Konuralp Journal of Mathematics, 7 (1) (2019) 16-24 Konuralp Journal of Mathematics Journal Homepage: www.dergipark.gov.tr/konuralpjournalmath e-ISSN: 2147-625X Characterizations of Inclined Curves According to Parallel Transport Frame in E 4 and Bishop Frame in E 3 Fatma Ates ¸ 1* , ˙ Ismail G ¨ ok 2 , F. Nejat Ekmekci 3 and Yusuf Yaylı 4 1 Department of Mathematics-Computer Sciences, Necmettin Erbakan University, Konya, Turkey 1,2,3,4 Department of Mathematics, Faculty of Science, Ankara University, Ankara, Turkey * Corresponding author E-mail: fgokcelik@ankara.edu.tr Abstract The aim of this paper is to introduce inclined curves according to parallel transport frame. This paper begins by defined a vector field D called Darboux vector field of an inclined curve in E 4 . It will then go on to an alternative characterization for the inclined curves “α : I ⊂ R −→ E 4 is an inclined curve ⇔ k 1 (s)  k 1 (s)ds + k 2 (s)  k 2 (s)ds + k 3 (s)  k 3 (s)ds = 0” where k 1 (s), k 2 (s), k 3 (s) are the principal curvature functions according to parallel transport frame of the curve α and also, similar characterization for the generalized helices according to Bishop frame in E 3 is given by α : I ⊂ R −→ E 3 is a generalized helix ⇔ k 1 (s)  k 1 (s)ds + k 2 (s)  k 2 (s)ds = 0” where k 1 (s), k 2 (s) are the principal curvature functions according to Bishop frame of the curve α . These curves have illustrated some examples and draw their figures with use of Mathematica programming language. Also, it is given an example for the inclined curve in E 4 and showed that the above condition is satisfied for this curve. Keywords: Parallel transport frame, Inclined curve, Bishop frame, Generalized helix 2010 Mathematics Subject Classification: 14H45, 14H50, 53A04 1. Introduction The curves are a very important topic in all disciplines. They appear in physical applications as well as medical sciences heart chest film with X-ray curve, how to act is important to us. Curves give the movements of the particle in Physics. Helical curves are the very important type of curves. Because, helices are among the simplest objects in the art, molecular structures, nature, etc. For example, the path, aroused by the climbing of beans and the orbit where the progressing of the screw is a helix curve. Also, in medicine DNA molecule is formed as two intertwined helices and many proteins have helical structures, known as alpha helices. So, helices are very important for understanding nature. Also, helices are called as inclined curves in higher dimensional Euclidean space E n (n ≥ 4). Therefore, recently researchers have shown an increased interest in the helices in the Euclidean space E 3 (See for details: [1, 3, 6, 10, 11, 14]). In 1802, M. A. Lancret first proposed a theorem and in 1845, B. de Saint Venant first proved this theorem: ”A necessary and sufficient condition of a curve to be a general helix is that the ratio of curvature to torsion should be a constant.” Another definition of the helix curve is that the tangent vector field at all points of the curve makes a constant angle with a fixed direction. Recently, many studies have been reported on generalized helices and inclined curves [1, 4, 9, 12]. The Frenet frame is constructed for the curve of third order continuously differentiable non-degenerate curves. Curvature of the curve may vanish on some points of the curve, that is, second derivative of the curve may be zero. In this situation, we need an alternative frame in E 3 . Therefore in [2], Bishop defined a new frame for a curve and called as Bishop frame which is well defined even when the curve has vanishing second derivative in 3−dimensional Euclidean space E 3 . Similarly, G ¨ okc ¸ elik et al. defined a new frame for a curve and called parallel transport frame in E 4 [7]. The parallel transport frame is an alternative frame defined by a moving frame. They consider a regular curve α (s) parametrized by s and they defined a normal vector Email addresses: fgokcelik@ankara.edu.tr (Fatma Ates ¸), igok@science.ankara.edu.tr ( ˙ Ismail G¨ ok), ekmekci@science.ankara.edu.tr (F. Nejat Ekmekci), yayli@science.ankara.edu.tr (Yusuf Yaylı)