The Gauss Map and the Third Laplace-Beltrami Operator of the Rotational Hypersurface in 4-Space

: 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.

Magid, Scharlach and Vrancken [9] introduced the affine umbilical surfaces in four-space. Vlachos [10] considered hypersurfaces in E 4 with the harmonic mean curvature vector field. Scharlach [11] studied the affine geometry of surfaces and hypersurfaces in four-space. Cheng and Wan [12] considered complete hypersurfaces of four-space with constant mean curvature.
General rotational surfaces in E 4 were introduced by Moore [13,14]. Ganchev and Milousheva [15] considered these kinds of surfaces in the Minkowski four-space. They classified completely the minimal rotational surfaces and those consisting of parabolic points. Arslan et al. [2] studied generalized rotation surfaces in E 4 . Moreover, Dursun and Turgay [6] studied minimal and pseudo-umbilical rotational surfaces in E 4 .
In [16], Dillen, Fastenakels and Van der Veken studied the rotation hypersurfaces of S n × R and H n × R and proved a criterion for a hypersurface of one of these spaces to be a rotation hypersurface. They classified minimal, flat rotation hypersurfaces and normally flat rotation hypersurfaces in the Euclidean and Lorentzian space containing S n × R and H n × R, respectively. Senoussi and Bekkar [17] studied the Laplace operator using the fundamental forms I, I I and I I Iof the helicoidal surfaces in E 3 .
In the present paper, we consider the rotational hypersurface with three-parameters and its Gauss map in Euclidean four-space E 4 . We give some basic notions of the four-dimensional Euclidean geometry in Section 2. We give the definition of a rotational hypersurface, and then, we calculate the mean and the Gaussian curvatures of such a hypersurface in Section 3. In Section 4, we obtain the mean and the Gaussian curvatures of the Gauss map of the hypersurface. Moreover, we introduce the third Laplace-Beltrami operator and calculate it in E 4 in Section 5. Finally, we give a conclusion in the last section.

Curvatures in E 4
We identify a vector − → α with its transpose. Let M = M(u, v, w) be an isometric immersion of a hypersurface M 3 in E 4 . The triple vector product of − → x = (x 1 , x 2 , x 3 , x 4 ), − → y = (y 1 , y 2 , y 3 , y 4 ) and For a hypersurface M in four-space, the first and the second fundamental form matrices are as follows: is the Gauss map (i.e., the unit normal vector). The product of the matrices g ij −1 and h ij gives the matrix of the shape operator S = 1 det I s ij 3×3 . Here, g −1 ij · h ij = s ij det I are the elements of S. Therefore, the formulas of the Gaussian curvature and the mean curvature are given by and respectively.

Rotational Hypersurface in E 4
We define the rotational hypersurface in E 4 . Let γ : I ⊂ R −→ Π be a curve in a plane Π in E 4 and be a line in Π.

Definition 1.
A rotational hypersurface M 3 in E 4 is defined by rotating a curve γ around a line . In this case, γ and are called the profile curve and the axis of M 3 , resp.
We now describe a rotational hypersurface M 3 of E 4 more precisely. Without loss of generality, we may assume that the straight line is the line spanned by the vector (0, 0, 0, 1) t . The orthogonal matrix Z(v, w) that fixes the above vector is The matrix Z can be found by solving the equations simultaneously. Since the axis of rotation is the x 4 -axis of E 4 , the profile curve can be put as follows where ϕ (u) is a differentiable function, u ∈ I. Therefore, the rotational hypersurface, which is spanned by the vector (0, 0, 0, 1) in E 4 , is given by where u, v, w ∈ R. Therefore, we obtain the parametrization of M 3 Theorem 1. The Gaussian curvature K and the mean curvature H of the rotational hypersurface (4) are given as follows, respectively, Proof. Using the first partial differentials of (4) with respect to u, v, w, we get the first quantities as follows We have det I = u 4 (1 + ϕ 2 ) cos 2 w.
Using the second partial differentials of (4) with respect to u, v, w, the second quantities are given as follows The Gauss map of the rotational hypersurface is given by See Figures 1 and 2 for some different projections and symmetries, from four-space to three-spaces of the Gauss map of the rotational hypersurface.  Thus, the shape operator of the rotational hypersurface is obtained as From these, we obtain the Gaussian curvature K and the mean curvature H of the rotational hypersurface as follows Therefore, we have the following corollaries: Proof. Solving the second order differential equation K = 0, i.e., we get the solution.
Proof. When we solve the second order differential equation H = 0, i.e., we get the solution. Taking z = ϕ , z = ϕ , we have Therefore, the solutions of ϕ(u) are given, by using Mathematica, as follows and, by using Maple, we get for the solutions ϕ(u): where EllipticF[φ,m] gives the elliptic integral of the first kind F(φ | m).

Gauss Map
Next, we calculate the curvatures of the Gauss map (5) of the rotational hypersurface (4). Proof. Using the first partial differentials of (5), we get the first quantities as follows We have where ϕ = ϕ(u), ϕ = dϕ du , ϕ = d 2 ϕ du 2 . Using the second partial differentials of (5) , we have the second quantities as follows The Gauss map of the Gauss map (5) of the rotational hypersurface (4) is The shape operator of (6) is Finally, we obtain the Gaussian curvature and the mean curvature of (5) as K = −1, and H = −1.

The Third Laplace-Beltrami Operator
Next, we introduce the third Laplace-Beltrami operator to the four-space. Then, we apply it for the hypersurface (4). See [28] for the Laplace-Beltrami operator in three-space.
The inverse of the matrix where e ij = (e kl ) −1 and e = det e ij and φ = φ(x 1 , is a smooth function of class C 3 . We can write ∆ III φ as follows Clearly, we can write the matrix of the third fundamental form where Here, e is the Gauss map (i.e., the unit normal vector). More precisely, we get Hence, using a function φ = φ(u, v, w), we specifically obtain We continue our calculations to find the third Laplace-Beltrami operator ∆ III R of the rotational hypersurface R using (9) in (4).