The Fourth Fundamental Form IV of Dini-Type Helicoidal Hypersurface in the Four Dimensional Euclidean Space

: We introduce the fourth fundamental form of a Dini-type helicoidal hypersurface in the four dimensional Euclidean space E 4 . We ﬁnd the Gauss map of helicoidal hypersurface in E 4 . We obtain the characteristic polynomial of shape operator matrix. Then, we compute the fourth fundamental form matrix IV of the Dini-type helicoidal hypersurface. Moreover, we obtain the Dini-type rotational hypersurface, and reveal its differential geometric objects. shape operator; curvatures; fourth fundamental form MSC: Primary: 53A07; Secondary: 53C42


Introduction
Rotational and helicoidal hyper-surfaces have attracted the attention of scientists such as architects, biologists, physicists, mathematicians, and especially geometers for almost 300 years.
Let us review some works about rotational and helicoidal characters in chronological order.
Moore [5] introduced rotational surfaces in a four dimensional space E 4 . Moore [6] considered rotational surfaces of constant curvature in E 4 .
Stamatakis and Al-Zoubi [26] considered surfaces of a revolution satisfying ∆ I I I x = Ax. Ji and Kim [27] worked on helicoidal CDPC-surfaces in Minkowski 3-space. Ji and Kim [28] introduced mean curvatures and Gauss maps of a pair of isometric helicoidal and rotation surfaces in Minkowski 3-space. Güler, Yaylı, and Hacısalihoglu [29] used Bour's theorem on the Gauss map in 3-Euclidean space. Arslan et al. [30] studied rotational embeddings with pointwise 1-type Gauss map in E 4 . Dursun and Turgay [31] worked on general rotational surfaces with pointwise 1-type Gauss map in E 4 . Arslan et al. [32] focused on generalized rotation surfaces in E 4 .
In this paper, we study the fourth fundamental form of the Dini-type helicoidal hypersurface in Euclidean 4-space E 4 . In Section 2, we offer some basic notions of fourdimensional Euclidean geometry. In Section 3, we define helicoidal hypersurface. In Section 4, we give Dini-type helicoidal hypersurface and calculate the fourth fundamental form. In addition, we provide a conclusion in the last section.

Preliminaries
In the rest of this paper, we identify a vector (a,b,c,d) with its transpose (a,b,c,d) t . In this section, we will introduce the first, second, third, and fourth fundamental form matrices, matrix of the shape operator S of hypersurface x = x(u, v, w) in the fourdimensional Euclidean space E 4 . Let x = x(u, v, w) be an isometric immersion of any hypersurface M 3 in E 4 . Let {e 1 , e 2 , e 3 , e 4 } be the standart base vectors of E 4 . The inner product of − → x = (x 1 , x 2 , x 3 , x 4 ), − → y = (y 1 , y 2 , y 3 , y 4 ), and the vector product respectively.
In 4-space, the first and the second fundamental form matrices of hypersurface x(u, v, w) are given as follows Theorem 1. The shape operator matrix S of any hypersurface x(u, v, w) in 4-space is given as follows Proof. We compute I −1 .I I, and it gives the shape operator matrix S.

Theorem 2.
The third fundamental form matrix I I I of any hypersurface x(u, v, w) in 4-space is given as follows Proof. We compute I I.S, and this gives the matrix of the third fundamental form I I I.

Helicoidal Hypersurface
Let γ : I −→ Π be a curve in a plane Π in E 4 , and let be a straight line in Π for an open interval I ⊂ R. A rotational hypersurface in E 4 is defined as a hypersurface rotating a curve γ (i.e., profile curve) around a line (i.e., axis) . Suppose that when a profile curve γ rotates around the axis , it simultaneously displaces parallel lines orthogonal to the axis , so that the speed of displacement is proportional to the speed of rotation. The resulting hypersurface is called the helicoidal hypersurface with axis and pitches a, b ∈ R\{0}. We can suppose that is the line spanned by the vector (0, 0, 0, 1) t . The rotation matrix is given by where v, w ∈ R. The matrix Q supplies the following equations Q. = , Q t .Q = Q.Q t = I 4 , det Q = 1.
When the axis of rotation is , there is an Euclidean transformation by which the axis is transformed to the x 4 -axis of E 4 . Parametrization of the profile curve is given by γ(u) = (u, 0, 0, ϕ(u)), where ϕ(u) : I ⊂ R −→ R is a differentiable function for all u ∈ I. Therefore, the helicoidal hypersurface, spanned by the vector = (0, 0, 0, 1), is given as where u ∈ I, v, w ∈ [0, 2π], a, b ∈ R\{0}. We can also write the helicoidal hypersurface as follows When the pitches a = b = 0, helicoidal hypersurface transforms into a rotational hypersurface in E 4 .

Dini-Type Helicoidal Hypersurface and the Fourth Fundamental Form
Next, for the sake of breviety, we use S u = sin u, C u = cos u, T u = tan u, C u = cot u. We consider Dini-type helicoidal hypersurface (see Figure 1) as follows where u, a, b ∈ R\{0} and 0 ≤ v, w ≤ 2π. Using the first differentials of (1) with respect to u, v, w, we get the first quantities and its determinant det The Gauss map of (1) is given by where Taking the second differentials of (1) with respect to u, v, w, with (2), we have the second quantities as follows Computing product matrix I −1 .S, we obtain the shape operator matrix of (1) as follows Theorem 4. Let D : M 3 −→ E 4 be an immersion given by (1). Then, characteristic polynomial of S is given as follows Proof. Computing det(S − xI 3 ) = 0, we get r, s, and t.

Corollary 1.
Let D : M 3 −→ E 4 be an immersion given by (1). Then, D has the following principal curvatures Proof. Solving characteristic polynomial of S, we obtain eigenvalues k i .
Hence, we can see the curvatures of (1), using the following formulas C 0 = 1, easily. See [59] for the formulas of the curvatures C i .

Corollary 2.
Let D : M 3 −→ E 4 be an immersion given by (1). Then, (1) has the third fundamental form matrix as follows Proof. Using I I.S of (1), we have the third fundamental form matrix.
Corollary 3. Let D : M 3 −→ E 4 be an immersion given by (1). Then, D has the fourth fundamental form matrix Proof. Using product matrix I I I.S of (1), we get the fourth fundamental form matrix.
Example 1. When a = b = 0 in hypersurface (1), we obtain Dini-type rotational hypersurface (see Figure 2) as follows Then its fundamental form matrices I, I I, Gauss map G (see Figure 3), shape operator matrix S, fundamental form matrices I I I, IV, and curvatures C i are given by as follows I = diag C 2 u , S 2 u C 2 w ,S 2 u , , I I = diag C u , −1/2 S 2u C 2 w , −1/2 S 2u , G = (C u C v C w , C u S v C w , C u S w , −S u ), S = diag(T u , −C u , −C u ,), and C 0 = 1, where S u = sin u, C u = cos u, T u = tan u, C u = cot u.

Conclusions
In this paper, we introduce the fourth fundamental form of Dini-type helicoidal hypersurface D(u, v, w) in the four dimensional Euclidean space E 4 . We compute its Gauss map G. We obtain the characteristic polynomial of shape operator matrix S. We calculate the fourth fundamental form matrix IV of hypersurface D. Taking pitches a = b = 0 of helicoidal hypersurface D, we have a Dini-type rotational hypersurface R(u, v, w), and reveal its differential geometric objects. Therefore, it can be seen that objects of D and R supply the following relation C 0 IV − 3C 1 I I I + 3C 2 I I − C 3 I = 0.
Funding: This work received no external funding.