Bour's Theorem in 4-dimensional Euclidean space
Bull. Korean Math. Soc. 2017 Vol. 54, No. 6, 2081-2089
https://doi.org/10.4134/BKMS.b160766
Published online November 30, 2017
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
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