Circular cone and its Gauss map

Tom 129 / 2012

Miekyung Choi, Dong-Soo Kim, Young Ho Kim, Dae Won Yoon Colloquium Mathematicum 129 (2012), 203-210 MSC: Primary 53B25; Secondary 53C40. DOI: 10.4064/cm129-2-4

Streszczenie

The family of cones is one of typical models of non-cylindrical ruled surfaces. Among them, the circular cones are unique in the sense that their Gauss map satisfies a partial differential equation similar, though not identical, to one characterizing the so-called 1-type submanifolds. Specifically, for the Gauss map $G$ of a circular cone, one has $\varDelta G = f(G+C)$, where $\varDelta $ is the Laplacian operator, $f$ is a non-zero function and $C$ is a constant vector. We prove that circular cones are characterized by being the only non-cylindrical ruled surfaces with $\varDelta G = f(G+C)$ for a nonzero constant vector $C$.

Autorzy

  • Miekyung ChoiDepartment of Mathematics Education and RINS
    Gyeongsang National University
    Jinju 660-701, Korea
    e-mail
  • Dong-Soo KimDepartment of Marthematics
    Chonnam National University
    Kwangju 500-757, Korea
    e-mail
  • Young Ho KimDepartment of Marthematics
    Kyungpook National University
    Taegu 702-701, Korea
    e-mail
  • Dae Won YoonDepartment of Mathematics Education and RINS
    Gyeongsang National University
    Jinju 660-701, Korea
    e-mail

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