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Rotational hypersurfaces satisfying ∆푰퐑 = 퐀퐑 in the four-dimensional Dört-boyutlu Öklid uzayında ∆퐼푹 = 푨푹 koşulunu sağlayan dönel hiperyüzeyler

Yazar(lar) (Author(s)): Erhan GÜLER

ORCID : 0000-0003-3264-6239

Bu makaleye şu şekilde atıfta bulunabilirsiniz (To cite to this article): Güler E., “Rotational hypersurfaces satisfying ∆퐼퐑 = 퐀퐑 in the four-dimensional Euclidean space”, Politeknik Dergisi, 24(2): 517-520, (2021).

Erişim linki (To link to this article): http://dergipark.org.tr/politeknik/archive

DOI: 10.2339/politeknik.670333

Rotational Hypersurfaces Satisfying 횫퐈퐑 = 퐀퐑 in the Four-Dimensional Euclidean Space

Highlights ❖ Rotational hypersurface has zero mean curvature iff its Laplace-Beltrami operator vanishing ❖ Each element of the 4×4 order matrix A, which satisfies the condition 훥퐼푅 = 퐴푅, is zero ❖ Laplace-Beltrami operator of the rotational hypersurface depends on its mean curvature and the Gauss map

Graphical Abstract Rotational hypersurfaces in the 4-dimensional Euclidean space are discussed. Some relations of curvatures of hypersurfaces are given, such as the mean, Gaussian, and their minimality and flatness. The Laplace-Beltrami operator has been defined for 4-dimensional hypersurfaces depending on the first fundamental form. In addition, it is indicated that each element of the 4×4 order matrix A, which satisfies the condition 훥퐼푅 = 퐴푅, is zero, that is, the rotational hypersurface R is minimal.

Aim We consider the rotational hypersurfaces in 피4 to find its Laplace-Beltrami operator.

Design & Methodology We indicate fundamental notions of 피4. Considering differential formulas in 3-space, we transform them in 4-space. Moreover, we use straight calculations by hand.

Originality All findings in the paper are original.

Findings We define rotational hypersurface using rotation matrix. We calculate curvatures of rotational hypersurface. Defining the Laplace-Beltrami operator (LBo for short), we compute the LBo of rotational hypersurface. Finally, we give the rotational hypersurface satisfying 훥퐼푅 = 퐴푅.

Conclusion Rotational hypersurface has zero mean curvature iff its Laplace-Beltrami operator vanishing. Then, each element of the 4×4 order matrix A, which satisfies the condition 훥퐼푅 = 퐴푅, is zero. Finally, the Laplace-Beltrami operator of the rotational hypersurface depends on its mean curvature and the Gauss map.

Declaration of Ethical Standards The author of this article declare that the materials and methods used in this study do not require ethical committee permission and/or legal-special permission. Politeknik Dergisi, 2021; 24(2) : 517-520 Journal of Polytechnic, 2021; 24 (2): 517-520

Rotational Hypersurfaces Satisfying ∆퐼퐑 = 퐀퐑 in the Four-Dimensional Euclidean Space Araştırma Makalesi / Research Article Erhan GÜLER* Bartın University, Faculty of Sciences, Department of Mathematics 74100 Bartın, Turkey (Received / Geliş: 04.01.2020 ; Accepted / Kabul: 21.04.2020)

ABSTRACT In this study, rotational hypersurfaces in the 4-dimensional Euclidean space are discussed. Some relations of curvatures of hypersurfaces are given, such as the mean, Gaussian, and their minimality and flatness. In addition, Laplace-Beltrami operator has been defined for 4-dimensional hypersurfaces depending on the first fundamental form. Moreover, it is shown that each element of the 4 × 4 order matrix 퐀, which satisfies the condition ∆퐼퐑 = 퐀퐑, is zero, that is, the rotational hypersurface 퐑 is minimal. Keywords: 4-dimensional Euclidean space, Laplace-Beltrami operator, rotational hypersurface, curvature. Dört-Boyutlu Öklid Uzayında ∆퐼퐑 = 퐀퐑 Koşulunu Sağlayan Dönel Hiperyüzeyler

