Hypersurfaces Admitting A Given Curve As A Line Of Curvature And Vice Versa

Abstract

In this paper, we develop a method to obtain the lines of curvature on parametric hypersurfaces in Euclidean 4-space. We show that such lines are the solutions to systems of first order triplet non-linear differential equations. Even if these curves cannot be obtained analytically in general, it is also shown that it is possible to compute all curvatures of a Frenet line of curvature by using the extended Darboux frame along the curve. We obtain the Frenet vectors and extended Darboux vectors of the line of curvature without encountering any singular case. In addition, we construct a developable ruled hypersurface whose base curve is always a line of curvature. We provide an example to show the applicability of the given method.