Translation Surfaces in the 3-Dimensional Pseudo-Galilean Space Satisfying: \(\boldsymbol{\bigtriangleup^{\mathrm{II}}\, r_i=\lambda_i r_i}\)

Azzi Ahmed, Bekkar Mohammed, Zoubir Hanifi

Abstract


In this paper, we classify translation surfaces in a \(3\)-dimensional Pseudo-Galilean space \(\mathbb{G}_{3}^1\)  under the condition  \(\bigtriangleup^{\rm II}\, r_i=\lambda_i r_i\), where \(r_i\) are the components of the position vector, \(\lambda_i\in\mathbb{R}\), \((i=1,2,3)\), and \(\bigtriangleup^{\rm II}\) denotes the Laplace operator with respect to the second fundamental form.


Keywords


Pseudo-Galilean space; Surface of finite type; Translation surfaces; II-Harmonic; Laplacian operator with respect to the second fundamental form

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References


Ch. Baba-Hamed and M. Bekkar, Helicoidal surfaces in the three-dimensional Lorentz-Minkowski space satisfying $triangle^{II} r_i=lambda_ir_i$, Journal of Geometry 100 (2011), Article number: 1 DOI: 10.1007/s00022-011-0074-2.

M. Bekkar and H. Zoubir, Surfaces of revolution in the three-dimensional Lorentz-Minkowski space satisfying $triangle x_i =lambda_ix_i$, International Journal of Contemporary Mathematical Sciences 24 (2008), 1173 – 1185.

B. Bukcu„ M. K. Karacan and D. W. Yoon, Translation surfaces of type 2 in the three dimensional simply isotropic space $I_1^3$, Bulletin of the Korean Mathematical Society 54(3) (2017), 953 – 965, DOI: 10.4134/BKMS.b160377.

B.-Y. Chen, A report on submanifold of finite type, Soochow Journal of Mathematics 22 (1996), 117 – 337, https://www.researchgate.net/profile/Bang-Yen-Chen/publication/264741832_A_report_on_submanifolds_of_finite_type/links/5e34c921458515072d73f9bb/A-reporton-submanifolds-of-finite-type.pdf.

B. Divjak and Ž. M. Šipuš, Some special surfaces in the pseudo-Galilean space, Acta Mathematica Hungarica 118(3) (2008), 209 – 226, DOI: 10.1007/s10474-007-6171-x.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th edition, Elsevier (2007), https://booksite.elsevier.com/samplechapters/9780123736376/Sample_Chapters/01~Front_Matter.pdf.

G. Kaimakamis and B. J. Papantoniou, Surfaces of revolution in the 3-dimensional Lorentz-Minkowski space satisfying $triangle^{II} r = Ar$, Journal of Geometry 81 (2004), 81 – 92, DOI: 10.1007/s00022-004-1675-9.

B. Senoussi and M. Bekkar, Helicoidal surfaces in the three-dimensional Lorentz-Minkowski space satisfying $triangle^{II} r = Ar$, Kyushu Journal of Mathematics 67 (2013), 327–338, DOI: 10.2206/kyushujm.67.327.

Ž. M. Šipuš, On a certain of translation surfaces in a pseudo-Galilean space, International Mathematical Forum 6(21-24) (2011), 1113 – 1125, http://www.m-hikari.com/imf-2011/21-24-2011/sipusIMF21-24-2011.pdf.

Z. M. Sipus and B. Divjak, Surfaces of constant curvature in the pseudo-Galilean space, International Journal of Mathematical Sciences 2012, Article ID 375264, p. 28, DOI: 10.1155/2012/375264.

T. Takahashi, Minimal immersions of Riemannian manifolds, Journal of Mathematical Society of Japan 18 (1966), 380 – 385, DOI: 10.2969/jmsj/01840380.

D. W. Yoon, Some classification of translation surfaces in Galilean-space, International Journal of Mathematical Analysis 6(28) (2012), 1355 – 1361.




DOI: http://dx.doi.org/10.26713%2Fcma.v12i2.1384

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