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J. Korean Math. Soc. 2009; 46(1): 215-223
Printed January 1, 2009
Copyright © The Korean Mathematical Society.
Helicoidal surfaces with pointwise 1-type Gauss map
Miekyung Choi, Dong-Soo Kim, and Young Ho Kim
Chonnam National University, Chonnam National University, and Kyungpook National University
The helicoidal surfaces with pointwise 1-type or harmonic gauss map in Euclidean $3$-space are studied. The notion of pointwise 1-type Gauss map is a generalization of usual sense of 1-type Gauss map. In particular, we prove that an ordinary helicoid is the only genuine helicoidal surface of polynomial kind with pointwise 1-type Gauss map of the first kind and a right cone is the only rational helicoidal surface with pointwise 1-type Gauss map of the second kind. Also, we give a characterization of rational helicoidal surface with harmonic or pointwise 1-type Gauss map.
Keywords: helicoidal surfaces, helicoid, right cone, pointwise 1-type Gauss map
MSC numbers: 53B25, 53C40
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