One way to get a curve on a sphere of radius 1 is to take an arbitrary curve Y(t) in three-space not passing through the origin and define X(t) = unit(Y(t)). The curve X(t) then lies on the unit sphere and the normal to that sphere is N(t). We then get X(t)*X(t) = 1 so 2X'(t)*X(t) = 0 and 2X''(t)*X(t) + 2X'(t)*X'(t) but X''(t) = s''(t)T(t) + s'(t)2(kg(t)U(t) + kN(t)N(t)) so X''(t)*N(t) = s'(t)2kN(t) + s'(t)2 = 0 so kN(t) = -1 for all t. Moreover N'(t) = X'(t) = s'(t)T(t) so N'(t)*U(t) = 0 = ?g(t)s'(t) and ?g(t) = 0.