Normal-like Curves with Respect to the Special Case of the ED-frame in Euclidean $4$-space

Abstract

The aim of this study is to present normal-like curves with respect to the special case of the ED-frame in Euclidean $4$-space. Furthermore, the relationship between geodesic torsion and curvature is given so that a curve lying on an oriented surface M in $4$-dimensional Euclidean space is congruent to a normal-like curve according to the special case of the ED-frame. Finally, an example of the study is presented.

Keywords

• Normal-like curve
• extended Darboux frame
• geodesic curvature
• geodesic torsion

References

1. Chen, B.Y. (2003). When does the position vector of a space curve always lie in its rectifying plane. The Amer. Math. Monthly, 110(2), 147-152.
2. İlarslan, K., & Nesovic, E. (2008). Some characterizations of osculating curves in the Euclidean spaces. Demonstratio Mathematica, 41(4), 931-939.
3. İlarslan, K., & Nesovic, E. (2008). Some characterizations of rectifying curves in the Euclidean space $E^ 4$. Turkish Journal of Mathematics, 32(1), 21-30.
4. İlarslan, K., & Nesovic, E. (2004). Timelike and null normal curves in Minkowski space $E_1^ {3}$ . Indian Journal of Pure and Applied Mathematics, 35(7), 881-888.
5. İlarslan, K. (2005). Spacelike normal curves in Minkowski space $E_1^ {3}$ . Turkish Journal of Mathematics, 29(1), 53-63.
6. İlarslan, K., & Nesovic, E. (2009). Spacelike and timelike normal curves in Minkowski space-time. Publications de l’Institut Mathematique, (105), 111-118.
7. Bektaş, Ö . (2018). Normal Curves in n-dimensional Euclidean Space. Advances in Difference Equations, 2018(1), 1-12.
8. Gluck, H. (1966). Higher curvatures of curves in Euclidean space. The Amer. Math. Monthly, 73(7), 699-704.

9. Guggenheimer, H. (1989). Computing frames along a trajectory. Computer Aided Geometric Design, 6(1), 77-78.
10. Darboux, G. (1896). Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal: ptie. Déformation infiniment petite et représentation sphérique. Notes et additions: I. Sur les méthodes d'approximations successives dans la théorie des équations différentielles, par E. Picard. II. Sur les géodésiques à intégrales quadratiques, par G. Koenigs. III. Sur la théorie des équations aux dérivées partielles du second ordre, par E. Cosserat. IV-XI. Par l'auteur. 1896 (Vol. 4). Gauthier-Villars.
11. Düldül, M., Uyar Düldül, B., Kuruoğlu, N., & Özdamar, E. (2017). Extension of the Darboux frame into Euclidean 4-space and its invariants. Turkish Journal of Mathematics, 41(6), 1628-1639.
12. Williams, M.Z., & Stein, F. (1964). A triple product of vectors in four-space. Mathematics Magazine, 37(4), 230-235.