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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.
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참고문헌 (11)
참고문헌 신청
E. Bour / 1862 / Memoire sur le deformation de surfaces / Journal de l'^Ecole Polytechnique 39 : 1 ~ 148
E. Guler / 2006 / Bour's theorem in Minkowski 3-space / J. Math. Kyoto Univ 46 (1) : 47 ~ 63
E. Guler / 2007 / Bour's theorem and lightlike profile curve / Yokohama Math. J 54 (1) : 55 ~ 77
E. Guler / 2010 / Bour's theorem on the Gauss map in 3-Euclidean space / Hacet. J. Math. Stat 39 (4) : 515 ~ 525
F. Ji / 2010 / Mean curvatures and Gauss maps of a pair of isometric helicoidal and rotation surfaces in Minkowski 3-space / J. Math. Anal. Appl 368 (2) : 623 ~ 635
E. Guler / 2006 / Bour's theorem in Minkowski 3-space / J. Math. Kyoto Univ 46 (1) : 47 ~ 63
E. Guler / 2007 / Bour's theorem and lightlike profile curve / Yokohama Math. J 54 (1) : 55 ~ 77
E. Guler / 2010 / Bour's theorem on the Gauss map in 3-Euclidean space / Hacet. J. Math. Stat 39 (4) : 515 ~ 525
F. Ji / 2010 / Mean curvatures and Gauss maps of a pair of isometric helicoidal and rotation surfaces in Minkowski 3-space / J. Math. Anal. Appl 368 (2) : 623 ~ 635
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