For background to this question much recent exciting related things, see this videotaped lecture by Alexander Gaifullin.
Consider a triangulation $K$ of a two-dimensional sphere and consider maps from the vertices of $K$ to $R^3$. A one-parameter family $\phi_t$ of such maps is called a flexing if the distance $d(\phi_t(v),\phi_t(u))$ is constant whenever $v$ and $u$ are adjacent vertices. A flexing is trivial if it keep the distances between every two vertices.
We can extend the maps of the vertices linearly to a map from $K$ considered as a geometric simplicial complex to $R^3$. When $\phi (K)$ is an embedding we can consider the solid body $L$ inside it.
The following fundamental facts are known:
1) (Bricard) If $K$ is the boundary complex of the Octahedron then there are maps of the vertices of $K$ with non-trivial flexes. (However when you extends linearly these maps to $K$ this is no longer an embedding.)
2) (Gluck) based on Alexandrov and Stainitz) A generic map of $K$ to $R^3$ is rigid.
3) (Connelly) There are embedded flexible spheres
4) (Sabitov) The volume of $L$ along a flexing of a sphere is constant for every flexing. (This extends to the case where we don't have an embedding by regarding the volume as a signed sum of volumes of bounded complements of the image of $K$.)
I would like to know what else might be fixed for flexing of 2-spheres into $R^3$.
Q1: (Perhaps this was asked by Bob Connelly) Are Dehn-invariants fixed for a flexing. In other words, are two embedded spheres along the flexing scissor-congruent?
Q2: Are the eigenvalues of the Laplacian of the (silid 3D-dimensional) body constant along the flexing?
Q3: Are lengths of closed geodesics in $L$ constant for a flexing?
Explanation (Added later, see Joe's comment): Here we consider $L$ as a 3-dimensional Billiard "table" and by "geodesics" we refer to billiard paths.
(Of course, positive answers to Q1 and Q2 would be a far-reaching if not far-fetched extension of Sabitov's theorem, and Q3 is also rather far-fetched.)
I don't know the answers to Q1, Q2, Q3 even for the known examples of flexible spheres. I dont know if Q2,Q3 can be extended to the non-embeddable case but I suspect Q3 could.