TRANSLATION SURFACES IN THE 3-DIMENSIONAL SIMPLY ISOTROPIC SPACE I13 SATISFYING
IIIxi = ixi

Abstract

In this paper, we classify translation surfaces in the three dimen- sional simply isotropic space I13 satisfying some algebraic equations in terms of the coordinate functions and the Laplacian operators with respect to the third fundamental form of the surface. We also give explicit forms of these surfaces.

Keywords

• Simply isotropic space,translation surfaces,Laplace operator

References

1. [1] L. J. Alias, A. Ferrandez and P. Lucas, Surfaces in the 3-dimensional Lorentz-Minkowski space satisfying
   x = Ax + B, Pacic J. Math. 156 (1992), 201{208.
2. [2] M.E.Aydin, Classication results on surfaces in the isotropic 3-space, http://arxiv.org/pdf/1601.03190.pdf
3. [3] M.E.Aydin, A generalization of translation surfaces with constant curvature in the isotropic space, J. Geom, DOI 10.1007/s00022-015-0292-0
4. [4] C.Baikoussis and L. Verstraelen, On the Gauss map of helicoidal surfaces,Rend. Sem. Math. Messina Ser. II 2(16) (1993), 31{42.
5. [5] M.Bekkar, Surfaces of Revolution in the 3-Dimensional Lorentz-Minkowski Space Satisfying
   xi = ixi;Int. J. Contemp. Math. Sciences, Vol. 3, 2008, no. 24, 1173 - 1185
6. [6] M.Bekkar, B. Senoussi, Translation surfaces in the 3-dimensional space satisfying
   III ri = iri; J. Geom. 103 (2012), 367{374
7. [7] S.M.Choi, On the Gauss map of surfaces of revolution in a 3-dimensional Minkowski space, Tsukuba J. Math. 19 (1995), 351{367.
8. [8] S.M.Choi, Y. H.Kim and D.W.Yoon, Some classication of surfaces of revolution in Minkowski 3-space, J. Geom. 104 (2013), 85{106

9. [9] B.Y. Chen, A report on submanifold of nite type, Soochow J. Math. 22 (1996),117{337.
10. [10] F. Dillen, J. Pas and L. Vertraelen, On surfaces of nite type in Euclidean 3-space,Kodai Math. J. 13 (1990), 10{21.
11. [11] F. Dillen, J. Pas and L. Vertraelen, On the Gauss map of surfaces of revolution,Bull. Inst. Math. Acad. Sinica 18 (1990), 239{246.
12. [12] O. J. Garay, An extension of Takahashi's theorem, Geom. Dedicata 34 (1990), 105{112.
13. [13] G. Kaimakamis, B. Papantoniou, K. Petoumenos, Surfaces of revolution in the 3-dimensional Lorentz-Minkowski space satisfying
    IIIr = Ar, Bull.Greek Math. Soc. 50 (2005), 75-90.
14. [14] M.K.Karacan, D.W.Yonn and B.Bukcu, Translation surfaces in the three dimensional simply isotropic space I13 ;International Journal of Geometric Methods in Modern Physics, accepted.
15. [15] H. Sachs, Isotrope Geometrie des Raumes, Vieweg Verlag, Braunschweig, 1990.
16. [16] B.Senoussi, M. Bekkar, Helicoidal surfaces with
    J r = Ar in 3-dimensional Euclidean space,Stud. Univ. Babes-Bolyai Math. 60(2015), No. 3, 437-448
17. [17] Z.M. Sipus, Translation Surfaces of constant curvatures in a simply isotropic space,Period Math. Hung. (2014) 68:160{175.
18. [18] K. Strubecker, Dierentialgeometrie des isotropen Raumes III, Flachentheorie, Math. Zeitsch.48 (1942), 369-427.
19. [19] T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan,18 (1966), 380{385.
20. [20] D.W.Yoon, Surfaces of Revolution in the three dimensional pseudo-galilean space,Glasnik Matematicki,Vol. 48 (68) (2013), 415 { 428.
21. [21] D.W.Yoon, Some Classication of Translation Surfaces in Galilean 3-Space, Int. Journal of Math. Analysis, 6(28) 2012,1355 - 1361.