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

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.

A new method for finding the shape operator of a hypersurface in Euclidean 4-space

“…Let T and N denote the unit tangent vector field of the curve β and the unit normal vector field of M restricted to the curve β, respectively. Then the extended Darboux frame (ED-frame) field along β is denoted by {T, E, D, N} [2], where…”

“…where κ i and τ i are the geodesic curvature and the geodesic torsion of order i, (i = 1, 2), respectively [2].…”

“…The computations of κ n (T), τ 1 and τ 2 are known for not only parametric hypersurfaces [6] but also for implicit hypersurfaces [2] in Euclidean 4-space. To make the matrix of the shape operator in the Proposition 3.1 understandable, we now show how the normal curvatures in the directions of E, D and E + D can be obtained for both parametric and implicit hypersurfaces.…”

“…Let M : f (x, y, z, w) = 0 be an orientable hypersurface and β(s) be a Frenet curve lying on M. Denoting the ED-frame field of β by {T, E, D, N}, the normal curvature of M in the direction of T is given by[2] …”

Smarandache Curves According to the Extended Darboux Frame in Euclidean 4-Space

“…In the light of the existing studies in this area, we define some special Smarandache curves such as TE , TD and TN -Smarandache curves in Euclidean 4-space according to the extended Darboux frame (or shortly ED-frame) defined in [9]. Then considering the extended Darboux frame of second kind, we obtain the Frenet apparatus of these special Smarandache curves depending on the extended Darboux frame invariants.…”