ÖZ Bu çalışmada, 4-boyutlu Öklid uzayındaki dönel hiperyüzeyler ele alınmıştır. Hiperyüzeylerin ortalama, Gauss eğrilikleri hesaplanıp aralarındaki minimal ve düzlemsel olma durumları gibi bazı bağıntılar verilmiştir. Ayrıca, 4-boyutlu hiperyüzeyler için birinci temel forma bağlı olarak Laplace-Beltrami operatörü tanımlanmıştır. Üstelik, dönel yüzeyin ∆퐼퐑 = 퐀퐑 koşulunu sağlayan 4 × 4 mertebeli 퐀 matrisinin her elemanının sıfır olduğu, yani 퐑 dönel hiperyüzeyinin minimal olduğu gösterildi. Anahtar Kelimeler: 4-boyutlu Öklid uzayı, Laplace-Beltrami operatörü, dönel hiperyüzey, eğrilik 1. INTRODUCTION Section 3, we define rotational hypersurface using After Chen [5], finite type submanifold, (i.e. for rotation matrix. We calculate curvatures of rotational Laplacian, coordinate functions are finite sum of eigen- hypersurface. Defining the Laplace-Beltrami operator functions) has been studied by [1-3,5-16,18,21-23,25- (LBo for short) in Section 4, we compute the LBo of 26]. rotational hypersurface. Finally, we give the rotational hypersurface satisfying ∆푰퐑 = 퐀퐑 in Section 5. We give In 피3, Takahashi [24] constructed spheres and minimal a conclusion in Section 6. surfaces are the only surfaces with Δ푟 = 휆푟, 푟 ∈ ℝ.

Ferrandez et al [12] found Δ퐻 = (퐴)3×3퐻 which are either an open part of sphere or of a right circular 2. PRELIMINARIES or minimal. Choi and Kim [8] classified the helicoid We introduce shape operator matrix 퐒, Gaussian depends on the first kind pointwise 1-type Gauss map. curvature (GC for short) 퐾, and the mean curvature (MC Dillen et al [9] gave Δ푟 = (퐴)3×3푟 + (퐵)3×1 which are for short) 퐻 of hypersurface (hypface for short) the circular , spheres, minimal surfaces. 4 퐌(푟, 휃1, 휃2) in 피 . Senoussi and Bekkar [23] introduced helicoidal surfaces Whole work, we identify its transpose with a vector. Let depends on three fundamental forms. Lawson [17] be an isometric immersion of a hypface 3 in 4. revealed general Laplace-Beltrami operator. 퐌 푀 피 ( ) General rotational surfaces were originated by Moore Definition 1. Inner product of 푥⃗ = 푥1, 푥2, 푥3, 푥4 , 푦⃗ = ( ) ( ) 4 [19,20] in 피4. Ganchev and Milousheva [13] gave the 푦1, 푦2, 푦3, 푦4 , 푧⃗ = 푧1, 푧2, 푧3, 푧4 in 피 is defined by 4 counterpart of them in 피1. 푥⃗ ⋅ 푦⃗ = 푥1푦1 + 푥2푦2 + 푥3푦3+푥4푦4. Arslan et al [2] worked generalized rotation surfaces, Definition 2. Triple vector product of 푥⃗ = Dursun and Turgay [11] considered pseudo umbilical, (푥1, 푥2, 푥3, 푥4), 푦⃗ = (푦1, 푦2, 푦3, 푦4), 푧⃗ = (푧1, 푧2, 푧3, 푧4) in 4 minimal rotational surfaces. Recently, Altın et al [1] E is given by worked Monge hypersurfaces with density in 피4. 푒1 푒2 푒3 푒4 We consider the rotational hypersurfaces in 피4. We 푥1 푥2 푥3 푥4 indicate fundamental notions of 피4 in Section 2. In 푥⃗ × 푦⃗ × 푧⃗ = 푑푒푡 ( ). 푦1 푦2 푦3 푦4 *Sorumlu Yazar (Corresponding Author) 푧1 푧2 푧3 푧4 e-posta : [email protected] 4 Definition 3. For a hypface 퐌(푟, 휃1, 휃2) in 피 ,

