Hyper-asymptotic curves of a Weyl hypersurface

Abstract

In this paper, firstly, we obtained the differential equation of the hyper-asymptotic curves in Wn with respect to a rectilinear congruence. With the help of it, we defined the hyper-asymptotic curvature vector field and the hyper-asymptotic curve in Wn. Secondly, we described an asymptotic line of order p in Wn in Wn+1 and a geodesic of order p in Wn+1. We gave necessary and sufficient condition to be an asymptotic line of a curve in Wn. And then we expressed the relation between geodesics in Wn and in Wn+1. Thirdly, we stated the relations among a hyper-asymptotic curve, an asymptotic line of second order and a geodesic of second order in Wn. Finally, we expressed the condition to be a geodesic of second order of a hyper-asymptotic curve. Mathematics Subject Classification: 53B25, 53A25