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

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 condition $\Delta^{J}x_i=\lambda_i x_i,$  with 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.

Keywords

• Invariant surface
• Laplace operator
• Simply isotropic space
• Surfaces of finite type

References

1. [1] B. Y. Chen, J. Morvan, T. Nore, Energy, tension and finite type maps, Kodai Math. J., 9 (1986), 406–418.
2. [2] B. Y. Chen, Total mean curvature and submanifolds finite type, World Scientific, New Jersey, 1984.
3. [3] O. J. Garay, On a certain class of finite type surfaces of revolution, Kodai Math. J., 11 (1988), 25–31.
4. [4] B. Y. Chen, A report on submanifolds of finite type, Soochow J. Math., 22 (1996), 117–337.
5. [5] T. Tahakashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan, 18 (1966), 380–385.
6. [6] F. Dillen, J. Pas, L. Verstraelen, On surfaces of finite type in Euclidean 3-space, Kodai Math. J., 13 (1990), 10–21.
7. [7] O. J. Garay, An extension of Takahashi’s theorem, Geom. Dedicata, 34 (1990), 105–112.
8. [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. [9] B. Y. Chen, Some open problems and conjectures on submanifolds of finite type: recent development, Tamkang J. Math., 45 (2014), 87–108.
10. [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. [11] M. E. Aydin, Constant curvature factorable surfaces in 3-dimensional isotropic space, J. Korean Math. Soc., 55 (1) (2018) 59–7.
12. [12] M. E. Aydin, A. Erdur, M. Ergut, Affine factorable surfaces in isotropic spaces, TWMS J. Pure Appl. Math., 11 (2020), 72–88.
13. [13] M. E. Aydin, I. Mihai, On certain surfaces in the isotropic 4-space, Math. Com., 22(1) (2017), 41-51.
14. [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. [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. [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. [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. [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. [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. [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. [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. [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. [23] M. E. Aydin, M. Ergut, Affine translation surfaces in the isotropic 3-space, Int. Electron. J. Geom., 10 (2017), 21–30.
24. [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. [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. [26] H. Sachs, Isotrope Geometrie des Raumes, Vieweg, Braunschweig/Wiesbaden, 1990.