B-Lift Curves in Euclidean 3-space

In this study, we introduce a new type curve in 3-dimensional space which called B-lift curve and we obtain the Frenet operators of the B-lift curve. Moreover, we consider the correpondence of Frenet operators between the B-lift curve and the natural lift curve. Finally, we investigate the B-lift curve according to the main curve is slant helix or darboux helix.

in Euclidean 3-space if tangent vector of the curve makes a constant angle with a fixed straight line. If both κ = 0 and τ > 0 are constant, a curve γ is called circular helix. [2] In 1802, M. A. Lancert proved that the ratio of the curvatures is constant along the curve in general helix (or cylindirical helix).
Many authors have written articles on helices and different types of helices. S. Izumiya and N. Takeuchi [9] defined slant helices as the principal normal vector makes a constant angle with a fixed direction. They have also introduced conical geodesic curve. L. Kula, N. Ekmekci, Y. Yaylı, K.İlarslan [11] gave some characterizations of slant helices and introduced Frenet operators of slant helices in R 3 . E. Zıplar, A. Şenol and Y. Yaylı [12] defined Darboux helix that Darboux vector makes a constant angle with a fixed straight line. Morever, they gave the relation between slant helices and darboux helices.
However, there is no research about B-lift curves. In this paper, based on Thorpe's definition, B-Lift curve is defined and some characterizations of the Frenet operators are given. We also investigate the relation between the Frenet operators of the B-lift curve and the Frenet operators of natural lift curve. Furthermore, we examine the status of the B-Lift curve according to whether the main curve is slant helix or darboux helix. Finally, some examples are given and we draw our curves with Mathematica program.

Preliminaries
In this section, some basic definitions and theorems in differential geometry are given. Furthermore, some properties of the natural lift curve, general helix, slant helix and Darboux helix are denoted.
Let A = (a 1 , a 2 , a 3 ) be a vector in R 3 . The norm is defined as || A|| = a 2 1 + a 2 2 + a 2 3 . If || A|| is equal to 1, then A is called unit vector in R 3 . For the vectors A = (a 1 , a 2 , a 3 ) and B = (b 1 , b 2 , b 3 ) in R 3 , the inner product is defined as < A, B >= a 1 b 1 + a 2 b 2 + a 3 b 3 .
A parametrized curve γ : I →R 3 is called regular curve if γ (s) = 0, where s ∈ I. Let γ be a curve in R 3 , if γ (s) is equal to 1 then the curve γ : I →R 3 is a unit speed curve.
Let γ be a unit speed curve in where X is a differentiable vector field on M [4].
Definition 2 Let γ : I → M be a unit speed curve. The natural lift curveγ : I → TM of the curve γ is defined as follows: Therefore, we can write Theorem 1 Letγ(s) be the natural lift curve of γ(s). Then there is the following equation between Frenet frames of the curves γ(s) andγ(s): Frenet frames of the curve and its natural lift curve, respectively. The Darboux vector W is presented as W = τ T + κB. Moreover, κ = W cos ϕ and τ = W sin ϕ are represented as curvature ans torsion. Here, ϕ is the angle between Darboux vector and binormal vector of γ(s), see [4].
Theorem 2 Letγ(s) be the natural lift curve of γ for a given regular curve γ. Then there are the following equations: whereκ andτ are the curvature and torsion ofγ(s), respectively [4].
Proposition 1 Let γ : I → R 3 be a unit speed curve with curvature κ and torsion τ . The curve γ is the general helix if and only if τ κ is a constant [6].
Proposition 2 For any unit speed curve γ : is a constant fonction [9].
is a constant function [12].
Theorem 4 Let γ be a curve in R 3 . If κ τ is not constant, then γ is a slant helix if and only if γ is a Darboux helix [12].

B-Lift Curves in 3-Euclidean Space
In this section, we describe the B-Lift curve which is the curve obtained by the end points of the binormal vectors of the main curve. We also obtain the where B is the binormal vector of the curve γ. Proof. Let γ B be the B-Lift curve of γ, then we can write: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 24 March 2021 doi:10.20944/preprints202103.0580.v1 (4) and (5) we get Using (2) and (6), we have From the equations (2), (6) and (7), the proof is completed.
Theorem 6 Let γ B be the B-Lift curve of a regular curve γ in R 3 . Then, we have the following formulas: where κ B and τ B are curvature and torsion of γ B , respectively.
Proof. From (5), we know Since is provided, we obtain the following equation: The torsion of γ B is given as Using (3), we get From (4), (7) and (11), we have Theorem 7 γ : I → R 3 is a slant helix if and only if γ B is a general helix.
Proof. Assume that γ is a slant helix. From Proposition 2, we have where κ and τ are curvature and torsion of the curve γ. We have to show if γ B is a general helix. From Theorem 6, we can have Then, γ B is a slant helix. Conversely, let γ B be a slant helix. Then we can write Since σ(s)=constant, the curve γ is a slant helix. Proof. Let γ be a general helix. Then the ratio of τ κ is constant. Hence ( τ κ ) =0 and consequently we can write Then the Frenet vectors of the curve γ are given as follows: ,  Therefore, we obtain Since the ratio of the curvatures are constant, the curve γ B is a general helix.   given. Based on the definition of B-lift, the situation of the curve being slant helix, darboux helix and general helix is examined.