  The paper is centered around a new proof of the infinitesimal rigidity of
smooth closed surfaces with everywhere positive Gauss curvature. We use a
reformulation that replaces deformation of an embedding by deformation of the
metric inside the body bounded by the surface. The proof is obtained by
studying derivatives of the Hilbert-Einstein functional with boundary term.
This approach is in a sense dual to proving the Gauss infinitesimal rigidity,
that is rigidity with respect to the Gauss curvature parametrized by the Gauss
map, by studying derivatives of the volume bounded by the surface. We recall
that Blaschke's classical proof of the infinitesimal rigidity is also related
to the Gauss infinitesimal rigidity, but in a different way: while Blaschke
uses Gauss rigidity of the same surface, we use the Gauss rigidity of the polar
dual. In the spherical and in the hyperbolic-de Sitter space, there is a
perfect duality between the Hilbert-Einstein functional and the volume, as well
as between both kinds of rigidity. We also indicate directions for future
research, including the infinitesimal rigidity of convex cores of hyperbolic
3--manifolds.