517 Erhan GÜLER / POLİTEKNİK DERGİSİ, Politeknik Dergisi,2021;24(2): 517-520

2 2 2 det퐼 = (퐸퐺 − 퐹 )퐶 − 퐴 퐺 + 2퐴퐵퐹 − 퐵 퐸, (1) cos휃1 cos휃2 − sin휃1 − cos휃1 sin휃2 0 sin휃 cos휃 cos휃 − sin휃 sin휃 0 det퐼퐼 = (퐿푁 − 푀2)푉 − 푃2푁 + 2푃푇푀 − 푇2퐿, (2) ( 1 2 1 1 2 ), (7) sin휃2 0 cos휃2 0 where 퐼 and 퐼퐼 are fundamental form matrices, 0 0 0 1 respectively, with coefficients: with 휃1, 휃2 ∈ ℝ. Matrix 푍 supplies: 퐸 = 퐌푟 ⋅ 퐌푟, 퐹 = 퐌푟 ⋅ 퐌휃 , 퐺 = 퐌휃 ⋅ 퐌휃 , 푡 푡 1 1 1 푍ℓ = ℓ, 푍 푍 = 푍푍 = 퐼4, det푍 = 1.

퐴 = 퐌푟 ⋅ 퐌휃2, 퐵 = 퐌휃1 ⋅ 퐌휃2, 퐶 = 퐌휃2 ⋅ 퐌휃2, Profile curve is given by

퐿 = 퐌푟푟 ⋅ 푒, 푀 = 퐌푟휃1 ⋅ 푒, 푁 = 퐌휃1휃1 ⋅ 푒, 훾(푟) = (푟, 0,0, 휑(푟)), ∞ 푃 = 퐌푟휃2 ⋅ 푒, 푇 = 퐌휃1휃2 ⋅ 푒, 푉 = 퐌휃2휃2 ⋅ 푒. where 휑(푟): I ⊂ ℝ ⟶ ℝ is 퐶 for all 푟 ∈ I. Rotational hypface, spanned by the vector (0,0,0,1)푡, holds: Here, the Gauss map is defined by 푡 퐑(푟, 휃1, 휃2) = 푍(휃1, 휃2). 훾(푟) . (8) 퐌푟 × 퐌휃 × 퐌휃 푒 = 1 2 . (3) In 피4, let us rewrite (8) as follows ‖퐌푟 × 퐌휃1 × 퐌휃2‖ Definition 4. Resulting matrix of (퐼)−1. (퐼퐼) indicates 푟cos휃1cos휃2 shape operator matrix as follows 푟sin휃1cos휃2 퐑(푟, 휃1, 휃2) = ( ), (9) 푟sin휃2 1 퐒 = (푠 ) , (4) 휑(푟) det퐼 푖푗 3×3 where 푟 ∈ ℝ − {0} and 휃 , 휃 ∈ [0,2휋). where 1 2 Using derivatives of (9) depends on 푟, 휃1 and 휃2, get 2 푠11 = (퐴퐵 − 퐶퐹)푀 + (퐵퐹 − 퐴퐺)푃 + (퐶퐺 − 퐵 )퐿, 1 + 휑′2 0 0 푠 = (퐴퐵 − 퐶퐹)푁 + (퐵퐹 − 퐴퐺)푇 + (퐶퐺 − 퐵2)푀, 12 퐼 = ( 0 푟2 cos2휃 0 ) (10) 푠 = (퐴퐵 − 퐶퐹)푇 + (퐵퐹 − 퐴퐺)푉 + (퐶퐺 − 퐵2)푃, 2 13 0 0 푟2 2 푠21 = (퐴퐵 − 퐶퐹)퐿 + (퐴퐹 − 퐵퐸)푃 + (퐶퐸 − 퐴 )푀, 2 and 푠22 = (퐴퐵 − 퐶퐹)푀 + (퐴퐹 − 퐵퐸)푇 + (퐶퐸 − 퐴 )푁, 2 ′′ 2 −푟 휑 cos휃2 푠23 = (퐴퐵 − 퐶퐹)푃 + (퐴퐹 − 퐵퐸)푉 + (퐶퐸 − 퐴 )푇, 0 0 2 √det퐼 푠31 = (퐵퐹 − 퐴퐺)퐿 + (퐴퐹 − 퐵퐸)푀 + (퐸퐺 − 퐹 )푃, 3 ′ 3 −푟 휑 cos 휃2 푠 = (퐵퐹 − 퐴퐺)푀 + (퐴퐹 − 퐵퐸)푁 + (퐸퐺 − 퐹2)푇, 퐼퐼 = 0 0 (11) 32 √det퐼 2 3 ′ 푠33 = (퐵퐹 − 퐴퐺)푃 + (퐴퐹 − 퐵퐸)푇 + (퐸퐺 − 퐹 )푇. −푟 휑 cos휃 0 0 2 Definition 5. The formulas of the GC and the MC of ( √det퐼 ) 푑휑 hypface are, respectively, as follows with 휑 = 휑(푟), 휑′ = , 푑푟 ( ) 퐾 = det 퐒 , (5) 4 2 2 det퐼 = 푟 (1 + 휑′ ) cos 휃2. 1 퐻 = tr(퐒), (6) Using (3) on (9), we find 3 2 ′ 2 where 푟 휑 cos휃1 cos 휃2 1 1 푟2휑′ sin휃 cos2휃 ( ) [ ( ) 1 2 tr 퐒 = −2 퐴푃퐺 + 퐵푇퐸 − 퐴퐵푀 − 퐴푇퐹 − 퐵푃퐹 푒퐑 = 2 ′ , (12) det퐼 √det퐼 푟 휑 sin휃2 cos휃2 2 2 2 2 −퐴 푁 − 퐵 퐿 + (퐸푁 + 퐺퐿 − 2퐹푀)퐶+(퐸퐺 − 퐹 )푉]. ( −푟 cos휃2 ) When 퐻 = 0 on 퐌, hypface 퐌 is called minimal. Hence, we have −휑′′ 3. ROTATIONAL HYPERSURFACE 3/2 0 0 푊 −휑′ Let us 훾: I ⊂ ℝ ⟶ Π be a curve, ℓ be a line in Π in 퐒 = 0 0 , 피4. 푟푊1/2 ′ 4 −휑 Definition 6. A rotational hypface in 피 is hypface 0 0 rotating a profile curve 훾 about axis ℓ. ( 푟푊1/2) ℓ is spanned by (0,0,0,1)푡 and orthogonal matrix where 푊 = 1 + 휑′2. Using (5) and (6), respectively, we 푍(휃1, 휃2) is defined by find followings

