Birotational hypersurface and the second Laplace–Beltrami operator in the four dimensional Euclidean space $mathbb{E}^4$

Erhan GÜLER, (Bartın Üniversitesi, Fen Fakültesi, Matematik Bölümü, Bartın, Türkiye)
Yusuf YAYLI, (Ankara Üniversitesi, Fen Fakültesi, Matematik Bölümü, Ankara, Türkiye)
Hilmi Hacısalihoğlu (Bilecik Şeyh Edebali Üniversitesi, Fen Fakültesi, Matematik Bölümü, Bilecik, Türkiye)
Turkish Journal of Mathematics
5 0
Yıl: 2022 Cilt: 46 Sayı: 6 Sayfa Aralığı: 2167 - 2177 Metin Dili: İngilizce DOI: 10.55730/1300-0098.3261 İndeks Tarihi: 09-12-2022

Birotational hypersurface and the second Laplace–Beltrami operator in the four dimensional Euclidean space $mathbb{E}^4$

Öz:
We consider the birotational hypersurface x(u, v, w) with the second Laplace–Beltrami operator in the four dimensional Euclidean space $mathbb{E}^4$. We give the i -th curvatures of x . In addition, we compute the second Laplace–Beltrami operator of the birotational hypersurface satisfying $bigtriangleup{II}$ x =Ax for some 4 × 4 matrix A .
Anahtar Kelime: Euclidean spaces four space birotational hypersurface Gauss map i -th curvature second Laplace– Beltrami operator

