  It is shown that $N$ points on a real algebraic curve of degree $n$ in
$\mathbb{R}^d$ always determine $\gtrsim_{n,d}N^{1+\frac{1}{4}}$ distinct
distances, unless the curve is a straight line or the closed geodesic of a flat
torus. In the latter case, there are arrangements of $N$ points which determine
$\lesssim N$ distinct distances. The method may be applied to other quantities
of interest to obtain analogous exponent gaps. An important step in the proof
involves understanding the structural rigidity of certain frameworks on curves.