518 ROTATIONAL HYPERSURFACES SATISFYING ∆푰퐑 = 퐀퐑 IN THE FOUR-DIMENSIONAL… Politeknik Dergisi, 2021; 24 (2) : 517-520

′2 ′′ ′′ ′3 ′ ′ ′′ ′3 ′ 휑 휑 푟휑 + 2휑 + 2휑 −휑 (푟휑 + 2휑 + 2휑 ) cos휃1 cos휃2 = 0, 퐾 = − 2 5/2 , 퐻 = − 3/2 . ′ ′′ ′3 ′ 푟 푊 3푟푊 −휑 (푟휑 + 2휑 + 2휑 ) sin휃1 cos휃2 = 0, 3 4 ′ ′′ ′3 ′ Corollary 1. Let 퐑 ∶ 푀 ⟶ 피 be an immersion in (9). −휑 (푟휑 + 2휑 + 2휑 ) sin휃2 = 0, Then, following results holds: 푟휑′′ + 2휑′3 + 2휑′ = 0. (a) 푀3 has CGC iff By using Remark 1, we find following corollaries: 5 4 2 4 ′2 (휑′) (휑′′) − 푐푟 (1 + 휑 ) = 0. Corollary 2. While cos휃푖 ≠ 0, and sin휃푖 ≠ 0, then 휑′ = 0. Hence, we obtain (b) 푀3 has CMC iff 휑 = 푐 = 푐표푛푠푡. ⇔ Δ퐼퐑 = 0. 3 ′′ ′3 ′ 2 2 ′2 (푟휑 + 2휑 + 2휑 ) − 9푐푟 (1 + 휑 ) = 0. ′ Corollary 3. When 휑 ≠ 0, cos휃푖 ≠ 0, and sin휃푖 ≠ 0, (c) 푀3 is flat iff then we see ′′ ′3 ′ 퐼 휑(푟) = 푐1푟 + 푐2. 푟휑 + 2휑 + 2휑 = 0 (i. e. 퐻 = 0) ⇔ Δ 퐑 = 0. 2 ′′ Proof. Calculating eq. 퐾 = 0, i.e. 휑′ 휑 = 0, we find the solution. 3 5. ROTATIONAL HYPERSURFACES (d) 푀 has ZMC iff SATISFYING ∆푰퐑 = 퐀퐑 IN 피ퟒ 푑푟 3 4 휑(푟) = ± ∫ + 푐2 = Elliptic퐹(푖푟, 푖), Theorem 1. Assume 퐑 ∶ 푀 ⟶ 피 be an immersion by 4 퐼 3 √푐1푟 − 1 (9). So, ∆ 퐑 = 퐀퐑 iff 푀 has ZMC. 휙 Proof. The Gauss map of (9) is where ( 2 )−1/2 is Elliptic퐹(휙, 푚) = ∫0 1 − 푚sin 휃 푑휃 ′ elliptic integral, and 휙 ∈ [−휋/2, 휋/2]. 휑 cos휃1cos휃2 1 ′ Proof. Solving eq. 푟휑′′ + 2휑′3 + 2휑′ = 0, we see 휑 sin휃1cos휃2 푒 = ( ′ ), solution. √푊 휑 sin휃2 −1 4. LAPLACE-BELTRAMI OPERATOR where 푊 = 1 + 휑′2. We use

