Differential equations and integral characterizations of timelike and spacelike spherical curves in the Minkowski space-time $E_1^4 $

Author M. Onder, T. Kahraman, H. H. Ugurlu
mehmet.onder@cbu.edu.tr, tanju.kahraman@cbu.edu.tr; hugurlu@gazi.edu.tr
Celal Bayar University, Muradiye Campus, Muradiye, Manisa, Turkey; Gazi University, Ankara, Turkey

Abstract In this paper we give differential equations characterizing timelike and spacelike curves lying on hyperbolic sphere $H_0^3 $ and Lorentzian sphere $S_1^3 $ in the Minkowski space-time $E_1^4 $. Furthermore, we give the integral characterizations of these curves in $E_1^4 $.
Keywords Minkowski space-time; timelike curve; Frenet frame
Reference
1. S. Breuer, D. Gottlieb, Explicit characterizations of spherical curves, Proc. Amer. Math. Soc., 27 (1971), 126–127.

2. V. Dannon, Integral characterizations and the theory of curves, Proc. Amer. Math. Soc., 81 (1981), ¹2, 600–602.

3. M. Kazaz, H.H. Ugurlu, A. Ozdemir, Integral characterizations for timelike and spacelike curves on Lorentzian sphere $S_1^3 ,$ Iranian Journal of Science and Technology, Transaction A, 32 (2008), ¹A1, 25–31.

4. E. Kreyzig, Differential geometry. – Univ. of Toronto Press, Toronto, 1959.

5. E. Kreyzig, A. Pendl, Spherical curves and their analogues in affine differential geometry, Proc. Amer. Math. Soc., 48 (1975), ¹2, 423–428.

6. U. Pekmen, S. Pasali, B.Y. Chen, Some characterizations of Lorentzian spherical spacelike curves, Mathematica Moravica, 3 (1999), 33–37.

7. M. Petrovic-Turgasev, E. Sucurovic, Some characterizations of the spacelike, the timelike and the null curves on the pseudohyperbolic space $H_0^2 $ in $E_1^3 ,$ Kragujevac J. Math., 22 (2000), 71–82.

8. M. Petrovic-Turgasev, E. .Sucurovic, Some characterizations of Lorentzian spherical spacelike curves with the timelike and null principal normal, Mathematica Moravica, 4 (2000), 83–92.

9. M. Petrovic-Turgasev, E. Sucurovic, Some characterizations of Lorentzian spherical timelike and null curves, Matemat. Vesnyk, 53 (2001), 21–27.

10. B. O’Neill, Semi-Riemannian geometry with applications to relativity. – Academic Press, London, 1983.

11. M. ¨ Onder, Timelike ve spacelike e.grilerin karakterizasyonu. – CB¨U Fen Bilimleri Enstit¨us¨u, Y¨uksek Lisans Tezi, 2002.

12. M. Sezer, Differential equations and integral characterizations for E4spherical curves, Turk J. Math., 13 (1989), ¹1, 125–131.

13. Y.C. Wong, A global formulation of condition for a curve to live in a sphere, Monatschefte fur Mathematik, 67 (1963), 363–365.

14. Y.C. Wong, On an explicit characterization of spherical curves, Proc. Amer. Math. Soc., 34 (1972), 239–242.

15. J. Walrave, Curves and surfaces in Minkowski space. – PhD. thesis, K.U. Leuven, Fac. of Science, Leuven, 1995.
Pages 30-37
Volume 40
Issue 1
Year 2013
Journal Matematychni Studii
Full text of paper PDF
Table of content of issue HTML