SPHERICAL PRODUCT SURFACES IN THE GALILEAN SPACE

Abstract

In the present paper, we consider the spherical product surfaces in a Galilean 3-space G3. We derive a classi cation result for such surfaces of constant curvature in G3. Moreover, we analyze some special curves on these surfaces in G3.

Keywords

• Galilean plane,spherical product surface,Gaussian curvature,geo- desic line,asymptotic line

References

1. [1] M. Akar, S. Yuce, N. Kuruoglu, One-parameter planar motion on the Galilean plane, Int. Electron. J. Geom. 6(2) (2013), 79-88.
2. [2] K. Arslan, B. Kilic, Product submanifolds and their types, Far East J. Math. Sci. 6(1) (1998), 125-134.
3. [3] M. E. Aydin, A. Mihai, A. O. Ogrenmis, M. Ergut, Geometry of the solutions of localized induction equation in the pseudo-Galilean space, Adv. Math. Phys., vol. 2015, Article ID 905978, 7 pages, 2015. doi:10.1155/2015/905978.
4. [4] M. E. Aydin, I. Mihai, On certain surfaces in the isotropic 4-space, Math. Commun., in press.
5. [5] A. H. Barr, Superquadrics and angle-preserving transformations, IEEE Comput.Graph. Appl. 1(1) (1981), 11-23.
6. [6] B. Bulca, K. Arslan, B. (Kilic) Bayram, G. Ozturk, Spherical product surfaces in E4; An. St. Univ. Ovidius Constanta 20(1) (2012), 41{54.
7. [7] B. Bulca, K. Arslan, B. (Kilic) Bayram, G. Ozturk, H. Ugail, On spherical product surfaces in E3, IEEE Computer Society, 2009, Int. Conference on CYBERWORLDS.
8. [8] M. Dede, Tubular surfaces in Galilean space, Math. Commun. 18 (2013), 209{217.

9. [9] M. Dede, C. Ekici, A. C. Coken, On the parallel surfaces in Galilean space, Hacettepe J. Math. Stat. 42(6) (2013), 605{615.
10. [10] B. Divjak, Z.M. Sipus, Special curves on ruled surfaces in Galilean and pseudo-Galilean spaces, Acta Math. Hungar. 98 (2003), 175{187.
11. [11] M.P. do Carmo, Dierential Geometry of Curves and Surfaces, Prentice Hall: Englewood Clis, NJ, 1976.
12. [12] Z. Erjavec, B. Divjak, D. Horvat, The general solutions of Frenet's system in the equiform geometry of the Galilean, pseudo-Galilean, simple isotropic and double isotropic space, Int. Math. Forum 6(17) (2011), 837 - 856.
13. [13] Z. Erjavec, On generalization of helices in the Galilean and the pseudo-Galilean space, J. Math. Res. 6(3) (2014), 39-50.
14. [14] A. Gray, Modern Dierential Geometry of Curves and Surfaces with Mathematica, CRC Press LLC, 1998.
15. [15] I. Kamenarovic, Existence theorems for ruled surfaces in the Galilean space G3; Rad Hazu Math. 456(10) (1991), 183-196.
16. [16] M.K. Karacan, Y. Tuncer, Tubular surfaces of Weingarten types in Galilean and pseudo- Galilean, Bull. Math. Anal. Appl. 5(2) (2013), 87-100.
17. [17] N. H. Kuiper, Minimal Total absolute curvature for immersions, Invent. Math., 10 (1970), 209-238.
18. [18] A.O. Ogrenmis, M. Ergut, M. Bektas, On the helices in the Galilean Space G3; Iranian J. Sci. Tech., 31(A2) (2007), 177-181.
19. [19] A. Onishchick, R. Sulanke, Projective and Cayley-Klein Geometries, Springer, 2006.
20. [20] H. B. Oztekin, S. Tatlipinar, On some curves in Galilean plane and 3-dimensional Galilean space, J. Dyn. Syst. Geom. Theor. 10(2) (2012), 189-196.
21. [21] B. J. Pavkovic, I. Kamenarovic, The equiform dierential geometry of curves in the Galilean space G3, Glasnik Mat. 22(42) (1987), 449-457.
22. [22] Z.M. Sipus, Ruled Weingarten surfaces in the Galilean space, Period. Math. Hungar. 56 (2008), 213{225.
23. [23] Z.M. Sipus, B.Divjak, Some special surface in the pseudo-Galilean Space, Acta Math. Hungar. 118 (2008), 209{226.
24. [24] Z.M. Sipus, B. Divjak, Translation surface in the Galilean space, Glas. Mat. Ser. III 46(2) (2011), 455{469.
25. [25] Z.M. Sipus, B. Divjak, Surfaces of constant curvature in the pseudo-Galilean space, Int. J. Math. Sci., 2012, Art ID375264, 28pp.
26. [26] D.W. Yoon, Surfaces of revolution in the three dimensional pseudo-Galilean space, Glas. Mat. Ser. III, 48(2) (2013), 415-428.
27. [27] D.W. Yoon, Some classication of translation surfaces in Galilean 3-space, Int. J. Math. Anal. 6(28) (2012), 1355-1361.
28. [28] D. W. Yoon, Classication of rotational surfaces in pseudo-Galilean space, Glas. Mat. Ser. III 50(2) (2015), 453-465.