Geometry of Timelike Normal Surfaces in De Sitter 3-Space

Abstract

In this paper, we characterize the normal surfaces of regular curves in de Sitter 3-space, which is a pseudosphere in 4-dimensional Minkowski space. The results obtained in terms of curvatures, which are the basic parameters of curves in differential geometry, have always been meaningful. Here we examine with the help of the given normal surface curvatures associated with a regular curve. First of all, we obtain the geometric components of the surface, such as fundamental forms and curvatures, in terms of the curvatures of the curve. Then, we examine the special conditions of this surface, such as minimalness and flatness. Finally, we obtain some new characterizations, theorems, and results with the help of the obtained equations.

Keywords

• Normal surface
• Weingarten surface
• minimal surface
• flat surface
• de Sitter space

Ethical Statement

The study is complied with research and publication ethics.

References

1. M. Do Carmo, "Differential Geometry of Curves and Surfaces", Prentice-Hall, Englewood Cliffs, 1976.
2. T. Mert and B. Karlıga, "Timelike Surfaces With Constant Angle in de-Sitter Space S_1^3", Cumhuriyet University Faculty of Science Science Journal, vol. 37(1), pp. 1-11, 2016.
3. G.U. Kaymanli, S. Okur and C. Ekici, "The Ruled Surfaces Generated By Quasi Vectors", IV. International Scientific and Vocational Studies Congress - Science and Health, November 2019.
4. R. Lopez, "Differential geometry of curves and surfaces in Lorentz-Minkowski space", Mini Course taught at IME-USP, Brasil, 2008.
5. F. Sakai, "Weil divisors on normal surfaces", Duke Mathematical Journal, vol. 51(4), pp. 877-887, 1984.
6. T. Körpınar and A. Sazak, "Binormal Surfaces of Adjoint Curves in 3D Euclidean Space", Journal of Polytechnic, vol. 26(3), pp. 1141-1144, 2023.
7. T. Körpınar and A. Sazak, "Normal Surfaces Associated with Adjoint Curves in E3", Bitlis Eren University Science Journal, vol. 11(3), pp. 779-783, 2022.
8. I. Protrka, "Harmonic Evolutes of Timelike Ruled Surfaces in Minkowski Space", 18th Scientific-Professional Colloquium on Geometry and Graphics, September, 2015.

9. I. Protrka, "The harmonic evolute of a helicoidal surfaces in Minkowski 3-space", Proceedings of the Young Researcher Workshop on Differential Geometry in Minkowski Space, Granada, Spain, pp. 133-142, 2017.
10. A. Sarıoglugil and A. Tutar, "On Ruled Surface in Euclidean Space E³", Int. J. Contemp. Math. Sci., vol. 2(1), pp. 1-11, 2007.
11. Y. Ünlütürk and C. Ekici, "Parallel Surfaces of Spacelike Ruled Weingarten Surfaces in Minkowski 3-space", New Trends in Mathematical Sciences, vol. 1(1), pp. 85-92, 2013.
12. G.Y. Senturk and S. Yuce, "Characteristic properties of the ruled surface with Darboux frame in E³", Kuwait J. Sci., vol. 42(2), pp. 14-33, 2015.
13. J.H. Choi and Y.H. Kim, "Associated curves of a Frenet curve and their applications", Appl. Math. Comput., vol. 218(18), pp. 9116-9124, 2012.
14. T. Körpınar, M.T. Sarıaydın, and E. Turhan, "Associated curves according to Bishop frame in Euclidean 3-space", Adv. Model. Optim., vol. 15(3), pp. 713-717, 2013.
15. N. Macit and M. Düldül, "Some new associated curves of a Frenet curve in E³ and E⁴", Turkish J. Math., vol. 38, pp. 1023-1037, 2014.
16. S. Yılmaz, "Characterizations of some associated and special curves to type-2 Bishop frame in E³", Kırklareli University J. Eng. Sci., vol. 1, pp. 66-77, 2015.
17. B. O'Neill, “Semi-Riemannıan Geometry with applications to relativity”, Academic Press, New York, 1983.