Abstract
We study and examine the rotational hypersurface and its Gauss map in Euclidean four-space E 4 . We calculate the Gauss map, the mean curvature and the Gaussian curvature of the rotational hypersurface and obtain some results. Then, we introduce the third Laplace–Beltrami operator. Moreover, we calculate the third Laplace–Beltrami operator of the rotational hypersurface in E 4 . We also draw some figures of the rotational hypersurface.
Highlights
When we focus on the rotational characters in the literature, we meet Arslan et al [1,2], Arvanitoyeorgos et al [3], Chen [4,5], Dursun and Turgay [6], Kim and Turgay [7], Takahashi [8], and many others.Magid, Scharlach and Vrancken [9] introduced the affine umbilical surfaces in four-space.Vlachos [10] considered hypersurfaces in E4 with the harmonic mean curvature vector field.Scharlach [11] studied the affine geometry of surfaces and hypersurfaces in four-space
General rotational surfaces in E4 were introduced by Moore [13,14]
Considered these kinds of surfaces in the Minkowski four-space. They classified completely the minimal rotational surfaces and those consisting of parabolic points
Summary
When we focus on the rotational characters in the literature, we meet Arslan et al [1,2], Arvanitoyeorgos et al [3], Chen [4,5], Dursun and Turgay [6], Kim and Turgay [7], Takahashi [8], and many others. Scharlach [11] studied the affine geometry of surfaces and hypersurfaces in four-space. Considered these kinds of surfaces in the Minkowski four-space They classified completely the minimal rotational surfaces and those consisting of parabolic points. For the characters of ruled (helicoid) and rotational surfaces, please see Bour’s theorem in [18]. Hieu and Thang [21] studied helicoidal surfaces by Bour’s theorem in four-space. Choi et al [22] studied helicoidal surfaces and their Gauss map in Minkowski three-space. We consider the rotational hypersurface with three-parameters and its Gauss map in Euclidean four-space E4.
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