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 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.b160766Published 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 Full-Text :