Vector fields and planes in E4 which play the role of Darboux vector

In this paper, we define some new vector fields along a space curve with nonvanishing curvatures in Euclidean4-space. By using these vector fields we determine some new planes, curves, and ruled hypersurfaces. We show that thedetermined new planes play the role of the Darboux vector. We also show that, contrary to their definitions, osculatingcurves of the first kind and rectifying curves in Euclidean 4-space can be considered as space curves whose positionvectors always lie in a two-dimensional subspace. Furthermore, we construct developable and nondevelopable ruledhypersurfaces associated with the new vector fields in which the base curve is always a geodesic on the developable one.

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