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

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

TURKISH JOURNAL OF MATHEMATICS, vol.41, no.6, pp.1628-1639, 2017 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 41 Issue: 6
  • Publication Date: 2017
  • Doi Number: 10.3906/mat-1604-56
  • Journal Name: TURKISH JOURNAL OF MATHEMATICS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, TR DİZİN (ULAKBİM)
  • Page Numbers: pp.1628-1639
  • Keywords: Curves on hypersurface, Darboux frame field, curvatures, CURVES
  • Yıldız Technical University Affiliated: Yes

Abstract

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.