On Minimal Surfaces in Galilean Space

Abstract

In this paper, we investigated the minimal surfaces in three dimensional Galilean space $\mathbb{G}^{3}$. We showed that the condition of minimality of a surface area is locally equivalent to the mean curvature vector $H$ vanishes identically. Then, we derived the necessary and sufficient conditions that the minimal surfaces have to satisfy in Galilean space.

Keywords

• Minimal surfaces,area of a surface,Galilean space

Kaynakça

1. [1] M. E. Aydin, A. O. Ö˘grenmi¸s, M. Ergüt, Classification of factorable surfaces in the pseudo-Galilean space, Glasnik Matematicki, 50 (2015) 441–451.
2. [2] M. Dede, Tubular surfaces in Galilean space, Math. Commun., 18 (2013), 209-217.
3. [3] M. Dede, C. Ekici, A. Ceylan Çöken, On the parallel surfaces in Galilean space, Hacettepe journal of math. and statistics, 42 (2013), 605-615.
4. [4] B. Divjak, Curves in pseudo-Galilean geometry, Annales Univ. Sci. Budapest., 41 (1998), 117–128.
5. [5] B. Divjak, Z. Milin-Sipus, Minding isometries of ruled surfaces in pseudo-Galilean space, J. geom., 77 (2003), 35–47.
6. [6] M. Hazewinkel, Encyclopaedia of mathematics, Vol 6., Kluwer Academic Publishers, 2013.
7. [7] J. Inoguchi, M. Toda, Timelike minimal surfaces via Loop groups, Acta Applicandae Mathematica, 83 (2004), 313-355.
8. [8] Y. W. Kim, S. E. Koh, H. Shin, S. D. Yang, Spacelike maximal surfaces, Timelike minimal surfaces, and Bjorling representation formulae, J. Korean Math. Soc., 48 (2011), 1083–1100.

9. [9] D. Kutach, A connection between Minkowski and Galilean space-times in quantum mechanics, International Studies in the Philosophy of Science, 24 (2010), 15–29.
10. [10] F. J. Lopez, R. Lopez, R. Souam, Maximal surfaces of Riemann type in Lorentz-Minkowski space, Michigan J. of Math, 47 (2000) 469-497.
11. [11] W. Meeks, J. Pérez, A Survey on classical minimal surface theory, University Lecture Series, vol. 60. AMS, (Providence, 2012).
12. [12] R. Ossernan, A survey of minimal surfaces, Van Nostrand Reinhold, (New York, 1986).
13. [13] G. Öztürk, S. Büyükkütük, ˙I. Ki¸si, A characterization of curves in Galilean 4-space G4; Bulletin of the Iranian Mathematical Society, 43(3) (2017), 771-780.
14. [14] O. Röschel, Die Geometrie des Galileischen raumes, Habilitationssch., Inst. für Math. und Angew. Geometrie (Leoben, 1984).
15. [15] D. C. Tkhi, A. T. Fomenko, Minimal surfaces, stratified multivarifolds, and the plateau problem, (Providence, RI: American Mathematical Society, 1991).
16. [16] V. I. Woestijne, Minimal surfaces of the 3-dimentional Minkowski space, World Scientific Publishing, Singapore 344-369, 1990.
17. [17] I. M. Yaglom, A Simple Non-Euclidean geometry and its physical basis, Springer-Verlag (New York, 1979).
18. [18] Z. K. Yüzbası M. Bektas, On the construction of a surface family with common geodesic in Galilean space G3, Open Phys., 14 (2016), 360-363.