Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2015; 52(5): 1661-1668

Printed September 30, 2015

https://doi.org/10.4134/BKMS.2015.52.5.1661

Copyright © The Korean Mathematical Society.

Ruled surfaces and Gauss map

Dong-Soo Kim

Chonnam National University

Abstract

We study the Gauss map $G$ of ruled surfaces in the 3-dim\-ensional Euclidean space ${\mathbb{E}^3}$ with respect to the so called Cheng-Yau operator $\square$ 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 $\square G=AG$ for some $3\times3$ matrix $A$ are the flat ones. Furthermore, we show that the only ruled surfaces with Gauss map $G$ satisfying $\square G=AG$ for some nonzero $3\times3$ matrix $A$ are the cylindrical surfaces.

Keywords: Gauss map, Cheng-Yau operator, ruled surface

MSC numbers: 53A05, 53B25