Turkish Journal of Mathematics
DOI
10.3906/mat-1604-56
Abstract
In this paper, by considering a Frenet curve lying on an oriented hypersurface, we extend the Darboux frame field into Euclidean 4-space $\mathbb{E}^4$. 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 $\mathbb{E}^4$. Finally, we compute the expressions of the new invariants of a Frenet curve lying on an implicit hypersurface.
Keywords
Curves on hypersurface, Darboux frame field, curvatures
First Page
1628
Last Page
1639
Recommended Citation
DÜLDÜL, MUSTAFA; DÜLDÜL, BAHAR UYAR; KURUOĞLU, NURİ; and ÖZDAMAR, ERTUĞRUL
(2017)
"Extension of the Darboux frame into Euclidean 4-space and its invariants,"
Turkish Journal of Mathematics: Vol. 41:
No.
6, Article 21.
https://doi.org/10.3906/mat-1604-56
Available at:
https://journals.tubitak.gov.tr/math/vol41/iss6/21
