Extension of the Darboux frame into Euclidean 4-space and its invariants

DÜLDÜL M. , Duldul B. U. , Kuruoglu N., Ozdamar E.

TURKISH JOURNAL OF MATHEMATICS, cilt.41, sa.6, ss.1628-1639, 2017 (SCI İndekslerine Giren Dergi) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 41 Konu: 6
  • Basım Tarihi: 2017
  • Doi Numarası: 10.3906/mat-1604-56
  • Sayfa Sayıları: ss.1628-1639


In this paper, by considering a Frenet curve lying on an oriented hypersurface, we extend 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 obtain two cases for this extension. For each case, we obtain some geometrical meanings of new invariants along the curve on the hypersurface. We also give the relationships between the Frenet frame curvatures and Darboux frame curvatures in E-4. Finally, we compute the expressions of the new invariants of a Frenet curve lying on an implicit hypersurface.