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Rotational Hypersurfaces Satisfying ∆ = in the Four-Dimensional POLİTEKNİK DERGİSİ JOURNAL of POLYTECHNIC ISSN: 1302-0900 (PRINT), ISSN: 2147-9429 (ONLINE) URL: http://dergipark.org.tr/politeknik Rotational hypersurfaces satisfying ∆푰퐑 = 퐀퐑 in the four-dimensional Euclidean space 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 geometry 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 cylinder 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 cylinders, 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 푧 푧 푧 푧 *Sorumlu Yazar (Corresponding Author) 1 2 3 4 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 tr(퐒) = [−2(퐴푃퐺 + 퐵푇퐸 − 퐴퐵푀 − 퐴푇퐹 − 퐵푃퐹) 1 2 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 plane curve, ℓ be a line in Π in 퐒 = , 4 0 1/2 0 피 . 푟푊 ′ 4 −휑 Definition 6. A rotational hypface in 피 is hypface 0 0 rotating a profile curve 훾 about axis ℓ.
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