Definition 7. For smooth 휙 = 휙(푥1, 푥2, 푥3)|퐃⊂ℝퟑ of −3퐻푒 = 퐀퐑, (15) class 퐶ퟑ of hypface 퐌, the LBo is defined by then we get 3 1 휕 휕휙 −(Θ 휑′ + 푎 푟) cos휃 cos휃 − 푎 푟 sin휃 cos휃 − 푎 푟 sin휃 ∆퐼휙 = ∑ (√푔푔푖푗 ). (13) 11 1 2 12 1 2 13 2 푖 푗 −푎 푟 cos휃 cos휃 − (Θ 휑′ + 푎 푟) sin휃 cos휃 − 푎 푟 sin휃 √푔 휕푥 휕푥 ( 21 1 2 22 1 2 23 2 ) 푖,푗=1 ′ −푎31 푟 cos휃1 cos휃2 − 푎32 푟 sin휃1 cos휃2 − (Θ 휑 + 푎33 푟) sin휃2 푖푗 −1 Θ where (푔 ) = (푔푘푙) and 푔 = det(푔푖푗). 푎14휑 퐼 Then, the LBo ∆ 퐑 of 퐑 = 퐑(푟, 휃1, 휃2) is as follows 푎24휑 = ( 푎 휑 ), 2 34 휕 (퐶퐺 − 퐵 )퐑푟 − (퐴퐵 − 퐶퐹)퐑휃 + (퐵퐹 − 퐴퐺)퐑휃 ( 1 2 ) 푎41 푟 cos휃1 cos휃2 + 푎42 푟 sin휃1 cos휃2 + 푎43 푟 sin휃2 + 푎44휑 휕푟 √det퐼 3퐻 2 퐼 1 휕 (퐴퐵 − 퐶퐹)퐑푟 − (퐶퐸 − 퐴 )퐑휃 + (퐴퐹 − 퐵퐸)퐑휃 where 퐀 is 4 × 4 matrix, and Θ = . The eq. ∆ 퐑 = 퐀퐑 − ( 1 2 ) . 푊 √det퐼 휕휃1 √det퐼 by (10) and (15) gives following ODEs system 휕 (퐵퐹 − 퐴퐺)퐑 − (퐴퐹 − 퐵퐸)퐑 + (퐸퐺 − 퐹2)퐑 푟 휃1 휃2 ′ ( ) −(Θ휑 + 푎11푟) cos휃1 cos휃2 − 푎12 푟 sin휃1cos휃2 − 푎13 푟 sin휃2 { 휕휃2 √det퐼 } = 푎14휑, ′ (14) −푎21 푟 cos휃1 cos휃2 − (Θ휑 + 푎22푟) sin휃1 cos휃2 − 푎23 푟 sin휃2

