Bulletin of the
Korean Mathematical Society
BKMS

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

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Bull. Korean Math. Soc. 2017; 54(6): 2081-2089

Online first article July 21, 2017      Printed November 30, 2017

https://doi.org/10.4134/BKMS.b160766

Copyright © The Korean Mathematical Society.

Bour's Theorem in 4-dimensional Euclidean space

Doan The Hieu, Nguyen Ngoc Thang

Hue University, Hue University

Abstract

In this paper we generalize 3-dimensional Bour's Theorem to the case of 4-dimension. We proved that a helicoidal surface in $\mathbb R^4$ is isometric to a family of surfaces of revolution in $\mathbb R^4$ in such a way that helices on the helicoidal surface correspond to parallel circles on the surfaces of revolution. Moreover, if the surfaces are required further to have the same Gauss map, then they are hyperplanar and minimal. Parametrizations for such minimal surfaces are given explicitly.

Keywords: Bour's theorem, helicoidal surface, surface of revolution, Gauss map, minimal surface

MSC numbers: 53A07, 53A10