Belge Türü: Makale Makale Türü: Araştırma Makalesi Erişim Türü: Erişime Açık
  • [1] Alias LJ, Gürbüz N. An extension of Takashi theorem for the linearized operators of the highest order mean curvatures. Geometriae Dedicata 2006; 121: 113-127.
  • [2] Aminov Y. The Geometry of Submanifolds. Gordon and Breach Sci. Pub., Amsterdam, 2001.
  • [3] Arslan K, Bayram BK, Bulca B., Kim YH, Murathan C, Öztürk G. Vranceanu surface in $mathbb{E}^4$ with pointwise 1-type Gauss map. Indian Journal of Pure and Applied Mathematics 2011; 42 (1): 41-51.
  • [4] Arslan K, Bayram BK, Bulca B, Öztürk G. Generalized rotation surfaces in $mathbb{E}^4$ . Results in Mathematics 2012; 61 (3): 315-327.
  • [5] Arslan K, Bulca B, Kılıç B, Kim YH, Murathan C, Öztürk G. Tensor product surfaces with pointwıse 1-type Gauss map. Bulletin of the Korean Mathematical Society 2011; 48 (3): 601-609.
  • [6] Arslan K, Milousheva V. Meridian surfaces of elliptic or hyperbolic type with pointwise 1-type Gauss map in Minkowski 4-space. Taiwanese Journal of Mathematics 2016; 20 (2): 311-332.
  • [7] Arslan K, Sütveren A, Bulca B. Rotational λ -hypersurfaces in Euclidean spaces. Creative Mathematics and Infor- matics 2021; 30 (1): 29-40.
  • [8] Arvanitoyeorgos A, Kaimakamis G, Magid M. Lorentz hypersurfaces in $mathbb{E}^4_1$ satisfying ∆H = αH. Illinois Journal of Mathematics 2009; 53 (2): 581-590.
  • [9] Barros M, Chen BY. Stationary 2-type surfaces in a hypersphere. Journal of the Mathematical Society of Japan 1987; 39 (4): 627-648.
  • [10] Barros M, Garay OJ. 2-type surfaces in $S^3$ . Geometriae Dedicata 1987; 24 (3): 329-336.
  • [11] Bektaş B, Canfes EÖ, Dursun U. Classification of surfaces in a pseudo-sphere with 2-type pseudo-spherical Gauss map. Mathematische Nachrichten 2017; 290 (16): 2512-2523.
  • [12] Bektaş B, Canfes EÖ, Dursun U. Pseudo-spherical submanifolds with 1-type pseudospherical Gauss map. Results in Mathematics 2017; 71 (3): 867-887
  • [13] Chen BY. On submanifolds of finite type. Soochow Journal of Mathematics 1983; 9: 65-81.
  • [14] Chen BY. Total Mean Curvature and Submanifolds of Finite Type. World Scientific, Singapore, 1984.
  • [15] Chen BY. Finite Type submanifolds and Generalizations. University of Rome, 1985.
  • [16] Chen BY. Finite type submanifolds in pseudo-Euclidean spaces and applications. Kodai Mathematical Journal 1985; 8 (3): 358-374.
  • [17] Chen BY, Piccinni, P. Submanifolds with finite type Gauss map. Bulletin of the Australian Mathematical Society 1987; 35: 161-186.
  • [18] Cheng QM, Wan QR. Complete hypersurfaces of $mathbb{R}^4$ with constant mean curvature. Monatshefte für Mathematik. 1994; 118: 171-204.
  • [19] Cheng SY, Yau ST. Hypersurfaces with constant scalar curvature. Mathematische Annalen 1977; 225: 195-204.
  • [20] Choi M, Kim YH. Characterization of the helicoid as ruled surfaces with pointwise 1-type Gauss map. Bulletin of the Korean Mathematical Society 2001; 38: 753-761.
  • [21] Dillen F, Pas J, Verstraelen L. On surfaces of finite type in Euclidean 3-space. Kodai Mathematical Journal 1990; 13: 10-21.
  • [22] Do Carmo MP, Dajczer M. Rotation hypersurfaces in spaces of constant curvature. Transactions of the American Mathematical Society 1983; 277: 685-709.
  • [23] Dursun U. Hypersurfaces with pointwise 1-type Gauss map. Taiwanese Journal of Mathematics 2007; 11 (5): 1407- 1416.
  • [24] Dursun U, Turgay NC. Space-like surfaces in Minkowski space $mathbb{E}^4_1$ with pointwise 1-type Gauss map. Ukrainian Mathematical Journal 2019; 71 (1): 64-80.
  • [25] Ferrandez A, Garay OJ, Lucas P. On a certain class of conformally at Euclidean hypersurfaces. In: Global Analysis and Global Differential Geometry; Springer: Berlin, Germany, 1990, pp. 48-54.
  • [26] Ganchev G., Milousheva V. General rotational surfaces in the 4-dimensional Minkowski space. Turkish Journal of Mathematics 2014; 38: 883-895.
  • [27] Garay OJ. On a certain class of finite type surfaces of revolution. Kodai Mathematical Journal 1988; 11: 25-31.
  • [28] Garay OJ. An extension of Takahashi’s theorem. Geometriae Dedicata 1990; 34: 105-112.
  • [29] Güler E. Fundamental form IV and curvature formulas of the hypersphere. Malaya Journal of Matematik 2020; 8(4): 2008-2011.
  • [30] Güler E. Rotational hypersurfaces satisfying $bigtriangleup{I}$ R = AR in the four-dimensional Euclidean space. Journal of Polytechnic 2021; 24 (2): 517-520.
  • [31] Güler E, Hacısalihoğlu HH, Kim YH. The Gauss map and the third Laplace − Beltrami operator of the rotational hypersurface in 4-space. Symmetry 2018; 10 (9): 1-12.
  • [32] Güler E, Magid M, Yaylı Y. Laplace − Beltrami operator of a helicoidal hypersurface in four-space. Journal of Geometry and Symmetry in Physics 2016; 41: 77-95.
  • [33] Güler E, Turgay NC. Cheng − Yau operator and Gauss map of rotational hypersurfaces in 4-space. Mediterranean Journal of Mathematics 2019; 16 (3): 1-16.
  • [34] Hasanis Th, Vlachos Th. Hypersurfaces in $mathbb{E}^4$ with harmonic mean curvature vector field. Mathematische Nachrichten 1995; 172: 145-169.
  • [35] Kahraman Aksoyak F, Yaylı Y. Flat rotational surfaces with pointwise 1-type Gauss map in$mathbb{E}^4$ . Honam Mathe- matical Journal 2016; 38 (2): 305-316.
