Darboux frame field and applications

In this study, by considering a Frenet curve lying on an oriented hypersurface, we extended the Darboux frame field into Euclidean 4-space E^4. Depending on the linear independency of the curvature vector with the hypersurface's normal, we obtained two cases for this extension. For each case, we obtained some geometrical meanings of new invariants along the curve on the hypersurface. We also gave the relationships between the Frenet frame curvatures and Darboux frame curvatures in E^4. Finally, we computed the expressions of the new invariants of a Frenet curve lying on an implicit hypersurface. In addition we examined osculating curves according to extended Darboux frame field.