Ruled surfaces and Gauss map
Bull. Korean Math. Soc. 2015 Vol. 52, No. 5, 1661-1668
https://doi.org/10.4134/BKMS.2015.52.5.1661
Printed September 30, 2015
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
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