Suppose that $M$ is an embedded sub-manifold of $D$-dimensional Euclidean space $E^D$, with embedding $\phi:M \hookrightarrow E^D$; the embedding is not *explicitly* known. 


And suppose that I know the induced Riemannian metric $g$ on $M$, which depends by construction on $\phi$ (which so happens that I don't explicitly know). 


Can I express $\phi$ in terms of $g$ (closed form or as a PDE maybe?), supposing that I want $\phi$ to be the embedding into $M$ which has minimal normal curvature and contains a predetermined (fixed) set of points. 



Recover Embedding from Metric
If the dimension of $M$ is greater than $2$, then the rigidity of $M$ can be due to the fact that the Gauss equations (giving the Riemannian curvature in terms of the second fundamental form) have a unique solution for the second fundamental form. In that case, it is, in principle, possible to reconstruct the embedding by first solving for the (unique) second fundamental form satisfying the Gauss equations. The embedding can then be "reconstructed" by integrating the system of partial equations that give the Riemannian metric and second fundamental form in terms of the embedding.


If $M$ is a closed $2$-dimensional convex surface in $\mathbb{R}^3$, then it is also known to be rigid. However, this is proved using an integral identity. In this case, it is not at all clear how to reconstruct the embedding. There are possible strategies, but they all seem impossible to carry out using known techniques.


