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Journals / Fundamental Journal of Mathematics and Applications / 2020 / Volume: 3 Issue: 2

Warped Translation Surfaces of Finite Type in Simply Isotropic 3-Spaces

Alev Kelleci Akbay(FIRAT ÜNİVERSİTESİ, FEN FAKÜLTESİ)0000-0003-2528-2131
Pages
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Publication Date
2020-12-15
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AI Summary & Key Takeaways

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⚡ TL;DR(Too Long; Didn't Read)

All graphics have been plotted with Maple14.

📌 3 Key Takeaways

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Objective & Hypothesis:In this paper, we classify warped translation surfaces being invariant surfaces of i-type, that is, the generating curve has formed by the intersection of the surface with the isotropic xz-plane in the three-dimensional simply isotropic space $\mathbb I^3$ under the condi tio n $\Delta^{J}x_i=\lambda_i x_i,$ w ith J=I,II .
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Key Findings & Data:Here, $\Delta^{J}$ is the Laplace operator with respect to first and second fundamental form and $\lambda_i$, $i=1,2,3$ are some real numbers.
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Conclusion & Impact:All graphics have been plotted with Maple14.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Invariant surface · Laplace operator · Simply isotropic space

Abstract

In this paper, we classify warped translation surfaces being invariant surfaces of i-type, that is, the generating curve has formed by the intersection of the surface with the isotropic xz-plane in the three-dimensional simply isotropic space $\mathbb I^3$ under the condi tio n $\Delta^{J}x_i=\lambda_i x_i,$ w ith J=I,II . Here, $\Delta^{J}$ is the Laplace operator with respect to first and second fundamental form and $\lambda_i$, $i=1,2,3$ are some real numbers. Also, as an application, we give some examples for these surfaces and also some explicit graphics of them. All graphics have been plotted with Maple14.

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Cite This Article

Akbay, A. K. (2020). Warped Translation Surfaces of Finite Type in Simply Isotropic 3-Spaces. Fundamental Journal of Mathematics and Applications, Volume: 3 Issue: 2. https://doi.org/10.33401/fujma.785781

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References (26)

  1. B. Y. Chen, J. Morvan, T. Nore, Energy, tension and finite type maps, Kodai Math. J., 9 (1986), 406–418.
  2. B. Y. Chen, Total mean curvature and submanifolds finite type, World Scientific, New Jersey, 1984.
  3. O. J. Garay, On a certain class of finite type surfaces of revolution, Kodai Math. J., 11 (1988), 25–31.
  4. B. Y. Chen, A report on submanifolds of finite type, Soochow J. Math., 22 (1996), 117–337.
  5. T. Tahakashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan, 18 (1966), 380–385.
  6. F. Dillen, J. Pas, L. Verstraelen, On surfaces of finite type in Euclidean 3-space, Kodai Math. J., 13 (1990), 10–21.
  7. O. J. Garay, An extension of Takahashi’s theorem, Geom. Dedicata, 34 (1990), 105–112.
  8. B. Senoussi, M. Bekkar, Helicoidal surfaces with DJ r = Ar in 3-dimensional Euclidean space, Stud. Univ. Babes-Bolyai Math., 60 (2015), 437–448.
  9. B. Y. Chen, Some open problems and conjectures on submanifolds of finite type: recent development, Tamkang J. Math., 45 (2014), 87–108.
  10. M. E. Aydin, A generalization of translation surfaces with constant curvature in the isotropic space, J. of Geo. 107(3) (2016), 603–615.
  11. M. E. Aydin, Constant curvature factorable surfaces in 3-dimensional isotropic space, J. Korean Math. Soc., 55 (1) (2018) 59–7.
  12. M. E. Aydin, A. Erdur, M. Ergut, Affine factorable surfaces in isotropic spaces, TWMS J. Pure Appl. Math., 11 (2020), 72–88.
  13. M. E. Aydin, I. Mihai, On certain surfaces in the isotropic 4-space, Math. Com., 22(1) (2017), 41-51.
  14. M. E. Aydin, A.O. Ogrenmis, Homothetical and translation hypersurfaces with constant curvature in the isotropic space, BSG proceedings 23, (2016) 1–10.
  15. A. Kelleci, L. C. B. da Silva, Invariant surfaces with coordinate finite-type Gauss map in simply isotropic space, J. Math. Anal. Appl., (in press).
  16. L. C. B. da Silva, The geometry of Gauss map and shape operator in simply isotropic and pseudo-isotropic spaces, J. of Geom., 110(2) (2019), 31.
  17. L. C. B. da Silva, Differential geometry of invariant surfaces in simply isotropic and pseudo-isotropic spaces, Math. J. Okayama Univ., (in press).
  18. B. Bukcu, M. K. Karacan, D. W. Yoon, Translation surfaces of type-2 in the three-dimensional simply isotropic space I13, Bull. Korean Math. Soc., 54 (2017), 953–965.
  19. A. Cakmak, M. K. Karacan, S. Kiziltug, Dual surfaces defined by z = f (u)+g(v) in simply isotropic 3-space I13, Commun. Korean Math. Soc., 34 (2019), 267–277.
  20. B. Bukcu, M. K. Karacan, D. W. Yoon, Translation surfaces in the three-dimensional simply isotropic space I13 satisfying DIIIxi = lixi, Konuralp J. Math., 4 (2016), 275–281.
  21. M. K. Karacan, D. W. Yoon, B. Bukcu, Translation surfaces in the three-dimensional simply isotropic space I13, Int. J. Geom. Meth. Mod. Phys., 13 (2016), 1650088.
  22. M. K. Karacan, D. W. Yoon, B. Bukcu, Surfaces of revolution in the three-dimensional simply isotropic space I13, Asia Pac. J. Math., 4 (2017), 1–10.
  23. M. E. Aydin, M. Ergut, Affine translation surfaces in the isotropic 3-space, Int. Electron. J. Geom., 10 (2017), 21–30.
  24. M. K. Karacan, D. W. Yoon, S. Kiziltug, Helicoidal surfaces in the three-dimensional simply isotropic space, I13, Tamkang J. Math., 48 (2017), 123–134.
  25. M. K. Karacan, D. W. Yoon, N. Yuksel, Classification of some special types ruled surfaces in simply isotropic 3-space, Analele Universitatii de Vest, Timisoara Seria Matematica – Informatica, 55 (2017), 87–98.
  26. H. Sachs, Isotrope Geometrie des Raumes, Vieweg, Braunschweig/Wiesbaden, 1990.