Hence, the LBo of (9) is given by = 푎24휑, ′ −푎31 푟 cos휃1 cos휃2 − 푎32 푟 sin휃1 cos휃2 − (Θ휑 + 푎33푟) sin휃2 퐼 1 휕 휕 휕 ∆ 퐑 = ( 퐔 − 퐕 + 퐖). = 푎34휑, √det퐼 휕푟 휕휃1 휕휃2 푎41 푟 cos휃1 cos휃2 + 푎42 푟 sin휃1 cos휃2 + 푎43 푟 sin휃2 + 푎44휑 = Θ. By derivatives of 푟, 휃 , 휃 on 퐔, 퐕, 퐖, respectively, we 1 2 Differentiating ODEs twice with respect to 휃 , we have get 1 ′ 푎14 = 푎24 = 푎34 = 푎44 = 0, Θ = 0. (16) − 휑 cos휃1 cos휃2 ′′ ′3 ′ 푟휑 + 2휑 + 2휑 − 휑′ sin휃 cos휃 From (16), we get Δ퐼퐑 = ( 1 2 ). 푟푊2 ′ − 휑 sin휃2 −푎 푟cos휃 − 푎 푟sin휃 = 0, 1 11 1 12 1 −푎21푟cos휃1 − 푎22푟sin휃1 = 0, Remark 1. When ∆퐼퐑 = 0, 푟 ≠ 0, the system of equation are as follows −푎31푟cos휃1 − 푎32푟sin휃1 = 0, 푎41푟cos휃1 + 푎42푟sin휃1 = 0.

519 Erhan GÜLER / POLİTEKNİK DERGİSİ, Politeknik Dergisi,2021;24(2): 517-520

Since the functions cos and sin are linear independent on [7] Cheng Q.M., Wan Q.R., “Complete hypersurfaces of ℝ4 3퐻 휃 , the coefficients 푎 = 0. Considering Θ = , it is with constant mean curvature,” Monatsh. Math. 118(3): 1 푖푗 푊 171–204, (1994). clear 퐻 = 0. Finally, 퐑 is a minimal hypface. [8] Choi M., Kim Y.H., “Characterization of the helicoid as ruled surfaces with pointwise 1-type Gauss map,” Bull. 6. CONCLUSION Korean Math. Soc., 38: 753–761, (2001). 4 [9] Dillen F., Pas J., Verstraelen L., “On surfaces of finite In this study, rotational hypfaces in 피 are investigated in type in Euclidean 3-space,” Kodai Math. J., 13: 10–21, detail by using its Gauss map, MC and the GC. (1990). According to findings, minimality and flatness cases are [10] Do Carmo M., Dajczer M., “Helicoidal surfaces with determined. Taking into account LBo on rotational constant mean curvature,” Tohoku Math. J., 34: 351– hypface in four space, 퐻 = 0 ⇔ Δ퐼퐑 = 0 are presented. 367, (1982). This means, rotational hypface has ZMC iff its LBo [11] Dursun U., Turgay N.C., “Minimal and Pseudo- equals to zero. This result can be seen in Corollary 3, Umbilical Rotational Surfaces in Euclidean Space 피4,” clearly. In addition, rotational hypface satisfies the Mediter. J. Math. 10: 497–506, (2013). condition ∆퐼퐑 = 퐀퐑, when matrix 퐀 occurs only 퐀 = [12] Ferrandez A., Garay O.J., Lucas P., “On a certain class of ퟎ4×4 for 4 × 4 matrix 퐀. That is, 퐑 is minimal hypface. conformally at Euclidean hypersurfaces,” In Global This study is important because rotation matrix, Analysis and Global Differential Geometry; Springer: rotational hypface, LBo could not defined before, clearly. Berlin, Germany, 48–54. (1990). Also, literature about the topic are limited to reveal and [13] Ganchev G., Milousheva V., “General rotational surfaces calculate the properties of this kind rotational hypfaces in in the 4-dimensional Minkowski space,” Turkish J. 피4. Math. 38: 883–895, (2014). [14] Güler E., “Bour's theorem and lightlike profile curve,” Yokohama Math. J., 54(1): pp. 55–77, (2007). ACKNOWLEDGEMENT [15] Güler E., Hacısalihoğlu H.H., Kim Y.H., The Gauss map The author is grateful to the reviewers and editor for their and the third Laplace-Beltrami operator of the rotational contribution and support to the article. hypersurface in 4-Space. Symmetry, 10(9): 398, 1–11 (2018). DECLARATION OF ETHICAL STANDARDS [16] Güler E., Magid M., Yaylı Y., “Laplace Beltrami operator The author(s) of this article declare that the materials and of a helicoidal hypersurface in four space,” J. Geom. methods used in this study do not require ethical Symm. Phys. 41: 77–95, (2016). committee permission and/or legal-special permission. [17] Lawson H.B., “Lectures on Minimal Submanifolds,” 2nd ed.; Mathematics Lecture Series 9; Publish or Perish, Inc.: Wilmington, Delaware, (1980). 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