kim & Young Ho Kim & Jae Won Lee, 2018. "Hypersurfaces with Generalized 1-Type Gauss Maps," Mathematics, MDPI, vol. 6(8), pages 1-14, July.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:8:p:130-:d:160017
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