Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
Minkowski space; timelike curve; Frenet equations; slant helix
Minkowski space; timelike curve; Frenet equations; slant helix
Summary:
We consider a unit speed timelike curve $\alpha $ in Minkowski 4-space ${\mathbf{E}}_1^4$ and denote the Frenet frame of $\alpha $ by $\lbrace {\mathbf{T}}, {\mathbf{N}}, {\mathbf{B}}_1, {\mathbf{B}}_2\rbrace $. We say that $\alpha $ is a generalized helix if one of the unit vector fields of the Frenet frame has constant scalar product with a fixed direction $U$ of ${\mathbf{E}}_1^4$. In this work we study those helices where the function $\langle {\mathbf{B}}_2,U\rangle $ is constant and we give different characterizations of such curves.
References:
[1] Barros, M.: General helices and a theorem of Lancret. Proc. Amer. Math. Soc. 125 (1997), 1503–1509. DOI 10.1090/S0002-9939-97-03692-7 | MR 1363411 | Zbl 0876.53035
[2] Erdoǧan, M., Yilmaz, G.: Null generalized and slant helices in 4-dimensional Lorentz-Minkowski space. Int. J. Contemp. Math. Sci. 3 (2008), 1113–1120. MR 2477940 | Zbl 1160.53331
[3] Ferrandez, A., Gimenez, A., Luca, P.: Null helices in Lorentzian space forms. Int. J. Mod. Phys. A 16 (2001), 4845–4863. DOI 10.1142/S0217751X01005821 | MR 1873162
[4] Gluck, H.: Higher curvatures of curves in Eulidean space. Amer. Math. Monthly 73 (1996), 699–704. DOI 10.2307/2313974 | MR 0198355
[5] Izumiya, S., Takeuchi, N.: New special curves and developable surfaces. Turkish J. Math. 28 (2004), 531–537. MR 2062560 | Zbl 1081.53003
[6] Kocayiǧit, H., Önder, M.: Timelike curves of constant slope in Minkowski space ${\mathbf{E}}_1^4$. J. Science Techn. Beykent Univ. 1 (2007), 311–318.
[7] Kula, L., Yayli, Y.: On slant helix and its spherical indicatrix. Appl. Math. Comput. 169 (2005), 600–607. DOI 10.1016/j.amc.2004.09.078 | MR 2171171 | Zbl 1083.53006
[8] Millman, R. S., Parker, G. D.: Elements of differential geometry. Prentice-Hall Inc., Englewood Cliffs, N. J., 1977. MR 0442832 | Zbl 0425.53001
[9] Önder, M., Kazaz, M., Kocayiǧit, H., Kilic, O.: $B_2$-slant helix in Euclidean 4-space $E^4$. Int. J. Contemp. Math. Sci. 3 (29) (2008), 1433–1440. MR 2514022 | Zbl 1175.14019
[10] O’Neill, B.: Semi-Riemannian geometry. With applications to relativity. Pure and Applied Mathematics. vol. 103, Academic Press, Inc., New York, 1983. MR 0719023
[11] Petrovic-Torgasev, M., Sucurovic, E.: W-curves in Minkowski spacetime. Novi Sad J. Math. 32 (2002), 55–65. MR 1949817
[12] Scofield, P. D.: Curves of constant precession. Amer. Math. Monthly 102 (1995), 531–537. DOI 10.2307/2974768 | MR 1336639 | Zbl 0881.53002
[13] Synge, J. L.: Timelike helices in flat space-time. Proc. Roy. Irish Acad. Sect. A 65 (1967), 27–42. MR 0208976 | Zbl 0152.45901
Search
Partner of
