Abstract— In this study, we introduce quaternionic type-2 Harmonic Curvatures and General Helices according to Frenet frame in 4-dimensional Euclidean Space and investigate its properties for two cases. In the first case; we use a constant angle between a unit and fixed direction vector field and the first relatively Frenet frame vector field of the curve, that is given in the paper.
where is the real quaternion inner product. Since the relatively Frenet frame vector field of the curve makes a constant angle with the unit and fixed direction vector field U, we call this curve as a General helix in 4-dimensional Euclidean Space . And then, in the other case, we define new type-2 harmonic curvature functions and we give a vector field which we call Darboux vector field for General helix. And then we obtain some characterizations for General helix in terms of type-2 harmonic curvature functions and the Darboux vector field .
Keywords— Euclidean spaces, General helix, type-2 harmonic curvatures, Quaternion algebra, Quaternionic frame.
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