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Classifications of Flat Surfaces with Generalized 1-Type Gauss Map in \({\mathbb L}^{3}\)

  • Dae Won Yoon
  • Dong-Soo Kim
  • Young Ho Kim
  • Jae Won Lee
Article
  • 44 Downloads

Abstract

In this paper, we study surfaces in the Minkowski 3-space \(\mathbb {L}^3\), such that \(\Delta G\) is a linear combination of the Gauss map G and a constant vector C, called generalized 1-type Gauss map. First of all, we prove that all cylindrical surfaces in \(\mathbb {L}^3\) have generalized 1-type Gauss map. Second, we classify conical surfaces with generalized 1-type Gauss map in \(\mathbb {L}^3\). After then, nonplanar tangent developable surfaces in the Minkowski 3-space \(\mathbb {L}^3\) with generalized 1-type Gauss map are open pieces of a Euclidean plane or a Minkowski plane. Finally, we show that a null scroll in the Minkowski 3-space \({\mathbb {L}}^3\) has generalized 1-type Gauss map if and only if it is either an open piece of a Minkowski plane or a B-scroll.

Keywords

Developable surface Generalized 1-type Gauss map Null scroll 

Mathematics Subject Classification

Primary 53A05 Secondary 53B25 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Dae Won Yoon
    • 1
  • Dong-Soo Kim
    • 2
  • Young Ho Kim
    • 3
  • Jae Won Lee
    • 1
  1. 1.Department of Mathematics and RINSGyeongsang National UniversityJinjuRepublic of Korea
  2. 2.Department of MathematicsChonnam National UniversityGwangjuRepublic of Korea
  3. 3.Department of MathematicsKyungpook National UniversityDaeguRepublic of Korea

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