Abstract
Abstract Null cartan curves have been studied by some geometers in both Euclidean and Minkowski spaces, but some special characters of the curves are not considered. In this paper, we study weak AW (k) – type and AW (k) – type null cartan curve in Minkowski 3-space E 1 3 E_1^3 . We define helix according to Bishop frame in E 1 3 E_1^3 . Furthermore, the necessary and sufficient conditions for the helices in Minkowski 3-space are obtained.
Highlights
Curves are one of the basic structures of differential geometry
In Euclidean 3-space E3, a general helix or a constant slope curve is defined in such a way that the tangent makes a constant angle with a fixed direction
For nature’s helical structures, helices arise in nano-springs, carbon nano-tubes, helices, DNA double and collagen triple helix, lipid bilayers, bacterial flagella in salmonella and escherichia coli, aerial hyphae in actinomycetes, bacterial shape in spirochetes, horns, tendrils, vines, screws, springs, helical staircases and shells of the sea [1, 2, 9]
Summary
Curves are one of the basic structures of differential geometry. It is safe to report that G. Monge initiated the many important results in Euclidean 3-space curve theory and G. In Euclidean 3-space E3, a general helix or a constant slope curve is defined in such a way that the tangent makes a constant angle with a fixed direction. Null curves of AW(k)-type are studied in the 3dimensional Lorentzian space by M. Ali and Rafael Lopez gave characterizations of slant helices in terms of the curvature and torsion and discussed the tangent and binormal indicatrices of slant curves in E13 [3, 10, 13]. F. Gökçelik and I.Gök defined a new kind of slant helix called W-slant helix in 3-dimensional Minkowski space as a curve whose binormal lines make a constant angle with a fixed direction [14]
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