  Let S be a complete surface of constant curvature K = + 1 or -1, i.e. the
sphere S^2 or the Lobachevskij plane L^2, and D a bounded convex subset of S.
If S = S^2, assume also diameter (D) < pi/2. It is proved that the length of
any steepest descent curve of a quasi-convex function in D is less than or
equal to the perimeter of D. This upper bound is actually proved for the class
of G-curves, a family of curves that naturally includes all steepest descent
curves. In case S = S^2, it is also proved the existence of G-curves, whose
length is equal to the perimeter of their convex hull, showing that the above
estimate is indeed optimal. The results generalize theorems by Manselli and
Pucci on steepest descent curves in the Euclidean plane.
