The task is to prove that a space curve and its spherical indicatrix of tangents coincide if and only if the curve is a circle.
Def. As a point moves along a space curve C envision a unit vector t located at the origin of the coordinate system moving in conjunction with the unit tangent T, always parallel to it. As one moves along C the unit vector t will trace out a curve Γ on a unit sphere centered at the origin. This curve Γ is called the spherical indicatrix of tangents of curve C.
It's obvious for me that if the curve is a circle, its indicatrix is the same circle turned by 90 degrees around normal to its center. It's also obvious that only spherical curves should be taken into consideration.
For each curve satisfying the task condition we have c(s) = c'(s + T). It seems, we need to prove that such curve always have zero torsion but I have no idea how to do it. So, please, help.