Elementary Approachs on De Sitter Space

Year 2019, Volume: 7 Issue: 2, 183 - 190, 15.10.2019

Emre Öztürk , Yusuf Yaylı

https://doi.org/10.36753/mathenot.583477

Abstract

In this paper, we characterize the de Sitter space by means of spacelike and timelike curves that fully

lies on it. For this purpose, we consider the tangential part of the second derivative of the unit speed

curve on the hypersurface, and obtain the vector equations of the geodesics. We find the geodesics as

hyperbolas, ellipses, and helices. Moreover, we give an example of null curve with constant curvature in

4−dimensional Minkowski space and we illustrate the geodesics of S11(r) × R .

Keywords

De Sitter space, geodesic, curve with constant curvature

References

• Baek, J., Kim, D.S. and Kim, Y. H., A characterization of the unit sphere, Amer. Math.Monthly, 110(2003), no. 9,830–833.
• Chen, B.Y., Kim, D.S. and Kim,Y.H., New Characterizations of W-Curves, Publ. Math. Debrecen, 69 (2006), no. 4,457-472.
• Duggal, K. L. and Bejancu, A.Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Kluwer 1996.
• Ferrandez, A., Gimenez, A.and Lucas, P., Null helices in Lorentzian space forms, Internat. J.Modern Phys. A,16(2001), no. 30, 4845–4863.
• Inoguchi, J. and Lee, S., Null curves in Minkowski 3-space, Int. Electron. J. Geom., 1(2008), no. 2, 40-83
• Kim, D.S., Kim, Y.H. and Lee, J.W. A Characterization of Hyperbolic Spaces, Bull. Korean Math. Soc., 55(2018), no.4, 1103–1107
• Lopez, R., Differential Geometry of Curves and Surfaces in Lorentz-Minkowski space, Int.Electron. J. Geom., 7(2014),no.1, 44-107.
• O’Neill, B., Semi-Riemannian Geometry with Applications to Relativity, Academic Press. Inc., 1983.
• Öztürk, E. and Yaylı, Y., W-Curves In Lorentz-Minkowski Space, Mathematical Sciences and Applications E-Notes,5(2017), no.2, 76–88.
• Sakaki, M., Notes on null curves in Minkowski space, Turkish Journal Math, 34(2010), 417-424, doi:10.3906/mat-0812-14