On the Quaternionic B 2-Slant Helices in the Euclidean Space E 4

Abstract

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 E4 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 E4 and we give a new characterization such as: \({``\alpha : I \subset {\mathbb R} \rightarrow E^4}\) is a quaternionic B2–slant helix \({\Leftrightarrow H^\prime_2 -KH_{1} = 0"}\) where H2, H1 are harmonic curvature functions and K is the principal curvature function of the curve α.