Both in the discrete and in the smooth context, we investigate two kinds of rigidity for surfaces in $\mathbb{R}^3$ : the one with respect to the induced metric (the first fundamental form) and the one with respect to the Gauss curvature parametrized by the Gauss map (the third fundamental form).

We discuss two different duality relations between the both and connect variations of the volume to variations of the Hilbert-Einstein functional. This allows us to interpret Blaschke's proof of the infinitesimal rigidity of smooth convex surfaces in the spirit of Minkowski's proof of the infinitesimal rigidity in his theorem.