A convex surface is a connected open subset of the boundary of a convex body in $\mathbb{R}^3$.

An "infinitesimal bending" of a convex surface $S$ is a deformation of $S$ given by a velocity field $v:S\rightarrow \mathbb{R}^3$ such that the length of every curve on the surface is preserved. Vector field $v$ is called the "bending field." A bending field is "trivial" if it corresponds to a simple translation or rotation of the surface. A surface is called "rigid" if all of it's bending fields are trivial.

A regular convex surface is one where every point has a neighborhood which can be parametrized by $r(u, v)$ with $r_u \times r_v \neq 0$.

Blaschke proved that a closed regular convex surface is rigid. Cauchy proved that all convex polyhedra (which are irregular) are rigid.

I've begun dabbling in A.V. Pogorelov's "Extrinsic Geometry of Convex Surfaces," where he proves and discusses both of these results in one of the chapters. However, the translation was published in 1973, so I'm curious about more recent research in this area. Namely, what is known about the rigidity of 1) regular convex surfaces which are not closed and 2) irregular convex surfaces which are not polyhedral?

global, rigidity of convex polyhedra, but his argument can easily be adapted to a proof of theinfinitesimalrigidity. The first to formulate and prove the infinitesimal rigidity was Dehn. $\endgroup$ – Ivan Izmestiev Nov 5 '18 at 14:08