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

In this paper we give a new definition of harmonic curvature functions in terms of B ₂ and we define a new kind of slant helix which we call quaternionic B ₂–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 B ₂–slant helix in 4–dimensional Euclidean space E ⁴ and we give a new characterization such as: "a: I ì mathbb R ? E⁴{``alpha : I subset {mathbb R} rightarrow E^4} is a quaternionic B ₂–slant helix ${Leftrightarrow H^prime_2 -KH_{1} = 0"}${Leftrightarrow H^prime_2 -KH_{1} = 0"} where H ₂, H ₁ are harmonic curvature functions and K is the principal curvature function of the curve α.