Translation surfaces in affine 3-space

Year 2020, , 1944 - 1954, 08.12.2020

Moahmd Saleem Lone , Murat Kemal Karacan , Yilmaz Tunçer , Hasan Es

https://doi.org/10.15672/hujms.564921

Cited By: 2

Abstract

In this paper, we study translation surfaces in three dimensional affine space. We characterize the finite type non-degenerate translation surfaces with respect to the first affine and the second affine fundamental forms.

Keywords

Affine space, Laplace operator, translation surfaces

References

• [1] M. Andrade, Calculus of affine structures and applications for isosurfaces (in portuguese), PhD dissertation, Rio de Janeiro, August, 2011.
• [2] M. Andrade and T. Lewiner, Affine-invariant curvature estimators for implicit surfaces, Comput. Aided Geom. Des. 29, 162–173, 2012.
• [3] K. Arslan, B. Bayram, B. Bulca and G. Ozturk, On translation surfaces in 4- dimensional Euclidean space, Acta Comment. Univ. Tartu. Math. 20 (2), 123–133, 2016.
• [4] M. Bekkar and B. Senoussi, Translation surfaces in the 3-dimensional space satisfying $\Delta^{III}r_i=\mu_i r_i$, J. Geom. 103 (3), 367-374, 2012.
• [5] W. Blaschke, Vorlesungen über Differentialgeometrie II, Springer, Berlin, 1923.
• [6] Ch. Baba-Hamed, M.Bekkar and H. Zoubir, Translation surfaces in the threedimensional Lorentz-Minkowski space satisfying $\Delta r_{i}=\lambda _{i}r_{i},$, Int. J. Math. Anal. 4 (17), 797–808, 2010.
• [7] B.-Y. Chen, Total mean curvature and submanifolds of finite type, World Scientific, Singapore, 1984.
• [8] F. Dillen, J. Pas and L. Vertraelen, On surfaces of finite type in Euclidean 3-space, Kodai Math. J. 13 (1), 10-21, 1990.
• [9] O.J. Garay, An extension of Takahashis theorem, Geom. Dedicata, 34 (2), 105-112, 1990.
• [10] W. Goemans, Surfaces in three-dimensional Euclidean and Minkowski space, in particular a study of Weingarten surfaces, PhD. Dissertation, September, 2010.
• [11] E.F.C. Huamani, Affine Minimal surfaces with singularities, Masters dissertation, Rio de Janeiro, September, 2017.
• [12] Y. Fu and Z.H. Hou, Affine translation surfaces with constant Gauss curvatures, Kyungpook Math. J. 50, 337–343, 2010.
• [13] M.K. Karacan, D.W. Yoon and B. Bukcu, Translation surfaces in the three dimensional simply isotropic space $\mathbb{I}_3^1$, Int. J. Geom. Methods Mod. Phys. 13 (7), 1650088, 9 pp., 2016.
• [14] A.M. Li, U. Simon and G. Zhao, Global affine differential geometry of hypersurfaces, De Gruyter Expositions Math., 328 pages, 1993.
• [15] M. Magid and L.Vrancken, Affine translation surfaces, Results Math. 35, 134–144, 1999.
• [16] F. Manhart, Die AffinminimalrückungsflÄchen, Arch. Math. 44, 547–556, 1985.
• [17] U. Simon, A. Schwenk-Schellschmidt and H. Viesel, Introduction to the affine differential geometry of hypersurfaces. Lecture Notes, Science Univ. Tokyo Press, 1991.
• [18] H. Sun, On affine translation surfaces of constant mean curvature, Kumamoto J. Math. 13, 49–57, 2000.
• [19] Y. Yang, Y.H. Yu and H.L. Liu, Linear Weingarten centroaffine translation surfaces in $\mathbb{R}^3$, J. Math. Anal. Appl. 375, 458–466, 2011.
• [20] D. Yanga and Y. Fu, On affine translation surfaces in affine space, J. Math. Anal. Appl. 440, 437–450, 2016.
• [21] D.W. Yoon, Some Classification of translation surfaces in Galilean 3-space, Int. J. Math. Anal. 6 (28), 1355–1361, 2012.