# DUAL SURFACES DEFINED BY z = f(u) + g(ν) IN SIMPLY ISOTROPIC 3-SPACE ${\mathbb{I}}{\frac{1}{3}}$

Cakmak, Ali;Karacan, Murat Kemal;Kiziltug, Sezai

• Accepted : 2017.12.21
• Published : 2019.01.31
• 253 8

#### Abstract

In this study, we define the dual surfaces by z = f(u) + g(v) and also classify these surfaces in ${\mathbb{I}}{\frac{1}{3}}$ satisfying some algebraic equations in terms of the coordinate functions and the Laplace operators according to fundamental forms of the surface.

#### Keywords

dual surfaces;simply isotropic space;Monge patch;Laplace operator

#### References

1. K. Arslan, B. Bayram, B. Bulca, and G. Ozturk, On translation surfaces in 4-dimensional Euclidean space, Acta Comment. Univ. Tartu. Math. 20 (2016), no. 2, 123-133.
2. M. E. Aydin, A generalization of translation surfaces with constant curvature in the isotropic space, J. Geom. 107 (2016), no. 3, 603-615. https://doi.org/10.1007/s00022-015-0292-0
3. Ch. Baba-Hamed, M. Bekkar, and H. Zoubir, Translation surfaces in the three-dimensional Lorentz-Minkowski space satisfying ${\Delta}r_i$ = ${\lambda}_ir_i$, Int. J. Math. Anal. (Ruse) 4 (2010), no. 17-20, 797-808.
4. M. Bekkar and B. Senoussi, Translation surfaces in the 3-dimensional space satisfying ${\Delta}^{III}r_i$ = ${\mu}_ir_i$, J. Geom. 103 (2012), no. 3, 367-374. https://doi.org/10.1007/s00022-012-0136-0
5. M. Bekkar and H. Zoubir, Surfaces of revolution in the 3-dimensional Lorentz-Minkowski space satisfying ${\Delta}x^i$ = ${\lambda}^ix^i$, Int. J. Contemp. Math. Sci. 3 (2008), no. 21-24, 1173-1185.
6. B. Bukcu, D. W. Yoon, and M. K. Karacan, Translation surfaces of type 2 in the three dimensional simply isotropic space $II_{3}^{1}$, https://doi.org/10.4134/BKMS.b16037.
7. B. Bukcu, D. W. Yoon, and M. K. Karacan, Translation surfaces in the 3-dimensional simply isotropic space $II_{3}^{1}$ satisfying ${\Delta}^{III}x_i$ = ${\lambda}_ix_i$, Konuralp J. Math. 4 (2016), no. 1, 275-281.
8. A. Cakmak, M. K. Karacan, S. Kiziltug, and D. W. Yoon, Translation surfaces in the 3-dimensional Galilean space satisfying ${\Delta}^{II}x_i$ = ${\lambda}_ix_i$, https://doi.org/10.4134/BKMS.b16044.
9. W. Goemans, Surfaces in three-dimensional Euclidean and Minkowski space, in particular a study of Weingarten surfaces, PhD. Dissertation, September 2010.
10. G. Kaimakamis, B. Papantoniou, and K. Petoumenos, Surfaces of revolution in the 3-dimensional Lorentz-Minkowski space $E_{3}^{1}$ satisfying ${\Delta}^{III}\;_r^{\rightarrow}$ = $A_{r}^{\rightarrow}$, Bull. Greek Math. Soc. 50 (2005), 75-90.
11. M. K. Karacan, D. W. Yoon, and B. Bukcu, Translation surfaces in the three-dimensional simply isotropic space $II_{3}^{1}$, Int. J. Geom. Methods Mod. Phys. 13 (2016), no. 7, 1650088, 9 pp.
12. H. Liu, Translation surfaces with constant mean curvature in 3-dimensional spaces, J. Geom. 64 (1999), no. 1-2, 141-149. https://doi.org/10.1007/BF01229219
13. H. Pottmann, P. Grohs, and N. J. Mitra, Laguerre minimal surfaces, isotropic geometry and linear elasticity, Adv. Comput. Math. 31 (2009), no. 4, 391-419. https://doi.org/10.1007/s10444-008-9076-5
14. H. Pottmann and Y. Liu, Discrete Surfaces in Isotropic Geometry, Mathematics of Surfaces XII, Volume 4647 of the series Lecture Notes in Computer Science, (2007), 341-363.
15. H. Sachs, Isotrope Geometrie des Raumes, Friedr. Vieweg & Sohn, Braunschweig, 1990.
16. B. Senoussi and M. Bekkar, Helicoidal surfaces with ${\Delta}^Jr$ = Ar in 3-dimensional Euclidean space, Stud. Univ. Babes-Bolyai Math. 60 (2015), no. 3, 437-448.
17. Z. M. Sipus, Translation surfaces of constant curvatures in a simply isotropic space, Period. Math. Hungar. 68 (2014), no. 2, 160-175. https://doi.org/10.1007/s10998-014-0027-2
18. K. Strubecker, Differentialgeometrie des isotropen Raumes. III, Flachentheorie, Math. Z. 48 (1942), 369-427. https://doi.org/10.1007/BF01180022
19. K. Strubecker, Duale Minimalachen des isotropen Raumes, Rad Jugoslav. Akad. Znan. Umjet. No. 382 (1978), 91-107.
20. D. W. Yoon, Some classification of translation surfaces in Galilean 3-space, Int. J. Math. Anal. (Ruse) 6 (2012), no. 25-28, 1355-1361.
21. D. W. Yoon, C. W. Lee, and M. K. Karacan, Some translation surfaces in the 3-dimensional Heisenberg group, Bull. Korean Math. Soc. 50 (2013), no. 4, 1329-1343. https://doi.org/10.4134/BKMS.2013.50.4.1329