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RULED SURFACES AND GAUSS MAP

  • KIM, DONG-SOO
  • Received : 2014.11.04
  • Published : 2015.09.30

Abstract

We study the Gauss map G of ruled surfaces in the 3-dimensional Euclidean space $\mathbb{E}^3$ with respect to the so called Cheng-Yau operator ${\Box}$ acting on the functions defined on the surfaces. As a result, we establish the classification theorem that the only ruled surfaces with Gauss map G satisfying ${\Box}G=AG$ for some $3{\times}3$ matrix A are the flat ones. Furthermore, we show that the only ruled surfaces with Gauss map G satisfying ${\Box}G=AG$ for some nonzero $3{\times}3$ matrix A are the cylindrical surfaces.

Keywords

Gauss map;Cheng-Yau operator;ruled surface

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Acknowledgement

Supported by : Chonnam National University