Rotational embeddings in E^4 with pointwise 1-type gauss map

Differential Geometry of 1-type Submanifolds and Submanifolds with 1-type Gauss Map

“…al. gave a complete classification of rational surfaces of revolution with pointwise 1-type Gauss map in Minkowski 3-space E 3 1 in [111], and they proved the following. Theorem 17.17.…”

“…4.3. 1-type surfaces in E 3 with respect to ∆ J (J = II, III ) S. Stamatakis and H. Al-Zoubi studied finite type surfaces M in E 3 via the Beltrami-Laplace operators ∆ J (J = II, III ) corresponding to the second and the third fundamental form respectively of a surface M in E 3 . In defining the operators ∆ II and ∆ III on M , they assumed that the surface M consists of only of elliptic points.…”

“…B.-Y. Chen, E. Güler, Y. Yaylı& H. H. Hacisaliho ǧlu (3) the curve c has zero spherical curvature and the meridian curve m is determined by the solutions of the following differential equation…”

“…[111] Let M be a rational surface of revolution. Then, M has pointwise 1-type Gauss map of the second kind in E 3 1 if and only if M is part of either a right cone or a hyperbolic cone. A. Niang studied rotation surfaces with pointwise 1-type Gauss map in three dimensional Minkowski space, and proved the following in [133].…”

Hypersphere and the fourth Laplace-Beltrami operator in 4-space

“…In E 4 ; Moore [39,40] worked general rotational surfaces; Hasanis and Vlachos [31] considered hypersurfaces with harmonic mean curvature vector field; Cheng and Wan [14] gave complete hypersurfaces with CM C; Kim and Turgay [35] introduced surfaces with L 1 -pointwise 1-type Gauss map; Arslan et al [2] worked Vranceanu surface with pointwise 1-type Gauss map; Arslan et al [3] studied generalized rotational surfaces; Aksoyak and Yaylı [32] worked flat rotational surfaces with pointwise 1-type Gauss map; Güler, Magid and Yaylı [29] introduced helicoidal hypersurfaces; Güler, Hacısalihoglu, and Kim [28] studied Gauss map and the third Laplace-Beltrami operator of the rotational hypersurface; Güler and Turgay [30] focused Cheng-Yau operator and Gauss map of rotational hypersurfaces; Güler [27] found rotational hypersurfaces satisfying ∆ I R = AR, where A ∈ M at (4,4). He [26] also studied fundamental form IV and curvature formulas of the hypersphere.…”