On the closed motions and closed space-like ruled surfaces

Abstract

The theorems about closed motion in Euclidean space is given in [2] and [3]. In this paper, using the real Steiner rotation and translation vectors, we have given some theorems about the real pitch and the real angle of pitch which are the integral invariants of the closed space-like ruled surfaces. Taking a space-like line x in Frenet trihedron {t,n,b}, the real angle of pitch of space-like ruled surface, which is drawn by the space- like line x, is calculated by the .real angle of pitchs of space-like ruled surfaces those are drawn by the space-like lines t and b. Then, we give some theorems about dral and harmonic curvature in $R_1^3$ .