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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.

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A magnetic field is defined by the property that its divergence is zero in a three-dimensional oriented Riemannian manifold. Each magnetic field generates a magnetic flow whose trajectories are curves called as magnetic curves. In this paper, we give a new variational approach to study the magnetic flow associated with the Killing magnetic field in a three-dimensional oriented Riemann manifold, (M3, g). And then, we investigate the trajectories of the magnetic fields called as N-magnetic and B-magnetic curves.

An isophote curve comprises a locus of the surface points whose normal vectors make a constant angle with a fixed vector. The main objective of this paper is to find the axis of an isophote curve via its Darboux frame and afterwards to give some characterizations about the isophote curve and its axis in Euclidean 3-space. Particularly, for isophote curves lying on a canal surface other characterizations are obtained.

In nature, science and engineering, we often come across helicoidal surfaces. A curve on a helicoidal surface in Euclidean 3-space is called a loxodrome if the curve intersects all meridians at a constant azimuth angle. Thus loxodromes are important in navigation. In this paper, we find the differential equation of the loxodrome on a helicoidal surface in Euclidean 3-space. Also we give some examples and draw the corresponding pictures via the Mathematica computer program to aid understanding of the mathematics of navigation.

In this paper, a one-to-one correspondence is given between the tangent bundle of unit 2-sphere, [Formula: see text], and the unit dual sphere, [Formula: see text]. According to Study’s map, to each curve on [Formula: see text] corresponds a ruled surface in Euclidean 3-space, [Formula: see text]. Through this correspondence, we have corresponded to each curve on [Formula: see text] a unique ruled surface in [Formula: see text]. Moreover, the relationships between the developability conditions of these ruled surfaces and their striction curves are analyzed. It is shown that the ruled surfaces corresponding to the involute–evolute curve couples on [Formula: see text] are developable.