In this paper we give a new definition of harmonic curvature functions in terms of B2 and we define a new kind of slant helix which we call quaternionic B2−slant helix in 4−dimensional Euclidean space E 4 by using the new harmonic curvature functions. Also we define a vector field D which we call Darboux quaternion of the real quaternionic B2−slant helix in 4−dimensional Euclidean space E 4 and we give a new characterization such as:where H2, H1 are harmonic curvature functions and K is the principal curvature function of the curve α.

In this paper, we define slant helices in three dimensional Lie Groups with a bi-invariant metric and obtain a characterization of slant helices. Moreover, we give some relations between slant helices and their involutes, spherical images. IntroductionIn differential geometry, we think that curves are geometric set of points of loci. Curves theory is important workframe in the differential geometry studies and we have a lot of special curves such as geodesics, circles, Bertrand curves, circular helices, general helices, slant helices etc. Characterizations of these special curves are heavily studied for a long time and are still studied. We can see helical structures in nature and mechanic tools. In the field of computer aided design and computer graphics, helices can be used for the tool path description, the simulation of kinematic motion or design of highways. Also we can see the helix curve or helical structure in fractal geometry, for instance hyperhelices. In differential geometry; a curve of constant slope or general helix in Euclidean 3space E 3 , is defined by the property that its tangent vector field makes a constant angle with a fixed straight line (the axis of the general helix). A classical result stated by M. A. Lancret in 1802 and first proved by B. de Saint Venant in 1845 (see [1, 2] for details) is: A necessary and sufficient condition that a curve be a general helix is that the ratio of curvature to torsion is constant. If both of κ and τ are non-zero constants then the curve is called as a circular helix. It is known that a straight line and a circle are degenerate-helix examples (κ = 0, if the curve is straight line and τ = 0, if the curve is a circle).The Lancret theorem was revisited and solved by Barros [3] in 3-dimensional real space forms by using killing vector fields along curves. Also in the same spaceforms, a characterization of helices and Cornu spirals is given by Arroyo, Barros and Garay in [4].The degenarete semi-Riemannian geometry of Lie group is studied by Çöken and Ç iftçi [5]. Moreover, they obtanied a naturally reductive homogeneous semi-Riemannian space using the Lie group. Then Ç iftçi [6] defined general helices in three dimensional Lie groups with a bi-invariant metric and obtained a generalization of Lancret's theorem and gave a relation between the geodesics of the so-called cylinders and general helices.

In this paper, we give the definition of harmonic curvature function some special curves such as helix, slant curves, Mannheim curves and Bertrand curves. Then, we recall the characterizations of helices [8], slant curves (see [19]) and Mannheim curves (see [12]) in three dimensional Lie groups using their harmonic curvature function.Moreover, we define Bertrand curves in a three dimensional Lie group G with a bi-invariant metric and the main result in this paper is given as (Theorem 3.4): A curve α : I ⊂ R →G with the Frenet apparatus {T, N, B, κ, τ } is a Bertrand curve if and only if λκ + µκH = 1 where λ, µ are constants and H is the harmonic curvature function of the curve α.

In this paper, we define Mannheim partner curves in a three dimensional Lie group G with a bi-invariant metric. The main result of the paper is given as (Theorem 4): A curvę W I R !G with the Frenet apparatus fT; N; B; Ä; g is a Mannheim partner curve if and only if Ä 1 C H 2 Á D 1 where , are constants and H is the harmonic curvature function of the curve˛:

Abstract. In this paper, we study spinor Frenet equations in three dimensional Lie Groups with a bi-invariant metric. Also, we obtain spinor Frenet equations for special cases of three dimensional Lie groups.