A NOTE ON SPECIAL CURVES IN E^4

Year 2016, Volume: 2 Issue: 1, 22 - 37, 30.06.2016

Süha Yılmaz Şuur Nizamoğlu

Abstract

This article has consisted of a part of doctorate thesis by Süha Yılmaz [7]. Firstly, Frenet formulas are given in E^4.

 Later, characterizations of regular and inclined curves are studied E^4. It has been given that a space-like curve is an inclined curve if and only if the

expression..........obtained using harmonic curvature functions in E^4. In addition, it has been observed that a fifth order vectorial differential equation of position vector of a space-like curve in E^4 been satisfied by means of Frenet formulas. Similarly, it has been denoted that a fourth order vectorial differential equation of tangent vector of a space-like curve in E^4 has been also verified using Frenet formulas. Moreover, we characterized tangent and trinormal indicatrices with one theorem. Finally, it has been denoted that if and only if spherical indicatrices of space-like curve with time-like trinormal vector are regular curves.

Keywords

Inclined curve, harmonic curvature, Frenet formulas, regular curve, space-like curve, time-like curve, spherical indicatrices

References

• [1] Ekmekçi,N., Lorentz manifoldları üzerinde eğilim çizgileri, Doktora Tezi , A.Ü Fen Bilimleri Enstitüsü, 1991.
• [2] Fox, L., Parker, I. B., Chebyshev Polynomials in Numerical Analysis, Oxford University Press, 1968.
• [3] Gluck, H., Higher Curvatures of Curves in Euclidean Space, Proc.America.Math.Monthly, 73,699-704, 1966.
• [4] O’Neill, B., Semi-Riemannian Geometry, Academic Press, NY, 1983.
• [5] Şemin, F., Diferansiyel Geometri-I,İstanbul Üniversitesi Fen Fakültesi Yayınları, 1983.
• [6] Turgut, A., 3-Boyutlu Minkowski Uzayında Space-Like ve Time-Like Regle Yüzeyler. A.Ü. Fen Bilimleri Enstitüsü, Doktora Tezi, 1995.
• [7] Yılmaz , S., Spherical Indicators of Curves and Characterisations some Special Curves In Four Dimensional Lorentzian Space, Dokuz Eylül Üniversitesi Fen Bilimleri Enstitüsü, Doktora Tezi, 2001.
• [8] Mağden, A., 𝑅4 Uzayında Bazı Özel Eğriler ve Karakterizasyonları, Atatürk Üniversitesi Fen Bilimleri Enstitüsü Doktora Tezi, 1990.