New Vector Fields and Planes Obtained by the Extended Darboux Frame Apparatus of the Second Kind

UYAR DÜLDÜL B.

WSEAS Transactions on Mathematics, vol.24, pp.8-15, 2025 (Scopus) identifier

  • Publication Type: Article / Article
  • Volume: 24
  • Publication Date: 2025
  • Doi Number: 10.37394/23206.2025.24.2
  • Journal Name: WSEAS Transactions on Mathematics
  • Journal Indexes: Scopus, INSPEC, zbMATH
  • Page Numbers: pp.8-15
  • Keywords: Darboux vector, developable ruled hypersurface, extended Darboux frame of the second kind, geodesic curvature of order 2, geodesic torsion of order 1, non-developable ruled hypersurface, normal curvature
  • Yıldız Technical University Affiliated: Yes

Abstract

In this paper, new vector fields are defined along a curve lying on an orientable hypersurface with nonvanishing extended Darboux curvatures of the second kind, and some new planes and curves are introduced using these vector fields in Euclidean 4-space. It is also shown that in 4-dimensional space, these new planes play the same role that the Darboux vector W = τgT − KnV + KgN plays in Euclidean 3-space. Besides, developable and non-developable ruled hypersurfaces associated with these new vector fields are defined.