Characterizations of Type-2 Harmonic Curvatures and General Helices in Euclidean space E⁴
Abstract
In this study, we introduce quaternionic type-2 Harmonic Curvatures and General Helices according to Frenet frame in 4-dimensional Euclidean Space 𝐸⁴ and investigate its properties for two cases. In the first case; we use a constant angle 𝜙 between a unit and fixed direction vector field 𝑈 and the first relatively Frenet frame vector field 𝑉₁ of the curve, that is,
𝐻 ( 𝑉₁,𝑈 ) = 𝑐𝑜𝑠 𝜙 =𝑐𝑜𝑛𝑠𝑡.
where ℎ (𝑉₁,𝑈) is the real quaternion inner product. Since the relatively Frenet frame vector field 𝑉₁ of the curve makes a constant angle with the unit and fixed direction vector field U, we call this curve as a General helix in 4-dimensional Euclidean Space 𝐸⁴. And then, in the other case, we define new type-2 harmonic curvature functions and we give a vector field 𝐷 which we call Darboux vector field for General helix. And then we obtain some characterizations for General helix in terms of type-2 harmonic curvature functions and the Darboux vector field 𝐷.
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Introduction
The curves are a part of our lives are the indispensable. For example, 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 very important type of curves. Because, helices are among the simplest objects in the art, molecular structures, nature, etc. For example, the path, arroused by the climbing of beans and the orbit where the progressing of the screw are a helix curves. Also, in medicine DNA molecule is formed as two intertwined helices and many proteins have helical structures, known as alpha helices. So, such curves are very important for understand to nature. Therefore, lots of author interested in the helices and they published many papers in Euclidean 3 and 4 - space [1-2],].
Helix curve is defined by the property that the tangent vector field makes a constant angle with a fixed direction. In 1802, M. A. Lancert first proposed a theorem and in 1845, B. de Saint Venant first proved this theorem: "A necessary and sufficient condition that a curve be a general helix is that the ratio of curvature to torsion be constant" [3-5].
In 1987, The Serret-Frenet formulae for quaternionic curves in ℝ³ are introduced by K. Bharathi and M. Nagaraj. Moreover, they obtained the Serret-Frenet formulae for the quaternionic curves in ℝ⁴ by the formulae in ℝ³ , [6]. Then, lots of studies have been published by using this study. One of them is A. C. Çöken and A. Tuna's study [7] which they gave Serret-Frenet formulas, inclined curves, harmonic curvatures and some characterizations for a quaternionic curve in the semi- Euclidean spaces 𝐸13 and 𝐸24 .