I'm trying to show that if alpha(s) is a straight line if and only if all its tangent lines are parallel.
Pf/ I know that I will need the Frenet Serret Theorem and my stab at it is:
Assume all the tangent lines of a(s) are parallel. So the tangent vector T is the same for all points xo on the curve a(s) and the values of T(s) of any two points on the curve are parallel. Thus T(s) is constant, and T'(s)=0 which implies that the curvature is zero, and thus a(s) must be a straight line.