  • [36] Kahraman Aksoyak F, Yaylı Y. General rotational surfaces with pointwise 1-type Gauss map in pseudo-Euclidean space E42 . Indian Journal of Pure and Applied Mathematics 2015; 46 (1): 107-118.
  • [37] Kim DS, Kim JR, Kim YH. Cheng − Yau operator and Gauss map of surfaces of revolution. Bulletin of the Malaysian Mathematical Sciences Society 2016; 39 (4): 1319-1327.
  • [38] Kim YH, Turgay NC. Surfaces in $mathbb{E}^4$ with $L_1$ -pointwise 1 -type Gauss map. Bulletin of the Korean Mathematical Society 2013; 50 (3): 935-949.
  • [39] Kühnel W. Differential Geometry: Curves-Surfaces-Manifolds. Third ed. Translated from the 2013 German ed. AMS, Providence, RI, 2015.
  • [40] Levi-Civita T. Famiglie di superficie isoparametriche nellordinario spacio euclideo. Atti della Accademia Nazionale dei Lince 1937; 26, 355-362.
  • [41] Moore CLE. Surfaces of rotation in a space of four dimensions. Annals of Mathematics Second Series 1919; 21: 81-93.
  • [42] Moore CLE. Rotation surfaces of constant curvature in space of four dimensions. American Mathematical Society 1920; 26: 454-460.
  • [43] Özkaldı S, Yaylı Y. Tensor product surfaces in $mathbb{R}^4$ and Lie groups. Bulletin of the Malaysian Mathematical Sciences Society 2010; 2 (1): 69-77.
  • [44] Senoussi B, Bekkar M. Helicoidal surfaces with $bigtriangleup{J}$ r = Ar in 3-dimensional Euclidean space. Studia Universitatis Babeș-Bolyai Mathematica 2015; 60 (3): 437-448.
  • [45] Stamatakis S, Zoubi H. Surfaces of revolution satisfying $bigtriangleup{III}$ x = Ax . Journal for Geometry and Graphics 2010; 14 (2): 181-186.
  • [46] Takahashi T. Minimal immersions of Riemannian manifolds. Journal of the Mathematical Society of Japan 1966; 18: 380-385.
  • [47] Turgay NC. Some classifications of Lorentzian surfaces with finite type Gauss map in the Minkowski 4-space. Journal of the Australian Mathematical Society 2015; 99 (3): 415-427.
  • [48] Yoon DW. Some properties of the Clifford torus as rotation surfaces. ndian Journal of Pure and Applied Mathematics 2003; 34 (6): 907-915.
APA Güler E, yaylı y, Hacısalihoğlu H (2022). Birotational hypersurface and the second Laplace–Beltrami operator in the four dimensional Euclidean space $mathbb{E}^4$. , 2167 - 2177. 10.55730/1300-0098.3261
Chicago Güler Erhan,yaylı yusuf,Hacısalihoğlu Hilmi Birotational hypersurface and the second Laplace–Beltrami operator in the four dimensional Euclidean space $mathbb{E}^4$. (2022): 2167 - 2177. 10.55730/1300-0098.3261
MLA Güler Erhan,yaylı yusuf,Hacısalihoğlu Hilmi Birotational hypersurface and the second Laplace–Beltrami operator in the four dimensional Euclidean space $mathbb{E}^4$. , 2022, ss.2167 - 2177. 10.55730/1300-0098.3261
AMA Güler E,yaylı y,Hacısalihoğlu H Birotational hypersurface and the second Laplace–Beltrami operator in the four dimensional Euclidean space $mathbb{E}^4$. . 2022; 2167 - 2177. 10.55730/1300-0098.3261
Vancouver Güler E,yaylı y,Hacısalihoğlu H Birotational hypersurface and the second Laplace–Beltrami operator in the four dimensional Euclidean space $mathbb{E}^4$. . 2022; 2167 - 2177. 10.55730/1300-0098.3261
IEEE Güler E,yaylı y,Hacısalihoğlu H "Birotational hypersurface and the second Laplace–Beltrami operator in the four dimensional Euclidean space $mathbb{E}^4$." , ss.2167 - 2177, 2022. 10.55730/1300-0098.3261
ISNAD Güler, Erhan vd. "Birotational hypersurface and the second Laplace–Beltrami operator in the four dimensional Euclidean space $mathbb{E}^4$". (2022), 2167-2177. https://doi.org/10.55730/1300-0098.3261
APA Güler E, yaylı y, Hacısalihoğlu H (2022). Birotational hypersurface and the second Laplace–Beltrami operator in the four dimensional Euclidean space $mathbb{E}^4$. Turkish Journal of Mathematics, 46(6), 2167 - 2177. 10.55730/1300-0098.3261
Chicago Güler Erhan,yaylı yusuf,Hacısalihoğlu Hilmi Birotational hypersurface and the second Laplace–Beltrami operator in the four dimensional Euclidean space $mathbb{E}^4$. Turkish Journal of Mathematics 46, no.6 (2022): 2167 - 2177. 10.55730/1300-0098.3261
MLA Güler Erhan,yaylı yusuf,Hacısalihoğlu Hilmi Birotational hypersurface and the second Laplace–Beltrami operator in the four dimensional Euclidean space $mathbb{E}^4$. Turkish Journal of Mathematics, vol.46, no.6, 2022, ss.2167 - 2177. 10.55730/1300-0098.3261
AMA Güler E,yaylı y,Hacısalihoğlu H Birotational hypersurface and the second Laplace–Beltrami operator in the four dimensional Euclidean space $mathbb{E}^4$. Turkish Journal of Mathematics. 2022; 46(6): 2167 - 2177. 10.55730/1300-0098.3261
Vancouver Güler E,yaylı y,Hacısalihoğlu H Birotational hypersurface and the second Laplace–Beltrami operator in the four dimensional Euclidean space $mathbb{E}^4$. Turkish Journal of Mathematics. 2022; 46(6): 2167 - 2177. 10.55730/1300-0098.3261
IEEE Güler E,yaylı y,Hacısalihoğlu H "Birotational hypersurface and the second Laplace–Beltrami operator in the four dimensional Euclidean space $mathbb{E}^4$." Turkish Journal of Mathematics, 46, ss.2167 - 2177, 2022. 10.55730/1300-0098.3261
ISNAD Güler, Erhan vd. "Birotational hypersurface and the second Laplace–Beltrami operator in the four dimensional Euclidean space $mathbb{E}^4$". Turkish Journal of Mathematics 46/6 (2022), 2167-2177. https://doi.org/10.55730/1300-0098.3261
Sayfa Oluşturulma Tarihi
24-04-2024 4:52

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