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.
Doan The Hieu, Nguyen Ngoc Thang
Hue University, Hue University
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
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