ABSTRACT
Let the curvature k(s) and torsion &(S) of a curve r, when viewed as functions of the length of arc S, be periodic functions with a period T. Denote by r(s) the position vector and suppose that r(0) = 0; by cp we denote a space rotation transforming the natural frame @(O) of at point r(0) into the natural frame @(g at the point with the position vector r(T). We will prove the following:
One can reformulate this statement in the following way. Let l be an axis of the rotation cp; this means that the vector a spanning l is a stationary vector of the rotation cp. If @(O) # @(T), then a is unique. The theorem says that r is bounded iff (r(T), a) = 0. If @(O) = @(T), then l? is bounded iff r(T) = 0, i.e. iff is closed.
